,.-----""~·-_._ ~-~-~. ~-"y~--~-';,....~-~~· .rx ~ .. }

;-'·

6V !295"" , [;t;~-

rs­!1[,(:;

·Copyright C 1962, 1966 by Edward 0. Thorp

AD righfi reserved under International and Pan-~CIO\ Copy­right Conventions. Published in the United States by Randcim H~. Inc., New York. Distributed i~ Canada by Random

- House of Canada Limited, Toronto. Originally pulllished by Random House, Inc., in 1966. The quotation from Paul ONeil's ankle in Ufo Magazine appears courtesy Lifo .Magazine @ 1964

Time Inc.

Library of Congress Cataloging in Publication Data

Thorp, Edward 0. Beat the dealer. Reprint of the 1966 ed. Bibliography: p. 1. Blackjack (Game) I. Tide.

[GV129,.B,T5 1973) 795.4'2 72-8006 ISBN 0-394-70310-3

Manu&ctuted in the United States of America

Vintage Books Edition, February 1966

\ tl \J' ~i 6}.

This book fs dedicated to

my wife VJ.VlAN and mY children~ ·-·,"Nand JEPPJU!.Y.

llUAN,~

~- ..

!'.

· Acknowledgments

I am. Indebted to Roger R. Baldwin, Wilbur E. Cantey. Herbert Maisel, and 1~ P. McDermott for making avale c

ablo the computational details of their work on blackjack and to the M~T. Computation Center for making available an mM 704 computer.

I wish to thank the many friends and colleagues who have mado valuable suggestions, particularly Protesson. Claude E. Sh8D110J1, Berthold Schweizer, Abo Sklar, an4 Elbert Walker. I am indebted to ViVian and James Thorp for the long hours they spent playing the part of "'tho house. • For showing me ~y of the metboda and devices 1l8ecl by casinos in cheating and for &Wing me a largo amouDt of . pneral information on the world of gambling, I owe a great deal to Michael MacDougall. former special inYCStigitor for the Nevada Gaming Control Board, to some of the old-tim.o.· Nevada •coUDt" playen, and to a certain crooked Nmda · card mechanic. Conversations with a certain federal invcsti­ptor have given me a largo amount of information about -~ the iDside 1Ctivitit8 and the out-of-state c:onnectioM of .cel'o tain Nevada casinos. Last, but not least, I wish to thaDt tho two milJjonaires who financed the highly tucCessful easiDO test of my SJSle.ll1 tbat is reported in ChapterS.

ffe··· [f ·viii ~ :

The resqlts of the first edition have been ~ al)41. improved by the extensive researches of Iulian Braun Of the IBM corporation. He has made most of the calculations

. for tho point-count method and has made numerous de­tailed an4 \'aluable suggestions. I am grateful to him for allowing his work to be incorporated into the second edi· tion.

I particularly wish to thank William B. Walden for tho related work he has done with moon Nevada baccarat.

Thanks to Paul O'Neil·for the integrity and journaJis. tic clarity with which he portrayed Beat the Dealer and its author in Life magazine. Thanks also to Life for its cour­ageous stand in printing the truth despite hostile Nevada mobsters and politicos. .

I wish to thank many leaden for. their helpful sugp dons, ideas, personal experionces. and testimonials, and for proving the book again and again in theusinos.

,_ Finally, thanks to the readmofthefilst edition of Beat f<.,lhe Dealer who, in their tnthusiasm, bought so many copies \. that they put it on the national best-seller list. !. r

r I· . i l ~ ''There Is a tide in the affairs of men, which.

~ taken tJt the flood, leads on to fortuM. .....

t --8HAKEsPEARB (Julius Caesar) ! '

Contents

f ' f ~ Acknowledgments

f 1. Introduction ~ . f 2. The Rules of Blackjack

3

8 Numben of player11. The pack. The deal. Betting. Na­merical wilue o/lhe carth.lutrd fUUisojt 'htmd8. Objecl oj 1M pla]er. Naturllb. The draw. The settlement. Split­ting pain. DoubUng down. lnsuranu. CIUIOliV tUUI lrtiCilcu: Shuffling. Shill~. N.w dM:kl.

~- 3. · TJJ.o ~B;lsic: Strategy 16 t / 1 _ The playu'11 decislonll. The btlllc lliilten for thwlng

or •tanding •. The btlllc mateo for doubling down. The baric strateKJ for spUttlng pain. WMI to up«:~ when ruing the balk maten. Comparison with 1untu pere untage agahut other blackjacl /ltrateglu tmtlln other· ~- So• common blllCkjtd erron. Finl ~ ment: drizwlng ver~~W llta1Uilng holdlng hard l6IIB'fblll 4111 Jfce. Secoml upet'lmal: tlollbllnl dowa oa hard 10 tzgaln/11 1111 Ac.. Thlr4 ~ 6J1Uitb18 a pfllr of 8/xu agolnllla Flve. Mlmldlng 1M dll4ler. 2'he player 1PIIo ....,. bllllll. X._ man who. trimmed hill barber.

-4. A Winning Strategy

Failure of the popular gambiing systenu. Thelmpoi'tllnce of th4 de~nce of trial4 in blackjack. The use of favl!rtlble 6itUiltions, A lint winning strategy: counting· Pwu. Counting dee ctJrds. An Improvement in the Flvu 11'16tltod. Preqwncy of favimzble -situations. Variations In bet slu~ Ca,ital required. utent of risk, rate of profit.

S. My ldclas Are Tested in Nevada 59 . : U. Pnpill'tld0111. rhe $10.000 banJ.:roll. The warm-up. A huntir6d ~we. a tltOIUllnd there. Nine hundred dollan kt on 11 6lngk hllnd. Tire twenty-/We-dollar inlnlmum game. an.,.,., thoiUand dollm-s In two houn.

· 6. The Simple Point-Count System 7S

The llmple point collllt, Refinements, Henry Morgan and I visit l'llerto Rico. Enter the Salmon. An lnten.rt- '} lng ld., for end play. Playing pTillegy when the deale~s hole C'tiTd is k110tvn. Value of knowing the deaie~s hole card. A 1G lorga: the Stzlmcm ww $SO,OOO, The rule~ tire "c!uzng.tr In Puerto Rico:

7. Tho Complete Point-Count System 93 . C01111dng th4 etlrds. The hers. Drawing tmd standing. Doubllng down. Pllir splitting. Insurance. AdvtUJtage l1lld

. 8. .· • 'fntluenc;y of /avOTilble sltuatiOM,_ _

A Winning Stta~gy Based on Counting Tens 102 Effect on pl4y.rs mlvantage M proportion of Tt'M . 'llal'iu. I.Aornlng to count. lmurance. Strategy tabla. Lettrtdng th4 strlllegy klhks. Rate of profit. lncbuling Accr In the c:oant. The l'elnllrkable pia /TOifl ~pu end play.

9. Beating the Casino Countermeasures 124

S/uif]le up. Dea/en who C()unt. Strike when the tleclc II hot. Punishing faker. Multiple decks. ~ules chonges. Rules variations. Camouflage. Disguises. The llUIOIIIIllil: Nllclcjw:k 11111chinu. Tlut /HlPIIN'out, teehni

11ue.

RefeJ:ences

List of Figures and Tables

I.adex

%OS

211

215

J.'

Beat the Dealer

r. ,· I' ~

~1 '~

Introduction

The game ol blackjact, ortwent,y-one, is one of the wodcrt 1llO$t widelJ played gambJing games. In the United States

· It is played in the state ol Nevada, in IeffeQOD Parish near . New Orleans. in the Galveston area, on and off iD Bot

r Spriags.· Arkansas, in White Sulphur Springs, West V1l'o · ~ pa. and in homes and private clubs throUghout ·. dJe J COUDtry. Blackjack can bo played in Puerto Rico. ~ ! Panama, the Grand Bahamas, and.other spots ill theCarlb-f bean. There is blac:tjac:t in Europe at the Lido casino JD ·I t VCDice. on the Isle ol Man, and In London~• ManDa. ill· I 1 the Philippine Islands, and the Portuguese colony of Macao.

near. Hong Kong. also have casinos which featum bJac:t-1 jack. r In England. blackjack is biown as -van-john• ad ia i Australia. as "'poll~ • BOth ol these are couuptioas f4 ~ the French term vlngt-et-un. In Germany it is ca1lccl BhJ.

• "- of this wrftiD&· many fonDa of pmhlfDg are leplla W A llUIIlber of the large LondOil samiJll cluba haVe bladt~ ndel tllat are esreatiall~ the IIDIO a 1boiO Ia 1.11 Vejat. Tho wiliaiD; ttrallsfll .. ddt llooJt apply. .,.

I,./'

-~ BBAT THB DBAI.BR

und-Zwandg or Achtuhn-und-Drel. Although the namo varies, the game is essentially the same. _ · In the modem casino game of blackjack. the player can gain a consistent advantage over the bouse by using the strategy presented in this book. Based on the mathe­matical theory of probability, this strategy was worked out With an electronic computer by tho author and others. ·It· is fortunate IUld. perhaps surprising that the system re­. 'i:lUce$ _ to .a few sbnple charts that c:ap1 be undetstood and :. ~rlzed by the _:*Verage player. Ii1 addition, the system ·.lends itself totl\0 rapid play usually encountered in casinos. .

The rutes ofblaCkjack"vary.somewhat from casino to casino. These variations, based on .my studies of many casinos, are listed in Table 9.2. The table shows how these rules vapations affect the player's chances and enables him to cOmpare any tWo. casinos and decide in which one to play. - - .. . · : No system can win when ·confronted with the chrOnic

diBeBso of gambling games: cheating. Blackjack gives ~e . ckaler an excellent opportunity to cheat. Aside trom not playing at ·all, the only sure protection seems to be to have the aemeesd a card expert. But, by taking the precautions described in Chapter 10, the average player can protect JDinself sufftciently in most· situations. · _

In tho following pages we begin by outlining and dis­cussing the rules of tho game and then proceed itep-by-stqp · to advance the reader to any level of playing proficiency be desires. The first step ~ to team what we shall can the -t;asic sft.ategy": a simple set of rules which tell the player · when· to draw or stand, when to double down, and when to spHt • pair. With the basic· strategy alone the player lias the slight odvantage of 0.1 per cent in moat Las Vega$ . caimOs. Blackjack-is the only casino gambHng·p.ae today whero you can consistently have. the edge. Other published ~tegi"' for blackjick commonly .give the player disad-,. vantages ranging from 2 to s per -~ ',l'he first substan-'

Introduction 5 dally correct wrsion of ~ ~ strategy .as ~ by Baldwin et al. and published in (2]. • There were slight inaccuracies both in this version and in the improved ver­sion published in ~ first edition of Beat the Dealtr. The correct version of the strategy for one deck and a ~ set of easino rules appears in Chapter 3· It was calcUlatOcl by JuHan Braun. In casinos with favorable ~~ tho baslo strategy actually gives the plflyer a ~er e<lp over.~ house. In casinos with unfavorable ~ tbe player JllB.1 have a slight disadvantage (up tci p.5 per cent or so). Bven so, he Js generally better o1f thari. in an? .other casino pne. including craps. · ·. ' ·

The basic ~tegy does not involv~ counting catds. · However, after mastering the basic strategy, the reader wtD Jearn a simple modification, using a card-counting system. that identifies many situations in which he has an a~ tage over tho casino of more than. S per cent. Most people who are advised to count cards say, "But I can't~ track of all the cards in the deck.. I can't-~ ~-~ phouc numbers." They may be surprised ta team fbat ~ must count only four cards (pet deck used by the ~l -the Fiw&-and that this additional inform~ _COJJlot bined with minor strategy cbanges. ~ enough tO &f.VcJtie player a comfortable 3 pet cent edge!. ·. . ..

Tho player who is willing and able to ~unt more tbaa four cards can go on to more sophisticated striltegies. Pint, there is the powerful new Thorp point-count ~ $ method was perfected by several people after thO firSt· edl~ tion of Beat the Dealer was published. It has proven Wft1 effective against assorted casino countermeasures. In pu-­ticular it is effective and easy to. use against sevetal deeD. It is as simple as the· famous Goren point count in brid~ Every card is counted +1, o, or -1 as it is seet1. Yoli simply need to keep track of the total number of pointS you have seen (thus you mnember ·one number only). The

• Numben fa ( J ilidicato nfercm:ea 1iatoc1 on pp. aos-g.

-_-;_~~--~----- --~--~----"' ----- --- B;O~-;-~T~-B D~~;;; 1 point-count method_ is effective because it takes into~~ !. aideration every card the plaF, see& 'This allowS Jtlu. lo Identify nearly half of an situations • favorablo tO Jami.. self.: In fact. half the time ~ has a sUght edge in the oni> deck game as soon as be sees a singlo cardJ

The Ten-count method was presented in detail iD the first edition. It was used so successfully by thousands of readers that the Las Vegas casinoso for tho first time in history, changed the rules of blackjack. £34]. (The change failed and was dropped. When I wrote the b9<>JC I foresaw the change and explained in Chapter 8 [of die first edition). Rules- Variations. how to continue winning.. The casino operators apparently only managed to Red ai far as Chap­ter S.)

For skeptics who do not believe a theory until the)' -~it w~. in Chapter S there is an accouut of the author's ~ginal test of the Ten-count method in Nevada. Backed by $Io,ooo, which was lent by two milUonaires interestecl in making a profit. I purposely played very cooservatively for about thirty hours. At the end of this time. the mil­~ Sxo,ooo had increased to ·over $20,000. As wo hecamo known, it became harder and harder to get the c:asiD01 to play a .. nice" game. The tactics they used in­eluded refusing to give us a private table. shuftUng every Jumd at two, switching dealers in and out. challgin& decks constantly (0110 casino switched in four new decks in five ~), and ~g to sen us largo-denomiaation chips. One Casino even mtroduced_ a cheating dealer when we sat dowa to play.

Despite- these. annoyances, we were stiJl able to get pmaa- whenever we wanted them. We finally quit because_ the millionaires had bigger business elsewhere. because my . facbing duties ~uired me to, and because the system had been sufficiently tested. ·

We think that this_ boot will pay for itself and for tlio time spent reading it many times over. Anyone who

I i.

Introduction '1

irequents the gaming tables or who plays b-· states.at hoo1o shOW~ be handsomely rewarded for hiS trouble. We bope also that some of the aura of superstition and mystery Slll"­rounding games of chance will. be dispelled for the reader as we study blackjack together.

2

The Rules· of Blackjack

I._,

,_first step in learning the game of blackjack is to master ~ . tba rute&-with emphasis on the .word "master." Even ~- · .}Dowing the exact meaning of the rules ia not ~gb. · · · . ThO reader Diust be able to uncterstand the effect of each ~:~ t1Jle and of each possible variation. Both experienced plaJ-

_jrs and beginners should study this chapter. · . • · ~ casino bas a set of blackjack rules which agree : with those of other casinos on the DUlin points ~t which .. ~differ in details. Later in.tbis bookwcman.analxze

· the effects of these variations; but~ for simplicity; wo consider a typical set of rules. A set that is c:Ommon. yet by no means universal. is listed below.

.. Number of Players A blackjack game has a dealer an4 from one to seven

players. We will see later that. generally speaking, the fewer the ~un1ber of ptayers at a table, the better it i8 for &be pm~~ .r

•••

The Rlllu of Blackjack

The Pack One ordl'Nll'j 52-card pack .Is commonly 11#4. Rcnf.

ever more and more casinos arc going· over to two aud even four pacb shuftled together, in an effort to maka card counting more difficult. • It turns out that an ia­crease in tho ·number of packs euts the player's advaatage s1igbtly. (In Puerto Rico two decks are genetaUy used, aad in London four decks dealt from a shoe are coiDIIlOD.)

The Deal Before play begins, the cards are shuffled by 1M

dealer an4 cut by a player. Next, a CDT'd is •~Jurnetr. (placell face up on the bottom of the deck). The. bume4 · t:ard may or may not be shown. The dealer then tleal8 two eards to himself an4 to each of the players. Playen Bc . both cards face down. The dealer receives one t:ard ~ ·· up 11114 one CDT'd face down. The two cards of the pf4yir · an4 the "down .. card of the dealer are called ... hole cards.·

Some casinos deal the player's hole cards face up. . This is the practice in Puerto RiCo. This is very convenient for players ~bo count cai'ds. On the other hand. l6e1ng · the pla~s ~ in no way helps the cfeaJer if, 8$ is usuany >

the case m casmos, he must act according to fixed rutes. Later we shall see that about half the time the abSeDe8 of the one burned card is _enough to give the playu a· sligllt· · edge- ovor the house. (This is not true when two or JDOJO-. decka ate. used.) .

Betting ,

· The players place aU bets except insura~e (discussed subsequently) before any cards are dealt. The house- estflb­li.shes a minimum. bet and· a maximum bet. The minimum

• In ·Nevada the· multipack dealing devices, whfdl an~ . more aac1 moro common. bavo ~ to bl taowa JaformaB u ~ c: :0 ~-rl ltOq b¥ ilarokJ Drab. ·rr!t Who ..... .;;:;

~:.···· 10 BBAT. TK'E DBA.l.lnt

bet is usually between 25¢ and $5 and·the maximum bet, between $zoo and $soo.

Our winning strategies involve varying the size of the player's bet- The player places larger bets in favorable ~itu­ations and smaller ones in unfavorable situations.

The size of the minim\llll bet is of greatest interest to the player with a smaU amount of capital. The size of the maximum bet is of interest to the player with a Jarge amount of capital because it limits the rate at which he can wid. (In Puerto Rico, a '$x minimum and a $50 maxi­mum arc <:9mmon. In. London a minimum of from ss. to '£!~proximately 701 to $2.8o-and a. maxnnum of ·£so--about $14o-are common.)

N.umeriCill Value of the Cards, Hard and Soft Hand$ . The player Clln choose either I Or 11 as the valae of

4111 Ace.. The numerical value of a f~ CIP'd ls'Io, and the ~ value of the othei- cards is simply theft' face

· .. --. W.- call a hantl"sotr i/ it contairu an Ace and that ~CII et~~~.H counted as 11 without causing the totlll to t!X•

. t;iitd: 21; we call all other hand8 "hard." Since -thbe are lwo possible tdtals for a soft hand, we shall define the .19talfor G sOft hand as the number obtained by counting the· Ace 113 11. · . ·

The distinction between bard and soft banda is im­, "'~ We shall see that the best strategy for 1r Jla.yer "'-·~Witk • soft hand of a 'Certain total usually differs tllatply · - trOJ;n his strategy with a hard hand of· the same totaL

Object of the Player·

, ,_ .. • . Each player tries lo cbtain a total that Is greater t1uut : · thtll-0/ tM detller but that does not exceed :u.

NGturai.J. i .· -11 ,..first two cards dealt either to lite player or to

1M 4lt!}R ·consist· of cn Ace and • 1fJJfllllllll card, they

. ! - -_- -..- "'""'~

The Rules of BlackFfc Jl ,

constitute what we shall call a "naturaf' or -&lackfdck. • II a player has. a natunil and the dealer. does not, tiN player: receives .1.5 times hU original bet from. the ~. If a player does not have a natural and the deOler ·doG, · the player loses his original bet •. lf both player and dealer have naturals, no money changes hands. · ,

In 1964 automatic blackjack machines were intro­duced into northern Nevada whi<:h paid 2 to 1 far aa untied player naturaL We analyze these machines io Chap- · ~~ . .

The Draw i The draw starts:at the kft of the dealer and proct!*b l In clockwise fashion. A player looks at his hole carib and ' may elect to "stand" (draw no additional COI'tb); otkmvise,

he can request additional cards from the.dealer, wlddUlR ,. dealt face up, .one at a time.

If the player "bustS' (goes over 21), he lmmediatelJ turns up his hole cards.and pays his ~et to the dealer. ~ft. each player has drawn his Cllrtb, the dealer tu1'n8 11p h& hole card. If his total is 16 or~ htJ must draw a Cllrd .. and continue to draw cards until id8 totizl Is 11 or mpre. tzt . ,, which point h6 must stand. If the dealer receive.r an-Ace. and .if counting it as 11 would bring his total to E7 or l!fPI'fl :·

. without exceedfng 2:1, then he must count the A.ce M zr , · and stand.

Many casinos alter this rule for soft hands so that the dealer draws on so_ft 17 or less and stands on ·sott 18 or.· more; in this way they gain a small advantage. Some casinos pin still more by other· variations of this ·type. · ·

·. It is common practice for the player to request .addi­tional ~ from the dealer either by saying·~~, or ~t me~ ,9f . simply by scratching the felt table. top widl · his cards. To: refuse additional cards the player places hJI bohtcards face d<>WJJ ud •Y also say "Stand." oq)~t:Jiia ~:ard.u•~)lis ~t. IUs ooDSidered bad form tor tJ1o..player . ·

12 B'B-AT THB DI!ALIB.

. to touch the bet itself after the deal has begun. One reason fOr this is that players have been known to auempt, by sleight of band. to alter their bet after seeing the dealer's up card.

, The. Rules of B14c1cjack '13 .. dealer.. even. though the dealer -may later bust also.~ Titus

the case wherein both the_ player and d~aler bU$t 1s:. au. .. example of a "tie" ~t is won_ by the dealer.. ·

·Splitting Pain

The Settlement lf the players hole cart1s are numerically Identical, · II the player does not go over~~ tmt1 the dealer tloel, they are called a [xzir. He may choose to tum them face up

1M play.r wins an amount- equal to his original 6et. II and to trellf them as the initial cards in two separate "twirl' neilher playlr nor dealer bllsts, ths person with ths higher hands. Thrs Is known ~ "splitting a pair." The original bet total wins an amount equal to the original bet of the player. goes on one of the spUt cards and an equal anuiunt is bet II dealer G1'lll player have 1M SlUM lt1tal, not exceeding u. on the other card. The player automaticaUy receives a

•110 lfll1M1 t:htmga 1umds. second card face down on each of the split cards. He then · A player-dealer tie is ca1led a -push. • When 1he!e k a plays his tM?In hands, one at a time, as though they were ~ the ~er removes the playt;r's cards without touchlnJ · ordl~ hands, with the following exceptiOJJS. In the_ COR JUs bet. This often sc;.ems to be CODfusing, 80 to bring the of splzt Aces, the player receivf!S only one. card • on eaa, -push" forcl"bly to the player's attentiog. 4ea1ers· sometimes .·Ace. Further, if a face card or Ten fall$ on one of the split · • hold the player's cards face up and atdke the table a couple Ac~, the . hand Is not counted. as a natural bid Of!I;: 116

· ·of times before temoving them. · ordinary 21. Simzlarly, II a player splits a. pair IJI fac6 . . ·In somo games the dealer takes 4Jl ties. This gives .· carda lf Tens and then draws an Ace, 1t counts only as 111m a horrible 9 per cent edge. Avolcl such gam.es. · (I1J ordmary 21. 11 a player splits a pair and tecelves a third · · · ·· · If tieS are a standoff, one might think that, except for · card o/ the same value, he is not permitted to ·split a8afn.

· · the effect of aatorals, the game is even if the player uses ··. Ac:es are the best pair to split The tempQ!llq .tu ;precisely the lline strattv u the dealer. HowevP.r lt baa : · Vegas rules-~ now off again. forbade sp~ Aces.

· · 1Jeea .. observed that the plaF who usea the strategy of the ,, _ 1he . first automatic blackja<?k machines iu~, kito dealer loses ·at 8lJ average rate of s to 6 per cent • The rea- . Nevada do not allow pair splitting. ·· · ·

· ton for this is that if the player busts, he loses his bet to the · Doubling Down • ~ St:llll'tW• eom"'* o.u- , ollltlblblg (S8] tt Js claimed oo I

f8&W ''ami 317 tblt tho llook'l audlor wu thdnt penoo to ca1ca1ate , ·. Af!R loOking at h& hole cards a player may elect to • balk .. fa¥orable ,._.... a& lll"ckJeat. Oa rJ: 317 It II aJso - .. ~ his bet and drtnf one, and only one, more Car4. :::S ::' .:.t',::Z. =::a :l., ':e. = 1ime ~ · ~This strategy is known as ~'doupling down." ..4 IJ.laytll' who '" 681) tile ..,.... 111te (I&. ~ or· ttae total 811101111l'1:) a& , . , lfOfll'lu down turns up his hole cards and r«ei'HS hl8 Wl*hoatbe~-=:-~ -=-~~~ tbat It II DOt feasible to alure . , thfrd ~d face down. A p~ who splitl any ,alr.ei'cept . dle ~ apl.a lilullwJdae1 P.Jen because their ltratigles . . . tf ot~· aail cfOea • . . . differ . • Allo - pap 536. ~ campositlaa Js Ulllllle4 1or r --Of lOsS? . . .DOt split p,afrs or- double doWn, "hat~ 1111 ·awrap lllo ~ '1'1.-.. - J!P ,,. • ilturtblr assumecl that the playw i "" ;> .. . . . • . . >: .

·'folleft. . tlaO aamo_ . ·ratea {te... '""*D_ . • u. tb8 tbllowiDg -~ itJoW> ! · ; -~- IrcDerm_.set the ~u.. ............. _fBI!.. 1FO mGfon tbat,Bal.dwlu. CMtey.,_· .. ·.· WaeJ. _ • .w; ..... That the IIOOk--la reality to be~ lho s01U- ! -~ . 0 1'- ._ tollUioD. to thia pr~ -~,... .t~QQ:eo.bprol!Ieal: If ~)!f- folfowa the same ltiate&Y • the J .. _5.__ .... _1..,_1!le,=-··. r~. ~iaatMt""'uaticalpper[z.p..4DJtiad · .... Joe. liaala OD d· 0C l'( er..,.... dfa'WI 00 aU tota1a of ~ ~ ,., •. , · " · · - P. 47); . .

. - - ~ . ' ··-

r------~= ==~=""'"',_~--- .. .,.~~~- ~- l " TM ~utes 0/ Blac1cfack. 15

I L ..., .. .,,., -g .In :,;;;,)~ :..· ;:,.•; I :=,::.~~:.::.:; = = ~ the split cards, double down on one _or both oJ his~ twin an4 there are therefore threo Shifts of employees.~ Puerl,o ~ hands. Rico the casinos are perally open from about 8 J'.V: ·to . ~ In Puerto Rico doubling down is permitted on a total 4 A.M.. so there is only cmo sbift.) Th~ customs and ! of 11 only. Some Nevada ~ particularly iJl the? Reno practices will have Httlo bearing on the basic strategy of ~ and Lake Tahoe areas, allow doubling down only on totals Chapter 3 but will. be of interest in connection with tho ! of 10 and n. 1be same was 1rUc of tho first automatic winning strategies to be discussed subsequently. ·. r · blackjack macbtnes. Tho temporary Las Vegas ~es Shuffling. It is a custom that the dealer can sbnftle at ~- changes. now off. again. also teStrictcd doubling ~ to any time between bands. Also the ·\'lealer sbuflles In· the

· · totals of bard rr. This was the only l'!lles change besid~ midst of tho play of a band if the" deck is exhausted. ·A · ·tho restriction on splitdng Aces mentioned al)ove. R.estrlC- dealer whO sbufiies in the mltlsl of tho play of a hand

lions on doubling down tend to increase tho house ad- when unused cards remain is probably a c~ The playel' untage. may request a sbuftle between bands.-Some dealers compf7

and some rcfuso. We shall refer to the practice of unnecess8rily fro. llf8U1'41U¥

- If the dealel's up card 18 an Ace, an tldtl/llotull wager 18 allowed before the draw. After c~klng his hole cards, , pJtqp may put up an tuldltlOnal side bet equal Ill most 10 hall his original bet. After the player has decided whdlter t11' not 10 do this, the tleoler checb hi.$. bole card. ·lf.lhe. Utzle~ has a natural, the Iitle bet wins tWice its ·11111tJU111. lfi/Je dealer does not _ltllve a l'llliUTtll, the side bet

quent sbufDing by 1be dealer as "sbufile up." . . ShiliL. A sbill is a-house employee who bet& moDe)'·

.l ·.and pretends to be a pJaJer in order to attract cust'omell ' or to stimulate play. Shills may or may oot be used in a

pen casino at a given ~e. . .

u 'lost tilld the plt,zy continues. The original bel is settled in the ll$l!lll way, regardless of the slde bet.

SupPose. for instance. that the player makes the side · ·bet. aDd that tho dealer hal a natural and the player. does

· DOt. The player then loses his original bet but wins ~e same amount back on the side bet, for no net loss or gam. This is why tho side bet is referred to as -msutance." Many of. the casinos. Jn northern Neyada do not allow inaurance. u0r do thO automatic blaOtjack macldnes.

CU81om8 mid Praf:dea. . . There are customs and practices ccmnected with tl1e

pne pt ·bJackjack which are not to be thought of as. part 1 of the ~-They 'YarJ_ enaticaDy from_ cuino to .casino, ' ~-in the same caaU1o oa different shifts. aad smno-

~-·

J

Shills generally follow "shill rules"; I.e., they uever doublo down, _split pairs, or insure, and they stand on ~

. totals of 12 or more. They often foUow Ulo dealer'• rules · for drawing or standing on soft totals. If the sbD1 does nat follow a fixed strat&!gy ho may be helping the dealer and/or · Jiouse to cheat the players (sec our 1atec discussim of -anchor men"). · ··

New ~. Tho player by custoiD. but uot ueces:. sariiy by law, is supposed to be able to reque$t a new deck whenever he wishes. Generally new decks are ~-spread face down. Among other tbfngl. this gives the. ·dealer a chance to check the backs of tho car(1s for hn~ODI that in. tum could be used by the player to identify cants when they are faee down. -Then tho cards aro -~ face up~ 'Ibis gives the player a Chance to see that no ~da llavo- been removed fcom or added to the cfeck(s).

- --... -~~~~~,.-.---~,_~·....---~-..--.- --~~-

·: .. ' ;&1

'.3 -~- ·:.~;~-~::?f' ~·-tu~-~~ -;;?M;;;. -=-. -;:BU;-:::-c-::StrtM:c .. "'"":c ..,..Jy---. -. -------17-l

·r· = silver dollars. 1 did not expect to w1n but wantect to .. how long my stack would Jut. as wen as to try out ibfs

_The Basic Strategy

During one Christmas vacation, my wife and I decided to relax from mj' teaching duties at tho University of California at Loi Angeles by spending a few days in Las Vegas. We -~ had been there before, but we were not gambJtrs. We ~ t,!le shows. the luxurious low-cost _mc8ls.. and· iD sea­IOD. the swimming poOls. -· . Before we took the trip, Professor · Sorgenfrey of U.c.L.A. told me of a recent artide in one of the mathe­matics journals [2). The article descn"bed a strategy for play­Ing _blackjack that allegedly limited the house :ro the tiny owi'alf edge of o.62 per cent. • Because thiS figure is so ._ly even, and So much better for the player than in any Other casino game,· I wrote tho- strategy on a little card and cainecfit on our trip. ,. . . _ · .

When I _arrived at the blackjack tables, I Purt?hasCd · , . . • Mr. Wilbert B. Caatey lias told us that an error in aritlunetfc.

4llaoverecl after [2) aa4 [3) were pu~ allows that the figure gfvea for 0. ·boase adYIIIItage should baWl belen 0.32 per Cent. rather thall .o.6a per.~ 'l'ho cprrcct· lpro for their llrl&te&Y ia· a p1tqtr advaDtap ~~~~ . . .

. . •16•

I strategy "under fire." - , I In a few moments tho slowness of my play and the 1 little card in my palm had attracted amused bystander& Tho . I dealer could not conceal his scom for one more •ayaraa•

player. These sentiments wero soon iaced with. pity whea these people saw· tho way I played. Who 'Plit a pair of lowly FJghts-and doubled tho amount of mouey being

. ~ed-when the dealet's up card was the powerful Ace? Had anyone ever ·seen a player who doubled dowD- on (A~) against a Five or who chose to stand on a piteous sa (hard) against a Four?

To add to this poor beginner's misery, tho dealer was having a very strong run of 1uct. Every player at the table was losing heavily. Surely my tea •Cl'1J~Jlbr would 100a ho swept away. Or would the.Y7 Somehow these weiR!-~ kept .tumiD! out right. As o~ !layers lost heaps of-~ my liUio stack held. It even iDchocl up ODCO. After twelltJ

1 · miiJiltes most of it was sdD thae. Bcgimleta tact. . . . ~ a strange thing happened. I wu dealt (A.2"J. I

drew a Two, and then a Three. I DOW~ (A,2,2,3}, a so{t 18. The dealer had a N'mo up. but ho might haw W It­Only a foc>l would dJaw again and • tho Clesiiucdon 4 ~a~ ~ I consulted my card and drew. With 110 Jitt1.e satisfactiOn and several ~~tsk's," the amu8od oD-~brs.. saw me .. ~w a Six. Hard 14. J "Serves. mo __ . _rig&_ . ~-• I ~ an .A£o which gave me liard 15. Tough luck;. f4tew ~.A six! I DOW held (A,2,2,S.6..4,6) or-~ ai. This JS an event so rare that it only u.ppeas OJXO De.r ~eral thousand hands. . . · - · "'-

' . · After a moment of ahock; some of the. bystanders saicJ I _bad a $25 -~ • • The dealer- .:..·~~ · · .. ~......... ....,;... - . couung . .. P1V -No" ....... was Ulll)' .r- at a few places fa .Reno. I wa$ ~Or ~a-bon~ But_ I thought lt_mlght be~-~ the DnpreSSion4bat I had saerific:ed my soft 18 ~-~

Ff:';-~~ - -- ;~-AT-THB DB-~~-;~---r fti: Baslc'Siiizi6gy It r . foresaw the seven-card 2I. "And who knows, they migh~ .· f / 1 even pay me." Of oourse they did not. But the amusement } De P~s DeclsloM · and._ ~nizing attitude of some _bystanders change4 to - . As we saw in the last chapter, the game begins Wid&-~ to attentiveness. and even to gopse pimples. · certain prelfmlnades. When the players arc seated, tho

· After another fifteen minutes-and after the oblitera- deck is shufDed by the dealet and cut by a player, and the i-. don by the dealer of all my fellow players-1 was .,ehind a cfealer burns a card. After the playm ·have plaeed their

total of eight and one-half silver doUars and decided to stop. , bets on the table in front of them. the dealer gha two But the atmosphere of ignorance and superstition that per• • cards to each player and to himself. As mentioned' pro.

· vaded my little ~ence securely planted in my mind the -1

· viously, one of the dealer's cards is up and the other dOw& ~n _that even "good~ players did not know the funda· · - At this point the player must make a number of decf...-•181$ of this game;. 'Ibetl: might be a way to beat it. • sions. The principal ones are whether to split a pair, if he,

if· When I returned home, I began an ~ve study of has one; whether to double down or DOt; and whether to --_• · thO:~. I was convinced at once that a winning system stand or draw. In general. whlit the player should do ~ ·

eQUid be devised with the help of a high-speed electronic pends on ,the cards he holds, Oli die dealer's up C8ld, au4 calculator. As the first step in_finding such a system, I used , on any other cards the player may have teen. However, an mM 704 computer to. improve the strategy discussed in . f· in . this chapter the player completely ·Ignores an- cardl

! the above episode. It is thiS revised version-which I call , Jae Jlas aeea except his own hole cards aud the deaiefs up

!, the "basic strategy': -that you will learn ·in this .chapter. It f 6l,td. no-basic strafe&Y, pen ill this chapter. is tho best •· •1he foundation for the winning strategies otla• chapters. + . pt)#lble way to play witla this iDfonnation. aso. Later, we

. . ~ show that iD. a typical casiRo ~pajer who , shall.l!nprove our atrategiea by using the 1cDoWJedp gaiDec1 - 1ises-•-1he comet basic strategy has an edge of 0.12 per_ cent t • bodl frOm the playet's seeiDg which can;fa were eonsnmecl

- over the housi In some casinos the player actually has a 1 --~ -~ ~- of play 8ll4 8lso- from his -., OD--i mwow advantage of -aS much as o.6 per cent. In casinos ~·- :-~---_ euneat_ . - -~-'IOillld. of .. play. exposed cards other dum Jds

I

I • .._ the ost ad -·t-.. th . 1 has di d · own hole cards and-~ ~~ up card. · 1 • Wlu-& m verse '"'- e p ayer a sa vantage i . . The player's. by decisioDs (pair splitting, ~Wiag l of ·-less· than. t pet cent. Against some- of the autOmatic i. ......._-_-_ . -$landin_ g .Qr drawina) Dnd the order in whicb M l: blackJack machines, the basic-strategy player should. theo- uy_w~ ···--o ~- retit;ally have an advantage of 1.6 per cent and be able to , lfl#kes them are illustrated ia Fipre 3-I.

l- win consistently. For details, see Chapter 9· Tlw· Bosic Strategy for Drawing or Standing Time and time again you will need to use this_ basic ·. In the great majority of hands ~ player will Deftlle&o

-_ strategy while "waiting" for more favorable betting situa- '· split ,a p~ nor double down. ~ his decision is~,. lions.~- arise. It must be ~ completely ~emorized that to·~ ho should draw or stand. Since this deciSion. is any decisiOn it calls for.can be made without hesitation. •. the ~~ana most important part of the. strategy, we . - •'111cre will ba DWlJerOIII anec:doles aDd fDcidents ~ our sball -~ li 1itst. -porarily negJectiog dae· poss,ibJiidet ~ ia .actual -,Jay. They are here to male tlaluJs "cGme a11vo.• · of.-_ ·.· · Uttin and dnt.J.rn.~ down. _ · · l1le .-dor • c:autiOnCcl tbat oae ~ a lew iDddeafi ltioul4 DOt Ill .c ~~a ~ ~~ Ta~ 3.1 in Orc1et to. do-

. ~be _CCJIIItnled .. ~ evfdem:e for 1M .,.... . ' c . .

l ;

BEAT .THB l>.i.ALl!R

, - 3.1. Tile Player•s Key Declsiold.

.• • Jteeal(•that•hel! a pair of Acts arc apUt, 7011 arc obliged to stand after 1Jei!la • deolt cme card on each Ace.

clde whether'- to stand on your current total or draw one or mo~ additional cards in an effort to improve your hand.

f -·The Bdifc· ifridegi -it · [ l1aid hand. You stand OB hud 1ofats of I7 or mote. )'011 l ckaw with hard totalS of 16 or leis. When the dealer 4owB :f a Six ·as an up card, the standing number drops tO 12! Now ·

p stand on 12 or more and draw on II or~ _

J

t

The persoa who is CODifortable ~the baste~ can add,the refinements no~ They''happen to wtor.-a . dealet's up card of Ten. The rCfinemeDt for totaJa of Jll1d 16 against a Ten actuaJ1y c:onsi4ers cards Jn additioa to tflo player's hole cards. This anticipates later.~ • · ·_

Nodce-IJso· that if you stand on a gtven total agaiDst ' a dea&'s given up card. you also '!aDd on an high« totala agaiDst that up card. SimDarly, if you draw on a givea total 8gaiDst a given up 'card. you also draw to an lower tota1l agaiDst tbat up Card.

....,.._ ~; · Notice that Table 3.1 recommeJKI~ drawing on aU ~~· . llard totals of I I or less. Th~ is reasonable because a player I ':";_~-->W!w dOe$- this cannot bust and must increaact Jli$· totaL

Jll· a· 4 s • · 7 ·a . .. ~ --------• / You Dealer shows

I-

~-; '

- t 11 4 5 6 7 i ·g 10 .. A 17 16 15 14 [3'

.. -·.-· .. • ,--_r_~, ~ ,;,~'ltandi{Ja mnnbetS

•Holding liard 16. 1lraw H you llold two. cards, -'1 (lOA or (9.7). and ltand if you hold three of mor. cards, fol' -mple. (6.4.4.2). ·

tStaad haklin& (7,1) a_gainsUO. ·

' ·, '!:able 3·1 is a pictorial Jist of "hard_standing numbers.• The harcl staJKiing number for a certain dealer's up _cant is simpty th~ smallest total you stand on ag~ tha~ up cat:t~. ·~or eXample, 'if the dealer. shows a Seven,. tlJejt_ Ta9le J.I ~ ihostand.ing number iS 17. ThiS is yoq{pu with a

l;: l 2::1 BBAT TBB DBALBK

~~.it can be countc:ct ~~-~.if nec:essUy~ ~;, oth~ caN wiU haYe a DUJ:Derica1 value-of 10 or less. .· ·

· ~ tho player with a soft tOtal of 16 0r lesi c8miot bust by drawing one moro card. he cannot make his total pOorer. Tbis is .because aD fbiaJ. or "standing. • totals of I 6 or less are equivalent. If you stand ·and tho dealet busts.

· you win ~ same amoun~ no matter what yOUr total is. Whether 1t lS 16 or less than 16 mnes no dift'erence. If you stand on 16 or less and tho dealer does not bust, then

. by. the ~les he must have _ended _up with a total between 17 and ~I. Thus he automatically beats all totals of 16 and

. uadcf. 'I'heiefore, ~you draw io soft 16 or less, you ~t · Jiarm yourself~ In fact, you may even be able to help your-·

self. For example, on holding (A,s}, you improve your dmnces to tic or win if the card. you draw is any one of tho group At2~.4.s: C?n .holding _(A,2,A), you improve your chances to tie or wm if tho card you draw 1s any one of the

I group 3.4,S,6,?.

~ _ ··· . When drawing to a soft 17~ there is a small possibility J efbls. If you stand and the dealer alsO hu 17, .:you will tie

Jdnt and -thus avoid losing your bet. However, lr you draw t _- to soft 17, you may convert your hand into a hard hand I that totals less than 17. If you then stand on· this, you are ! worso off than before, for the dealer may ~nd up with exact­

ly 17 and n~ you lose, whereas y~ would b~ tied. If you draw to thiS hard band, you may bust and .,_ at once.

I" Po~ example. with (A,3.3)=soft 17, suppose a Five is , draWn to make (A,3.3.SJ=bard 12. If the dealer shows a [ · Rve. Table 3·• recoDunends standing. If tbe dealer shows t. an Ace, tho tal:ile recommends drawing. tf a T-eA is drawn,

Mfttach (AJ,3,s,zo)=22 (even counting thO Abe as one) and bust. '• -~te this chance of making your hand poorer by

drawmg to a soft 17, calculations show that this risk is more ~ o~ by the possibility of improving your han(l1'Jul$. .with (A,6), you may draw an A,2,J, or 4, all of~hiGh ill}-

1. :. ~ tr ~ ;r ,... draw • s.6.7J1.9 • .,. .~ p do JJ4t bust. You still have another draw~ if you wislt..

. . with which to try for a good total. . · · ' . It sometimes. takes a little· wiD power to follow th~

'Instructions. More than once I have been confronted with heart-stoppers like this. I was playing a "big:.money" gan:ao in ~ certain Nevada casino. By tho card-counting methodS of later chaplm, I knew...:.tbat I bad as per cent ecJ&e on tho 1mt ~ of play. Therefme, I had placed the maQ.

,_ mum bet of Ssoo. The ~~ up card was a 7· I was dealt (J4,6), a soft 17. Since the remaining~ in tho deck consisted largely of Tens, I was fairly certaiB- that tho dealer had 17. Since thecc were only four canfa that would help me-the A.2.3.4-8Dd- five tha.t would. hurt · me-the s.6 .. ?,8,g,1 was reluctant to draw and was.inclhled to play for .a_ de. Neverthc!lees, I grittect my- teeth and,~

I once, rec:dring an 8. I now held hard IS. I held my -~tla. ' and drew again; this time I RCeived an Ace. I now held

._.d 16. Resignedly, I drew again. ~my lllllaZC)o

ment-a 3· I now decided to stand with my hard 19. \VbeJl the deaior exposed; his handJ to my surprise he held • happeaed to be the only Ace yet UllaCCOUDted for (oQOlaad .. already appeared on an earlier round of play).~- -· to tho rules-Oi the game he wauequired to stand. The basic strategy not only produced. the one line of play .that c:oqkf-ave the $soo. it doubled the PlO!ley besides. ·

We see from Table 3.2 that there are minimum staJtd­Jng numbers for 80ft hands similat to those.for. har4 ha1lds. · ~you should draw if your sott total is less tbaa the _soft ~number given for tho current up cant of the dealer, and~ if your soft total is greater tban or equal to this.

._ . ,bcr. The reader w)J.o practices with the basic_ .. 'SOOI)lcnow the ~g uumbeis well enougll --

tq _ _ .. with TAbles 3.1 JQ~d 3-2. . . -_. -· . . : :J,Jiti~ llQW. that you go iQ.to a ~- tq practice Ulijjlf_(lltlmding uumbers. You never double down, never

i l l ! I l

i ! '

~----~--~-----.-

BBAT THE DBALB:R < •

'split pairs, and never h1sute. How will you do? Suiprisingly «iough, tho casino edge win be only about z·percent. Your game is alteadyctose to~ It is better than tbe methods recommertded. until recently by die worlcfa fotemoSt ·card experts (see. for example. [8]).

1 The Basic Strategy for Doubling Down . · The part of the strategy which is next in importance, !

u well as next in simplicity, is hard doubling down. It is , probably DlOl'O couvenient to. postpone memorizing the ft ·IOft.doubling-d strategies untU after pair splitting has

· · beea learned.· But- for cOmpleteness, we $hall aTso discuss 10ft doubling down now. · r

AI indicated in Figure 3.1, the decision about whether · · or not to double down must be made before that about

drawing versus standing. This· decision is made by using Table 3·3· The poJSible up cards ot the dealer are aga~ listed across the top of the table, and the player's totals f~--_-· are listed in the column on the teft. In order to decide .mether to double down, first see if· your total~ in tho. column on the left. If it does not appear~· ,OU 'Should \ .

. DOt double down; instead, proceed to the next decision. t

whether to draw or stand. If your total does appear. run f clown the calumn belqw the dealer's up card until yo~ ~ . reach the row in which your total appelll'S on·the left. If r the· sqww at this location is -shaded, ctome -down. Note ( that the table bas two parts. One part is for soft hands • .only; the other is for hard hands only.

To illustrate the use of Table 3.3, suppose that the dealer shows a 3 and you have been deaft (A,6J, or soft 17. On locadng the· appropriate square you find ihat it is ahaded. Therefore you double down. . ;

Tbero are S8'Vetal tbings to notice in Table 3·3- Ptrst. ' there is no total on which the player -doubles down regard- -~ less of whether it iS hard or soft. Second, hard doubling ' .cfown is done only on totals of I 1 or less._ and soft dOubling

. f . .

, Tbe Bqsit: S~rategy - , . -~S · ·~ is done only on totals of 13 or more. Sof(d.b~

dOwn .on a total of 12 is sometimes better thaa. draWing. - But-.soft 12 means a pair of Aces. It i&:mnch better to split

the Aces instead of doubling down with t:hem. Observe that the player always doubles dowa. on bard

u. With hard 10, the player doubles down except against :an Ace or a Ten. Hard 10 is a less favorable tOtal t1ian hard II, except when an Ace is drawn, becauso tJ» total the P.layer obtains when doubling down on hard 10. is ()ne less than the total that he would }lave obtained by doubling down on hard II. Hard 9 is even less favorable thali hard. xo, and with hard 8 the player rarely doubles down. In fac:t.

-1he situations where you doub~. d~ with hatd 8 are so tate; and the gains are so slight, fh:at you can neglect them ·with practically no loss. ·

. • Cauble down

D no nor double-

~dOwn wilt (A,A) only if Aces cannot ba,.,

IIlii Double down D oit not doulll8 dowiJ

.·:,._.·~~>.,~dolon.-Pt"""(6JO. . ~:

rc--;,.-.. UU' TBa--.. DhLlfll

. --· -~A. conspicuo111 feature of. the table is that soft doubling .«twn is never recomDiended against 7.&.9,10; or A.

It is hard to explain the doubling-down strategies without using mathematics. Biit experience in actual play 10011 engraves them on the memory. I always remember to double down on soft 13 against a Five because of a hand 1-played at the Silver Slipper in Las Vegas. On this occasioa, my friends and I bad gone to see whether this casino would cOntinue to play when we began to win sub­&Wltial aunounts. 1 was varying my bets from $1 to $10.

' ,{A variation of $1 to $3 would be wiser at present, now f': that thOusands of readers of Beat the Dealer have infticted f . bloody losses on ~e casinos. r Because I had frequently

!.

.•. bet_ooly $1, we agreed that I was not to raise ·my bets

. above $1o-to do so wouk1 attract attention. However, a juicy situation arose (a 6 per cent advantage). 1 could not teSist! I shoved out $30. To my satisfaction the dealer's up card turned out to be a Five, tho most favorable card

· . fOr the player. Confidently I tutnccl up my (A,2) hole cards and doubled my bet. I did not bother to loot :at the down QU'd that was dealt to me beCause I expected the dealer _to..have a~ down and then to draw another card, busting bimself. To my horror tho dealer's hole card was a .Four.

·· Be drew the CJq*ted Ten for a total of 19. I was resigned to • loss when the dealer turned up my hole card to settle

. the·bct. It was a Seven! · . - There was a strange expression oil the dealer's face.

· · Luck. by itself was one thing. but my huge bet in adv~ naade it &eei.Jl- like I could foretell the future (which 1 ~-of course. to a limited extent, although iD this ease I-was quite wrong about the detail$). ·What the deater did not realize•was that he was as lueky to have a Fout under~ neath u I was to draw a Seven.· Orie of the characteristics of the basic strategy is that it m~ those who use it considerably .. luckier" than the average player. In thi$ m. &taDce. my "luc:It- proved embarrassing~ ·

---.-,

..... -s~ it-'< • Oneo you have.raasteted the ltr8tegy fer ltar4 doubt­mg. JOD furthei cat thocaSiao'aecJge tom thaa 1 per cat.

Tbi Boslc Sttrltegy for Splitting Ptllr.f After memorizing tho strategy for drawing and stancJ.

ing and for bud doubling (soft doubliug too, if tho rest il easy), you am mady to add pair splitting to your repertoire. Tho detalJal_ pair-splitting str8tegJ will be deacribed,' fol.. lowed by a simple way to 1eam it. ·

If )'OU have a pair. Figure 3-1 shows that the fttst decl­sion you have ~to mate. before both doubling dowa and drawhig or stJmding. ia w~ or nOt to spHt it.. You C8A

4ecido this by using Table 34 In that table tho possible ·r· up aida of tho dealer am listed in a lOW llCl'OSI tlle- top

and tho possible pain of tho plaJar·am-liated in tho columa on f:ho left. If you havct a pair. IUD &nm tho column be­low tho ~ up caret until you gel to tho mw.labelo4

. with your.,. If tho square at this locadOa ta b1aDk, do .. JlOt split your pair. Proceed immediately to Table 3·3· If

tho .. js shaded, tint split JOUr pair aud thell,go 01110 Table ·3·3- If you have no pair, as is tho c:aao about six

1 1fmea out of sevea, skip Table 34 altogether aDd go 6 zectty to 4'8bte 3·3· .

· If Table 34 seems imposing. you may replace it by -~- t~pproxlmate set of ruleL They am: .alW&JI split Aea

i · aDd Bights; never split Foun, Fives. or Tens: split tho Other t parr._ whoa· tho dealer shows 2 tluough 7· Tile bea'V)' 1iDea

in Teblo 3-4 DKficate this set of rules. Notice that,_., .. . ~iDtroducesonly1iveerron. Some of tbeseerroa .--quito large. but becauao. the sttuatious arise iafrequently,­... elf~'l.is to add only o.13 per c:ent to tho Oftl81l hoUse -~,.().aeo you haft Jeamecl to lise. theae· .approxt. ... ~~·~_with ywr-doub~·--

. ~ Jlrategy .. you .. ready fo.Jeaar

li~~~~~-•~u 1

~h-

28 -B-:&A'r THB :DB-ALia

~;~er to team by•visualizinfthe ammgc!uu:nt of the ~quares. For ·example, the . infonnatioq in forty of- the squares is contained in the.. rule "Alwa~ split Aces and Eights, never split Fives and Tens ... There are "reasons .. tor these rules that may help yau to retain them. • Aces ~ould be. ~!it because there is a very good_ chance of get­ung a wmntng. hand-even 2I-with each of the new hands, whereas the original hand (A,A) is only fair for

· doubling down or for drawing or standing.

A

31

·-,ulO~ from the book and used while playing., As. )'OR·~ ~' ·Oeano more expert you will consult the card lesS frequent~)' ~--·~ finally not at alL

· What to Expect When Using the Basic Strategy You are now familiar enough with the basic strategy

to try it out in aCtual play. If a casino is not available and you play at home, be sure that tho set of casino rules we have adopted are in force. This will mean a CODSkterably differentprocedure from that usually employed in a home _game, but perbaps your friends will go along in the interest of learning something new about blackj*

The following data may enCO\U'Ilgo yon to tty tho basic strategy at the ~ in. spite Of tho fact that whea you use it you are still, in general.· s,lmply _playing about even with the house. Table 3.6 descn"bes the possible out­comes that can be expected if_ IOO hands (~ from

· : thirty mitmtes to one and one ball -ho1U8 of··playfag time, - . depending on the speed of the· dealer and tho .J~Uin1)er. of

• Spiii_Piir players at the tabl~). are played at $I pet hand, an~ also if CJDonatsp&r,_ I,oOo hands (generally from five to :fifteen hours of

·· If the dealer has a 1.8,g,ro, or A up, Eights should be playing timo, depending on conditions) are played ·llt split, not so much because a good total will be· obtained · $I per hand. If the amoun~ ~ per hand is ditiereDt. just

· · ,With each new hand, but rather because I 6 is, in ge~l, a multiply all dollat values· by the appropriate DUIJlber •. Por bad total to hold. The reason that I 6 is unfavorable iS this. example,· if you bet Ss per hand, multiply by s; if }011

. When the dealer's up card is 7 or high~, be is·n6t likeiy ~sol per hand, multiply by o.so (or divide by 2) •. Tho . to- bU$1; ·and if be does not bust, he is sure to beat I 6. Thus, .. average gain after one thousand $I bets is $I. Afte.t ·one

tho splitting of Eights against z through A "breaks up" a hUndred $1 bets, the average gain is IO#.· Thus we regard · ~ •l:timd. · the basic ~tegy as essentially even: ~ real advallttge

I~ turns out that the new hands _are not:Vf!rJ unfavor- ' for either side. a?le (1n fact, they have about an av.e,rage chance Of win- Bat~ et al. (3] report the results in Table 3· 7 with· JW1g), and even 1hough you are staking more money, y9ur their strategy (it ~ essentially tho samo · as the· basic

~). . • These *reasons• are onty a yery crude picture of the actual state · ,- · -Who number of hands· play'ed in each. group is near

. of affairs. &be precise. situation .is given ill U4e Ap~ We give ~ to 000 that. · if n-tMad the .. -t-- of .hands · .. reasons" here t? help you 1ix the rules iif your mind witJiou~ M.flng • ~ . I, SO . • we r~ nu..u.u,.- ·

.to worry .about mvotved mathCilUWcal Rofnts. no. wJIO .......... . _ ...... -·. . . iii-.-(JOO each time hl Table 3·7• we caa '&180 the Ulistance might c:onsUk [14}. . ' - .·

l . I

-·

33

-

... - - ------~~--~~:-:---:c--

~ -:;.·

BBA'f· TiiB Ji~A.L'ltll ' ..

. .. TABU! 3.6 •. RalllU V.slng tb BaSii:' Straleg1. If 100 hands ~ playcd,at $1 per ha~

approximate the result. is* per cent of time that between and

0.01 -$19.9 'below 0.1 -149 -$19.9 2.1 - 99 -149

13.6 - '49 - 4.9 -34.1 0.1 - 4.9 34.1 S.1 0.1 13.6 10.1 S.l 2.1 lS.l 10.1 0.1 20.1 15.1 0.01 abovo 20.1

If 1,000 hands are played at $1 per band-

approximate the result is* per tent of time .that between and

0.01 -$62.2 below 0.1 -46.4 -$62.2 2.1 -30.6 -46.4

13.6 -14.8 -30,6 34.1 r.o .:.. 14.8 34:1 16.8 1.0 13.6 32.6 16.8 2.1 48.4 32.6 u 64.2 ~.4 0.01 above 64.2

......... lllllllliiD ln4lcale .....

TAB~ ]·'· Ruulu tJ/ Blzldwln, et lltl. Nu,mbll' of hands playecl Gain

930 $38.50 ( 770 -.56.00

1140 -4.!10 690 -4.00 -'3S30 -426.00

-~ ; ---"?"-.;·-..:-;.;~ /~;.:i; ~:.;;::""-<?.:--- < .·-- ;

~~-· &zslc .. ··~ .. ~.- ~_~~~--~-- -. -~

.. ~- -. ' 33~ .-•.·

.,

.

.

' ·,

secon d part of Table 3.6 for· a rough check of ~ JeSUlts able 3·1· Everything is normal- except for the awing Of'I1

fiof4 s6. This awing, jf it is not the result of causes .other chance, is a. rare event. Table 3.6 tells us that ~ than

1,00

more

less. at $1 Ss6 that·

o hiuads are played, the chance of a awing of -$56 or is less than o.I per· cent. With a smaller number of ..

hands, in this case 770, the chance of such a swing is even In fact, calculations show that if 770 hands are played

per hand, the probability that tho player will lose or more turns out to be approximately o.oi per cent; IS, the odds against this are about Io,ooo to I.

Com parison with House Percentage Agm'nst Other Bla clcjack Strategies and in Other Game1

than We remarked earlier that the basic strategy is better other blackjack strategies and that it is also better any published strategy for any other gambling game.

es 3·8 and 3·9 illustrate how much better it is. than T~l

TABLB 3.8. Thtt Ba.dt: Stl'(ltt/81 Comporeil wldJ Oihttr Blackjtlck Stl'(ltttgitl~o ·

BlacJtjack strategy Player's advantage (in per cent) . +0.1 is typical; rangoa roughly fJOGl

-1 to +1 card -3.2 .

-5.1 -2.00 to -15.00 -6.0 (estimated)

' .

Some Common Blackjack En-oil The basic strategy for blackjack was first published

(wi th a few insignificant• errors) by Baldwin et al. [21 years before .this book was begun. Nevertheless black­strategies containing gross errors continue~ appear.t

fQur jack

• We caD them insl~ because thoy cost tho player on tho average a mom 0.04 pee ceut of hil action. . · fin tliJa aectiOil we are ~ tho basic strategy_ with only ... ~

atratog1es fJl which completli composition is assumed. tha& ~-which do aot COUDt card~.

f ... ~~ c -~ :.>. --..

.TABLB 3.9. Till Btalc Strtrt•gy· Compol'ftl with But Plfl1 in Other Casino Gamu• . -. _ .

Game

. crap JOUfette (Europe)

.• roulette (United States) Nevada~lll'llt

Player•s advantage (besfplay) (in . per ee~)

+0.13; rallge$ roughly from --1 to +i

-1.40 (may drop as low • -o.6: see [80]) ·

-J;3S -2.70 to -5.26 -1.06 (banker); see [70] -1.24 (player); see [70] -1.52 b average; ranges fi'Olll

-30.0 to 0; see [80)

• Card C01IlltCn Wbo -'stently make 8'ftiiMIIOIIeY bets (0 per CCDt edge) -frawtaed \II>OIJ 111 llle operaton et Ibis vlllllshlns pme. The variation In advantage wbldl Is li'fell - 1111& tile piQW II aware of lbe cue keeper't RCOrd al Cll'ds plaJed. .

In the discussion to follow, we shall formulate experiments to demonstrate several of the more obvious of these errors. Some of the experiments will take the reader less than an hour~ Bach _reveals the error in one of the commonly recom·

-, mended syst~ by comparing with it a feature of the basic umte~. _ _

. The ~periments should convince anyone wJJ.o tries them that the basic strategy is correct on these points and that the other strategies are seriously in error. Study of these experiments should enable you to formulate your own ~riments for testing the gross differences between

lf the basic strategy llJ!d any :()ther strategy. We can, in prin-ll ciple, test tmy discrepancy, not merely the large ones;· in -~ the case of close decisiops, however, the experinients are li unpleasantly. long. 11 Tables teferred to in the rest of this chapter are in . , the Appendix. Th~ is no n~ now to refer to or ~--~ • _stand these tables. The important thing is to get the idea ] of how to ~ ~ stratcay by uperimeat. ~

~

!:r i

r;f ~~~-,"'--~,__,__: ~

3S

· FintExpqimmt: Drawing Versus Standing HoT.ding · Bard_ z6 Agai11St an Ac~

Table 2a shows us that the player who draWs rather than stands on hard 16, when the dealer bas an Ace show-. ing, gains· an average of 14.6 per cent in such situations. Put another way, to stand rather than diaw on luu'd 16 costs the player an average of 14.6 per cent The fOllow. ing experiment is designed to verify this. Reuiove a... Ace ~a complete deck and place it face up on the tal)Je.·-pua. represents the dealer's up caret Next, write the number 16 on a _card or paper and place it in front o~ yourselt This represents your hard total. · · .

Of oourse this does not correspond exactly to the. real . situation. In a game the cards that the player actually has drawn to make up his total of hard. 16 will alter tl\e. ad- . _ vantage in drawing. Conceivably, if enough small cards .­used to -c;omprise the tata1. it may even be. wise to stand. For exmnple, consider the very close decision as to wliether. 10 atand or draw when holdiiJg hard 16 and the dealer's up cant is a Ten. According to Table 2a, drawing is favored o-Yer standing for an·average gain of 2-9 per cent. But·Wbea the player's bard 16 totalis composed of (4. 4, 4, 4),-* precise figure is 6.382 in favor of standing. acc:ordi!lg_to l.H. Braun. ·. · · · ·, The objection to-_the use of a. paper total in our ~

meat is answered as follows. A player could replace our atperiment ~one in which he played a reiJl game of black­jack and kqit a record of the re.sults of standing vemia drawing in this situation. His average long-run result W01lld be. within a few tenth$ of a per eent of the 14.6 per cent figure based on paper totals. Therefore much time and ·· trouble C8ll be saved by a paper-total experiment. The same CODSid.erations apply to the other experiment& ·· .· · • -· . · ,; Let'• ~m to- our -eXperiment. S1utftle the • deck 1Uid ~ 200 dealer "JwldsP as folloWs.· Assume you stantl' on

~, _____ -~----:--~-- -'

16 and deal a card to the dealer (his hole card). If he gets a natural, discard the Io-value card and dc:f'not reeei'd the

. result. We do this because .the question of whether or not . ·to draw on hard 16 arises only if the dealer has already checked and found he has no natural. If the hole card is not a Ten, continue dealing until the dealer ·either busts or achieves a total, soft or hatd, .of 17 or more. If the. ~r busts, you win. If he does not bust; you lose.. Record the result. Discard the used cards and deal another hand. When 100 "hands" are dealt in this fashion the player w111 win, on an aver11ge, about I7 of the hands and lose the

.rest. This follows from the assertion Of Table 3, that the player wJ,.o stands on I 6 when the dealer &bows an Al:e loses.at the rate of .66 per cent.

. · Next, deal 200 hands as folio~ Give the dealer one card (his .hOle card). If it is a Ten, discard it and deal

· anOther card, for the same reasons as before. Now assume 1,.

you draw e~ctly one eard to a total of I 6. If you bust, you l. !Qse. Discard the card and record the loss. If yoti do not !.·· bust you have ahard total between I7 and 2I~ Stop draw­j ing to. your own hand and, if necessary, prOceed tosfvo tho j . dealer further cards until he either busts or gets a total of a 17 or more. Record whether you won, tied, or lost, and il continue.

1

,, Your percentage of "wins" should be figured as the 1 ~~.·. number of wins plus one half of the ties (to tie e~ry hand,· r. for example, would be substantially.~ the same as winnincr li -o il half and losing half, for no net gain). ln1 this part of the p· experlinent, the average number of. "wins" per IOO hands . H should be 24·3· Thus, with 200 hands the average separa-li tion between the two ways of playing harcl I 6 against an \] As:e will be 2 x (24.3 - 17.0) or 14.6 hands. In each. li part of the experiment, however, there will be chance deVia·

tions from the cited average totals. In -fact, I time in so, 11 standing on hard 16 against an Ace will produeo better }. 1 han !L resu ts, over 200 ds, than drawing. · i, r!i

~:.Co~

·: "'s,cond Experimmt: Doubling DOwn on Hard 10 ,,~8afnst an Ace "''~ •· · This eiperiment is eonducted in much the same way .·'tii the previous one. Select as hole cards (8,2) for the max• 1mum error of 6.1 per cent, in order to shorten the expedo ment. This figure of 6.1 per cent is obtained from Table 4J, where we see that, holding (8,2) versus an Ace, if " ·limply draw untU we reach a SUitable total, in the long l'11lr

We win about 8.6 per cent of our bet. However, if 'We ·double dOWn ~win only 2.5 per cent of our original bet in the long run. The difference is· 6.1 per cent. Play about

· 400 hands in which you double down. Afterward, subti'act the number of hands lost from the number of bands won.-· Then double this number to take into 8CCount your doublecl ttet on the double-down hands· This is your total profit iD the 400 hands by doubling down against the Ace. Rem.em· ~ as before to disregard all dealer naturals in yaur talll·

·If the·deller has a Ten under, give him another hole car4-- · · Next, play 400 hands in' which you follow the correct draWing and standing strategy versus an Ace (Table 3.5);. ¥ottt wins minus your losses give your profit for the 400 1wlds. ·0n the average, in 400 handS you· will have~ excess o(-wms over losses of about 17.2 hands with draw­ing and Standing. With doubling down, you will have an .aYetage excess of wins·~ losses of s.o hands. . _

Third Experiment: Splitting 4 Pair of Sixea A.gaiiut 4 Five . ·~.. According to Table 4f. the gain here by splitting

- ra~r than s~ding is 17.2 + 10.2 or 27.4 per cent. if JQU·.$114. you have a net lQss of 10.2 units per 100 bet&

· lf.yousplit, your 100 hands become 200 and you will win ,OOUt 17.2 more of those 200 hands than you loSe. You !lave an average net gain of 27.4 units per 100 original ~ by splitting rather tbm,. ~ding. Fifty original hal$ .of each t)'pe should be decisive.

BEAT :FHB J)EALB:a

Mimlc1cing the Dealer , -~ -. _ i. · To quote Baldwi~ et at. [2·, · p. 439], WJbe ~layer who t iDhnics the dealer, drawing to 16 or 1~. standmg on 17. *i or. more, never doubling down or splitting pairs, has an ~; ' clpectation of --o.os6 •• That is, the @aler. has a s.6 per ·' Cent edge. ·

11 Let Us illustrate the use of Table I by computing .a ef figure for ·the player who mimics the dethaler. First. notice 111 that when t1te player follows these rules, e game 1S sym· ~l metric except for two situations. If both the dealer and , \ player bust, the dealer wins. Count the dealer as a bust if .j he would have busted supposiilg that, althousJl the player 1 bUsts and bets are settled, the dealer plays out his · hand · anyway. This favors the dealer. The edge it giv~ him is I the pi'obabiUty that both dealer and player bust. Smce the. ! dealer • player are assumed to be using the s.ame stra· J tegies, Table 1 (Dealer~s Probabilities) applies to both of

thcn\.. .. The overall probability of. each busting. is therefore o.2836,and (asswning stochastic independence. DOt stricti? Wlid but good. in this instance to a high degree of appr?~ IllAtion. whea the deck is nearly complete) the probability

f of both busting is 0.2836 X 0.2836, or 8.04 per cent, ia 1· 'favor of the dealer, as a result of this factor. The second I· nonsymmetry in the game is the fact- that the player wins

1·1

1

:·::,- 1.5 units when he has a natural and the dealer .does DQt. The dealer, on the other hand, only wins one unit from tho

t · piayer.when.he has a natural and the player does not. Thia <,t ·happens 4.68 per ~t of the time for ea<:h side; so the 1\l player gains ·one half that amount, or 2.34 per cent from ll~Jl this factor. Thu,s. the net 4ealer's edge is (8.07 .- 2.34).

1 or S·13 per cent. ·

• The Player Who Never Brist8 l· . · Another interesting figure to calCUlate . is th~ ac:lvan· ~· tago which the casino has against a player who never ;:·:.

tl

~

---',

·>'-;.,.- ii · ... - .. . " 3n .. :.·-:tbe· Basic Strategy 7

;~ draws to a posst'ble buSt hand. FirSt we note~ tlduneaas, · t4e_player's hard standing. numbers are ~ 12. H~er, · tJtO ·son standing numbers aro not. det~ ThUs ~·

problem is meaningless as .stated. ~mce 1t iS unanswerable as it standS.. we assume soft standing numbers Of 1.7 au4 proCeed. As we pointed out earlier. cOmmon senso dictates that the· soft standing number should always be -at least 17• We ~ that 18 is at~ys better than. 17, . so 17 g(ves the player a greater average rate of loss ~ he

- wcSuld have with soft standing numbers of 18~ We wiD call a ptayer using this curlous stra~ "co~tive. ~ · ·we assert that the true figure for the house advantago 8g8lnst a co~ve player is between 5 and 8 per cent. Our evJdence comes from three sources. First, we ran. an

. eipctiment In which six groups of xoo ~ds each wero . played with the conservative strategy. The number of pla~ units lost ranged from 13 to 2. with an a~ge of 7· Tbi. -apemezit with our figure of s to 8 per cent JS good. ~ the figure of 6oo hands was selected in advance and llot. IDftUenc:ed by the results of the early hands, ~dard ~,

· · mulas from the mathematical theory of probability apply to this data. We conclude that the true figure· for the ho_ulo. advantage almost certainly JiCs between 3 and ~I per .Cent. Second. we ran a hand calculation (which iS c:Omparatively

· · e8.sy because of the low hard standing numbeD) tllat proved that the true figure was wdl_below IP per ~ TJiird lind beat. BaldW. and his coauthors give a ftgum. of 4-25 F ow for the house advantage againSt a. player wbo stands on hard- 12, ·never· doubles down, and spD.tl A£es and Eights only. (They fail ~o specify the soft ~

. .. · ing miinDCts.) It can be shown that splitting Aces.~ • Eights adds Jess than 1 per cent to the player's adVa.ntage. · ··. The 1:01TCCtion, if any, for ~t soft standing-aumbem . b also, overall, of the order of 1 or 2 per cent. ~ tho

true AgUre estimated frqm this ~co is between 5 or 6 F cenc ana 8 per cent. - ..

l .1

.li t Ji . "i

'! ! I

t I.

.l

BBAT THB ~BALBB.

The Man Who Trimmed His Barber The disadvantages of such conservative play are amus­

ingly illustrated by the experience ~f "the man who trim· med his barber,'' my friend Professor John Blattner of the mathematics department of San Fernando Valley State College.• ·

Blattner and his barber feU to talking about blackjack . one day. When Blattner told the barber of his friend who wrote a book' on how to beat blackjack consistently, the barber scoffed. "Why that's easy," said he. "Anyone can win by just refusing ·to go bust (standing on· hard 12 always). mattner tried in vain ~o oonvince.the barber that he was wrong. Eventually the barber touted Blattner into ·a little game after closing hours. Blattner brought $x6o. At Ss and $xo a hand,' the barber quickly lost an equal amount. He constantly exclaimed that Blattner was tho

.luckiest man he ever saw. After losing the $r6o, he refused

.' SO quit. He demanded the chance to get hi$. .money back. Tliey played on at $20 a band. When he was behind $10200, the barber's luck turned. He won back $300 of this'· loss. Then it was all over. He fell behind $x,soo and

·quit. The barber still belieVes Blattner is lucky. He delayed

· paying his loss. Fmally he decided to ~ve Blattner free . baircuts. After .a year of these he cried that hard times were upon him and went back to chargiilg Blattiler •. (ThC barber

"fnsi4ts that he will pay Blattner some day.) Question: Did Blattner tritn his barber or didn't he?

• There is a bit of mathematical irony Ia thiS story, as we shaD .. IH. Aa .background for the noamathematical reader, . we mentioa BerlraDd llussell's. famous paradox. ~ that a barber iii a certain town triDII the hair of an those pet'IOIIIo ~ Oniy these pe!IODI. who do not trim: their own :hair. (We aaume that a person's .hair .. always trimmed by the same ~) Wbo trims the barber's1lair'llf someone .

· else triiDI the barber's hair, then it :must be tho barber who trims 1he &rber's hair. Impossible! ·If 1he barber. trims bls own bait, then il cannot be the barber .who trims the barber'a hair. ~~ WJJO· trim~. the barber's hair? · · · · .

4

.A Winning Strategy

Ganl'blers soon teamed through experience that g8mee of .chance could be run in such a way that a certain . ..,... .. centage".favored one side -' the expense of the other ·aide.

.'J:'hat ~ if a game was played a BUtlicient aumbet of tiiael (the "long run"), the winniDgs of the favored lidO wuuJ4 pnerally .bo uear a c:e:rtaiD ~ percentago of tho. total

. ~t of all bets placed by the opponent. 1110 IDOlfora. ·~bling cas1no takes the side in its games tbat hat pRtVCa . m. practb. to bo tavorabJe. If necessary, the calino otten ~ rulea· of tho game so that the casino advantage fa ID1Bcient fO cover .expenses and also yield a desirab1D rate . ~ profit oil the capital that the owners have invested. ··,. . Tho~ aniouu,t of bets~~ called "actioa." Fgr

. ~Ie, if ~ p~ bets of $3, '2, aD.d $i1, I have ·-ru; ·~. of action. • A player whO has a cei1ain amount ·of ·.•eapitat em geueraJly get many times that amount ia·acb .~~~tely Josing- capital. to the.lwusO. 'This ecm.-)~,greatly to the=~ of gambling. .

. •.p•

r __ .,.,......, __ ,.'*'.,.,_""'"'--""""=~. ~--'="....,;~-· --~-=-- .--,....~__, ............ ,_- ~ ...... ,--;--~--~-.-.. -., .... ~-

FaiJzll'e of the Popular Gmnbling Sy$lenu

There have been many attempts to ~ the casino advantage. A· freqUent approach has been- to vary tho amount that is bet from play to play according to various methoda, some of which me simple and some of which are -very complex. By way of illustration, in the Small Martingale, better known as the "doubling-up," sys­tem, the player makes an initial bet of, say, $1. If he loses, be beta $2. Thea he wagers $4. $8, $16, and so on, dou­bting the bet each time until he wins. Then the proeess is repeated starting with $1 again. The bet placed following a .string of losses equals the entire amount lost in the string, plru one. A winning bet either _is a $1 bet, or has been

. placed after a -string of losses. Thus each win results in a net profit of $1, counting from just after the last win, and the player k~ wUming a dollar every few bets. However, this sjstem has a fiaw. The casino always sets a limit to the

·:,amount that may be bet. Suppose the limit is Ssoo and we have started by betting $1. If there- is a string of nine losses

;-- ($I, $a, $4; $8, $16, $32, $64. $128, $256), the next bet

1. · caJkd for by the "doubling-'!p" system is $512, and tbis bet is not permitted. -.- It seemed in practice that, with this limit on bets, the

caSino won the same· percentage of the ectic;>n it normally • cv~ though a player was using the doubling-up sys­tem •. Thus the doubling-up system provided no advantage whatsoever to the player. The other complicated betting

. _, _ tchemes all seemed to have the same fiaw. It was no sur-~! . prise; $en, when .it was later proven, by the: lbathematical \ tlieory of probability, that for most of the -~ •pll· tr bJing games no betting scheme can ever be devised that. ,~ will-have _ the slightest effect upon the casino's long-run :;' advantage. . . - . -; The games fOr which this is an establishecl fact include-

those games that mafu.ematicians call 41independent- trials ·

.0· / processes." (Craps and roulette are such ~·) What

•.·· this means is that each play Of the game- is ~ by past outcomes and, in turn, has no influence on futwe C)IJtcOJn.es. For example, suppose we shu1Be a deck of _cards and draw one cant. which happens to be the. Four of Spades. We now return this card to the deck and shuftle · tlwroughly. If we draw one c8Id again, the chance_that.tt will be the Four Of spades is no greater than and no leu

·thaD the chance of its being any one of the~ 51 ~-- Tllis f8ct has made popular the phrase "The cards have no ·memory." -

The ImpOrtance of the Dependence of Trials llf Blackjack In contrast to the previous situatioa, in casino black~

jack the cards do have a memory! What happens _in OliO· round of -play may infiuenco what happens both later Ja that round and m succeeding rounds. Blackjack. theiefoie. maybe exempt from the· mathematical arguments _wldcJI rule out favorable gambling systems for in~ ~ · pnes. -_ · - Suppose, for example, that the four Aces appear Oil

· tho fir$t round that ia dealt from a fresh, thoroughly shnftfecl_ deck. Attt:t that IOUDd is over, the cards are placed f&ceup _oa the ldtom of _the deck an4 th6 second round- ia ~ frOm the remaining unused card& Now on the second. rowirf_ 110 Aces can appear; there will be no blackjacks, no soft• · -IWlds, 8ncl no 11~na .:of Aces (splitting Aces is- highly favorable to~_p19er)o; This situation of having no AceS in ptay·(which Is. On the average. almoSt 3 per cent agaiQs( ,tlie ·player as we shall see tater) continues. in succeedidg rounds -until the deck Is reshuftled and the Aces are broupt back into play. . .

. A few years ago one cashio made a practice of rem.ov-•we 8ssumo ~dice and a~ .. rou1etto wheeL For aa ~~ f1f attemp. fA> bell bialed rouloUe wbeeJs. ..

•. - -~ ...... . ~· ~ .

~"··. """"~ "~-""''""_," __ .,,. .. v~---- ~---.--~- -- . ." .. .. . . -c"ci:;i.'ivtnnlng StrtJUg • 45

t :44 . BE A~ THE DEALE It -~ ~ that are p~ Suppose }'011 ate playing du\~-~ .lbgfout'f~daNme.fromthcdec~ Fromour_~Ja.' :: --~~~-d;this meaD~ tbat )'O'i·are tho drdl:'-~ a(tbo , . , tions, we,~w this added 2.5 per cent to their adva.atage. ·--.~ SUppose· also that you. have bear_-~1 C8nW1 ~ . This ~~was spotted by theNevada Gaming Control _ ; -,ttack of~ cards played and you know t1Jat'ihe. unp1ayed

Board aJJd.1Jie casino was brought to triaL Bveutually the '· 'cards, from which tho next round wiJI be ~"'ODSfst pro­-casinO's 1iCease was .-evoked. However, there was one ironic - dsety of two Sevens ancf four Eights. • Hovr much should

.. sidelight tG the . trial. The casino operatOrs were practical · -JOll bet? Answer: Placet tho maximum bet the casino ~-

,--

i J

men through and through and not at all theoreticians. They · -:anow. liven borrow money if you have to, for you are l:new that their short deck helped them. but. they. did not ·- certain to win if you simply stand 911 the two· can:ls ~ know how much; Thus they had no answer for the damning -• ·will be dealt. · · asse~ of an expert witness that they were putting the ·., --. • · Here "is the analysis. If you stand on yottr, two- altds, -

. ptayer,.I_IOt at_ a 2.5 per cent disadvantage but at a 25 per · · ~p do not bust and are temporarily safe. When· the dealer · Cent disadvantage! - picks_up his hand, he finds either (7.7), (7,8), or (8,8).

· · Since his total is below 17, he must draw. If he holds (7,7), . . . • there are no Sevens left so he will draw an Eight and bust. The Use of Favorable Situations · If he holds (7,8) or (8,8) ho will bust if he dritws either

The winning strategies to ~ given in ~book depend ·. a. ScveD. or an Bight-the only -choices. Thus tho. dealer largely on the fact that, as the compos1ti0n of a deck : · -busts end you win. · -changes during play, the advantage iri blackjack will Shift · , · This brings us to the central problem that I had to back and-~ b~n player and casino. The advantage . ''IOl'fe':iri analyzing the game of blaekjack: How can a player

·· . ~ ~ beyQnd 10 per. cent for one IJide or tho other :ewtusto the depleted deck in general to deteiminO ~ and on occasions even reaches 100 per cent We watch ·it not it is favorable, and if it is favorablo. precisely how

_the cards tfiat are used up on the fitst round of play. The · 'tDucb ril?' This problem was · solved•• {)y. asking _tho . fact that thcs~ cards are now missing from the deck will, <IBM 704:; high-speed electronic computet • • seriet_. of .Jn p2csaft~sbift the house advantage up or down on the -···~ The first question was: SUppose blaCkjack . ~~"'that will be dealt on the secondrpund from -now ·fs-ptayed with a deck from which only the four _ACes ._depleted deck.. . . . are iemoved. What is the best possible strategy · for

. ~ successive rounds continue to_ be dealt ~ the ·Cho player tG follow and what is the bouse (or p~) increasmgly depleted, deck, and the advanta~ shifts back . · advantage? In other words, the computer was to do .

, ~ · fpith between player and house, a· make latge bets · · .exaetty the samo thing it bad done in finding the basic when the player has the advantage ~ very small bets . •'the esseDtlat thiDs fa tbat there be at least~ ~~~-lit .

· when the .~use ·has the advantage. The result is that. the : · 11t01t two SeveDa aetuany tJWJIIaltr. for p~tq. Por example, il tho -~ . payer usually wins a majority of hiS large favorable bets, :. tfoea I\Ot cleal tho last -c81d <• ~ practice), two Seftlll aDi1 tbzee

and although be generally ·lOSeS a majority cf his· small : ; ~~~!"to -::r: degree of~-~ anfawrable -... he bas .a considerable net ........ f;t. · · ·eqct recU:a1atioDs were iDade tarer bJ' Juliaa Braua of tho IBM c:or-

-. · · . · · . · . t"":.,.. · .~ '11irou~ thia mile4 eclidoa wo have uso4 1111 figvRa Ill

.· ~-:~ !~:f ;:ry~~edext;p~i!~~:: .<~of our~ ODelt .-.ver poaible. .

l j I {

1. \

. :.:~~---- BEAT· THE DB A X. E i "·c~:JflMing ~ 4'7. ...., -~8)' promptly should plan to use the point-eouut ~ .ttategy, -with. one.diffei.ence. It had to solve-~ prob~ tV'Of the next chapter as their first winning approac4 ~ with a de<:k fiom which ~ four ./v;es were DU$SU1g. : - - tbo ,,.me. ~t o1e.rs many adftatages owr tbe Five-cOunt · · ·· The resu.lt was noteworthy. Wh= playing with a deck , ·; 1trareg with ODly a moderato increase in the level of diftl. t.bat has four Aces missing. the player is at a disadvantage i c:alty. These leaden pmbabJy should not spend a great of 242 per cent, unJ,ier best play •. It may seem that the .. ~- deal of time .practlclng the Five-count strategy. Howevere removal of the four Acer. should affect matters much more · iinco the various dJscusaloDS in the remainder of tbiJ cllapo ~-the remov'al.ofany·other four cards, since Aces play tee ate important to the Jacer strategies, it should still be such a unique role in the game. They are essential for a th0iougblytea4 and UDderstood. by those who are PI Oil natura111,nd ·for soft. bands, and they mak¢. the .most faV9r· ~ them~ powerrw strategies. · able pair. Wherever they appear, they seem to help the

_player. Thus some players may $UpPose that fluctuations - A First Winning St'l'alegy: Counllizg Fives in _the proportion. of Aces in the deck would have a muc~ Table 4-1 shows that when four cards of one kind ate greater effect on things than fluctuations in the propprtion iemoved from the deck. the greatest shift in the relative' of any _of the other cards and that we ought simply to advantager. of player and house is caused by ,mnovfng the . study Aces alone •. However, we will see that Aces alone four Fives· from the cteck. The e1fect is even greater .tbaa aro ~ot overwhelmingly important. . _ when the four Aces are removed. More important,.~

· 'I'he computer was now asked to compute the p~.s 1ng ·the Fives gives an advantage of 3;6 per cent ~ tho adv. anta.ge or disadvantage, using the best -s~tetrr~en player. . playjng With decks from which were removed m --~ "~ · , . · __ Now, suppose that the depleted deckcontaiDs no~ Twos. four Threes. etc. The reslllts for the$e: qj(i~· -- IJ.ut· does contain enoup cards for - next round of p]q. o~ speclat decks are listed in Table 4.1.·The COli·--·-~_\ aD4 :bat therefore no Fives will appear during tho neXt ing ~t strategies •were computed but have been_ , ~ It caa be abowD that theso ~tuatioDI may ~ c:Oa--to sa\re SJ*e. . ,"~-- ~. lille.red as mathanatically identical with those that arise ·_· T~ble~.I suggests that a $ortage of--,_ · .-handsaredealtfromadectwhichiscompleto~

. iaJues 2 tlu"ough . 8 might give the player ~- ~ _·, ·_ · Cbat the four F~WS have been remoftd. Without ~ - .wbile a relative excess of such cards might hurt the' ~· · tO gms the cfetaiJed cxplaDatioD. for_~ we simply point

Qmversely, a shortage of Nmes, lfent, and Aces ough~ to :JUt that this meaus that if the player knows that no F~ hurt.tbe play~. while an excess of~ _should help him. caa appear on tho _am round of play. and.if he tileD fof A variety of winning strategies may be based on counting lows what we shall can the "FJve..count'" strategy, on that .OJIC or ~ore type!! of cards. A good, simple: winning ltra&- ~of play be Will enjoy the 3.6.per cent advantage~ ~gy is based on counting the Fives. It. will' be described .ill IIJiveli in Table 4-L · clotal1 in the rest of this chap~. The readers who find the '=:'. ''Tia8 P'~ ~ is given in Table 4-2._ TlJo baaic strategy in Chapter 3 di1Ucult should plan to-~ · ,..,._ tstt.aiofTable 3·S. · _ .. · ~ Five-count strategy as thc;ir &st winning approach ~ · ~.i';;<O'bserve that the Fi~t strategy is Very similar to l).c ~e. . . , . · , thOiirlsio fWl-deck l&rateg, whieh eases the burdoll on ODe'&

· ·On the other· tumd, readers "'bo kamed the -~- · :-::.. -:· ·

r· ~---··--~~--

i 48 BEAT THE D.EALEJi. i: _: II D ~ memory. In particular; note that the soft standing ~'bets li :; i

~ are the ~e, that all the· basic doubling.:ctown 1ituati0ns ~~ it also call for doubling down when the Fives are gone, ana .t;O

·' tbat the same statement is tiue for pair splitting except that '!!'§

~~ .

I fl 65 I' ~ a pair of Sixes is not spllt against a dealer's up card of !J

Seven. It I Jl . If -~ f ll Ut ii TABLB 4.1. Pltrytr'z Advantage or Dlstltlv11111age lor Certain ,,

Special Decks. !I l: 10 ~ii ~ Description Advantage (in per cent) II '2~

0~ deck With best strategy l ~ t It ~~;

~ c:ompleto 0.13 ~· . - "11 Q(l)=O -2.42 '• 9 Q(2)=0 1.75 Q(3)=0 2.14 *I Q(4)=0 2.64 la Q(S)=O 3.S8 · .h Q(6)=0 2.40' u Q(7)=0 2.0S Q(S)=O 0.43 ma

. Q<9)5oe -o.41 ! .... ! lfl!lf Q(lO}=O 1.62 '"! ~ cleck o.ss [093] j two.clecb -o.lS • ri~.

four' dec:ks -oAt I ·h ~& ! '· sooo dec:ks -o.ss

' l t f!lfj' i Q(10)=4 -2.14* 1 1Hfi~ 1 . -- . Q(l0)==8 -3.13- Ia 'Q(lt>)=l2. -i.as 10 10 ~~.:J!l [ Q(10)=20' l~89D.21l Q(10)'=24 3.51£4:241 ~~u • Q(I0)=28 5A>6• [6.10*] J~8iUf Q(10)=32 6.48• [7.7S*l lj'il~ 11 1 Q(10}=36 7.66{9.11) l!l,;l! .. Q(9}=Q(10)=0 9.92* a l'la Q(8)-Q(9}=Q(10)=0 19.98•--'

~I Q(S)= ••• :;:QOO)=O 78.14 ~I t -Key: Q(X) = Y means that a panicular deck - altered by cbaDJing 0111}1

!! die quantity Q or carda that haft 1111medcal 'IIIIUe X so that there are now Y IUdl cards. For aample. Q(2) =3 woulcl mean that in the deck there are a111J

~; ~ Twos insteacl or the ~ toar. "'TWo cleeb" - the cana. aro deal& bolia ,_ ordinary 5l-<:arcl decb 11tat laaft IIcea miXed IOielber as oae. Tlae ~

r! wbh Insurance Is O.ll per cent J11U1« for Q(l) :::0 to Q(9) =0• 1'be player lasures only If neither or Ills bole canll Is Ten. For· Q(l0) E; » the adYan&alt .

1i ,. wbh Insurance folknls Ia squua ltrllcbll. ~ '- Wllill'·QUO) ii;Z. ' ii • Apjll<llllmate. .. . . ~ .. f-1, ·~ l;

"·--

,,, ,;>.

~;,:i~. ;\\'·.

.• BEAT THE DBA1.1Ul

· As a matter of fact. when all the Fives al'C gone it is ped~tly acceptable to use ~nly the .. chaD~~ in the ~ standing -numbers llDd otherwJSC play ~ccording to the bwc strategy. The errors thus in~uced p~arily invo~e·

-neglecting to split pairs o~ double down m several _m­stances. Their effect is quite small. The player's a~vantag~ is decreased from 3.6 to 3·4 per cent. I suggest this to de­crease the load on yo}lr memory. We shalt adopt .this "'silnplified Fives strategy" in all our calcula,tions and dis­tussions of the Fives m~od.

We now outline a simple method for winning at _casino blackjack. Begin by making "small" bets and using the standard strategy. Watch the cards that are played and keep track.of the Fives. When you see that all four of them liave been used, check to see that the next round of play . will come entirely from the remainder of the deck, and thus that no Fives will appear.

Now, you must place your bet before any of the cards lre dealt in this next round. However, you know that you ha\re better than a 3 per cent advantage on wl1atever you · bet. Therefore. place ·a bet 'that is "Jar~· ·in comparison. to the ones you have been placing. When the cards uo d~t, employ the simpruied Fives strategy. .

We have been recoinmending tbat -if the FIVes are used up before a certain round is dealt, the pla~r should make a large bet and usc: the simplified Fives &triite8f. ~. it may happen . that some Fives remain w~en a round of play begins and·that all of thtm appear ~unng that round. At the mstant tbis happens the player should change to the simplified Fives strategy. For example, s~pPo.e he is dealt llllrd. 7 and the dealer shows a Two. Suppose that when the player draws he receives the last remaining Five. He now has hard-I2. Tho basic strategy says to draw. How­ever, the ·Fives strategy Jl()W applies, and according to it be sbPUld stand. ., . __

. This 'is to be considered a refinement and Ja DOt es-

A . Winning Stmtegy S:t . ·Seadal. for .Wbmbig wl1h tbe Fives strategy. It improve~ tho player's dJances ohrioning some of his 8111all bets, uam.ely., some of those placed at tbe beginning of the round oa which the last FJVeS appeared. .

Suppose that you continuo over _many deals to placo large bets when Q(S) = o, and small bets otherwise. Ia those situitiona in which you made largo bets. J011 w1n ill

· the long run at a rate of above 3 per ceDL With your aaia1l . bets,-you Joso ill a rate of about -o.2 per cent. • If.tllo largo bets are big enough c:om.pared to the small bets and if tho favorable situations occur often enough, the profits from the big bets should both otfaet .the losses from the small bot& and leave a comfortab1o overall'profit. .

There are several questions that we must now ~ In detail in order to make our instructions complete. ..

(1) How can you tell whether the remaininJ.CIIda wiJl·be adequate for the next round Of play? · (2) How often do favorable situations arise? · ~

· (3) How much larger than the small beta shoWd tho. large bets be? . ' . . · (4) How fast will you make money?

(S) How much risk is there? (6) How much capital is required to start? ~

We will take these up in the order listed.

Counting the Cart/a

The check as to whether the remaining cards aro acfeo. quate can be made in several ways. The SURSt ~ s ~ actUally coUnt how many cards have been used ·lll_paJ-

• Oue migftt 'WOJlder why, fD tbo Rvea strategy. die small ..,.... 110& fDstca4 win at'tfae rate of o.JO pw CCDt. the ... 'Wia.rafe-. .. ~·bide atrategy. The ·ftiiiOII .. tllat ..u -- .. ::.::a--= ===-==-!!r~~-...... ~-,:~~~-=-.. ~par-;:.~= ... ~_-·. of pta,_. at dill_ ·tate. W. telectol{a liDj1e milalllr • ' ... --~. w. .. it pesaimt.tic.

- \:·

.~ BBAT, THE DBA~ER

For. example, after each round, you might ~y to yourself something like "Eleven cards have been played, and I have seen'~ Five ... Count every card that is used as. "played" but only count the Fives that you have seen. For example, if a card iS burned, be ~ure to count'it whether or not you see what it is. It is not necessary that you see every card that is used in play. If, however, you miss seeing any of the played cards, some of them may be Fives, in which case you will miss some favorable siwations. For example, sup­pose after a certain round you see that seventeen cards have been played and that three Fives have been used. Suppose also that a Five has been burned and that you do ·~ know this. nen, as far as you know, one Five may yet -aJ>Pear, so you will make a small bet and miss the oppOrtunity of exploiting a favorable situation. _ - If· your dealer habitually conceals the burned card, ~ may. wish to request that lle show it to you. It is acimetimes difficUlt to know whether to make this request. It shOuld not be made jf you think that it will arouse the asino's suspicions that you are playing one of our win­aiDs, strategies, for they may_ take countermeasures that ...0 01ore costly to you than not seeing· the burned card.

If the casino does not use the last card, _incorporate thiJ into your count from the beginning. The reason.for this ia that the particular count, when subtracted fr<>m 52, is IUpposed to give the number of cards yet to be played. Tab.!e 4-3 is a rough guide as to when the remaining cards

• will fail to be adequate for the next round. . · Counting the used cards has other advantages· besides

Jelling the player whether the unused cards are. adequate for the next round. FirSt, the training in card counting is pMparation for the more powerful, and also more difticu1t winning systems to be presented in succeeding chap~ •. Secon4, the count is an inValuablo asset in the detection of chea~g. because a common device is to ·remove one. or more caros from the deck. (One might wonder at tldJ pojnt

d Winning Strategy TABLE 4.3.

53 ,,.,. t1ie Dd Prohlb JVIli • ~tktpulte tor • Fall Rot.~.

of Pltzy. JC«ordbtg to • Coltlll of UMd CGnlr '

Remaining carcJa Ula8Jly Number of adequate if caaat of mecl

player~ cards fa DO more thaD: . l ~

I 41 s 38

" 34 5 31 6 2.1 1 24

whether casinos have also tried adding Cilrds to ~ dect. When two or more ~ are being used, this can be doDo easily. I have only seen it done once when one deCk wil in use. That was risky. Imagine the shock and fury of a player who picks up his hand and sees that, not onlY an both his cards Fives, but they are also both S~) ·

Another well-known method of cheating which· can ·often be detected by a card counter iS ca1Jed the' -mm<mll"." Though the name is apt. the experienc:O is not. a cfes1a;t treat. In the weak form of tho turnover, the-dealer Wlltche8 to see whether the first half of the deck seems to fMor tha .house strongly. If. it does not, he continues nonllaJl7• lii:Jp-­ing that the latter half will. However. if tho deck'dooslnor · the house during the first half, he secretly turns ·the do4 o\rer SO· tlult the used cards are on top and are lepJayed during the secand half. In the strong form of the mmover. · the dealer stacks the used cards from tho first 'half of tbo· deck -as he picks them. up after' play. When tho decl: ft_ about balf-used, he turns it over and deals out stacbd haJKJst . · · ... The unwary player generally doeS not remember w11fc1l

cards he has just seen. Howo~ if the us_ed portioa · c:oa­taill$ a number other dian 2lJ ·cards, the total deck will .. toone who counts eatds, to contain twice tho DUJDblr

54 BE AT .THE DEALER· .

in the used portion instead of 52 Cards. Further, even .if. the_ used portion contains 26 cards, 'the. fra~!d. may be de-_ tected unless it also contains two Fives. For the number of Fives in the total deck also seems to be, to. one who is counting them, tWice the num~. in the used portion.

For readers who do JlOt want to count the used cards, there is. a less satisfactory method for determining about bow many cards remain to be played. It can be used if the dealer checks to see bow near the end of the deck he is. He does this by pUshing the lower car~ slightly forward so that the upper edges of all the cards show slightly. Then the J,JSed cards, which are face up, appear ''whiter" than the unu!fe(l cardS-provided that the unused cards, . which are .face down, do not have borders that make their edges white also. Th8 relative thickness of the two portions makes it easy to estimate the number of unused cards remaining.

If you have a deck without borders, place a portion of tlle deck face up· underneath the remainder. Then skew the deck ·by pushing the bottom cards forward slightly. There should be a clear line of demarcation between the tw(). portions. From this you can estimate the amounts in · each portion. With a tittle practice you can become quite skillful. If you attempt the same thing with a deck having borders it is harder, since the clear line of demarcation usually does not appear. .

1 Here is a warm-up experiment that can be done with any deck; it should convince you that estimating the number of ·cardS in a portion of the deck is not so difficult. F'J.rst, square ibe deck by striking its edge against a smooth table tep~ N;ow try to ,break the deck into two equal portions .. If neeessary, transfer cards from one portion to the other until they appear equal. Do not place the two stacks side by side

, . . on the table top and match their height. That would~ the purpose of tbis experiment, which·· is to introduce you to estimating by_ eye alone. After a few attempts you ~11 find that rarely, if ever, are you "off" more than tvlo carck •

A Wlnnlng Stiatttgy 5~ ,­Many peop1e soon learn to divide the deck iJlto· ~ pte-_· cisely equal parts almost every time. .. . . . .

An Imprpvement in the Fives Method Suppose you keep track not only of the number of

Fives remaining but also of the total number of remaining unseen cards. Then you can estimate whether or not the . deck is Five-rich or Five-poor. One way is to diVide the number U of unseen cards by the number F of unseen Fives. Normally U/F = 13. When U/F is greater than 13: the deck is Five-poor. (In the extreme case where F = o, .

· that is, when the deck bas no Fives, U IF doesn't make sense~ But you already know what to do filen.) - ·

The larger U fF, the larger your edge. When U /F is ·. 26, for exampl~ the edge ·js about 1.9 per cent (balfw~y from 0.13 per-cent to 3.,8 per cent. You should bet 2 ~ 3>- -; units. -

When UIF is less than 13, the deck is FIVe-rich. The casino bas the advantage and you should make SDUl11 bets. __

The advantage of using U IF is that you find and ex-.:. · ploit many additional favorable situat!:on5. The method works without change against any number of decks. ·

Frequency of Favorable Situations The rate. at which money is won depends upon hoW ··

often favorable situations arise and is influenced by how many players are at the table. This dependence !II shoWn in Table 4-4- ·

It clearly strengthens the player's advantage, when be. -is Using the Fives strategy, to play iJl·games iJl which there · are no more than five players.

Y tll'.ii:ldotv in Bet Size · ·1fte institictive answer to the questi~ "How mndi ·

tarp,rlt.n the small bets should the· large bet be?" is "As JaJp·aa poSsible." for· it _is the 1arge favor~ bets that ·

•, 56 BBAT.TllB D'BAt.Blt

. are responsible for the profit. However ·lh~ are some cir­CQmStances that· need to .be considered.

TABLE 4.4. Variation ln"the Number.of K110wn Favorable Situations, When Fives Only Are Counted, u a

Function of the Number of Players. ·

Number of players

1 2 3 4

. ' 6 7

Approximate number of times ·Fives are gone

per hundred hands

9.8 5.9 6.5 3.5 6.0 0.9 1.7

Average amount iD .. ta:rge units, won

per hundred hands

0.33 0.20 0.22 0.12 0.20 O.o3 0.06

If a player goes along steadily betting $ r and then suddenly, every once in :a while bets S,soo, he may soon be the object of study by the casino operatoG. If be is win­ning, they are very likely to take countermeasures. One· simple and effective method is to shuftle the deck after the player bas made his large bet and before any cards are dealt. Although the player can then remove his large bet, his favorable situation is destroyed.

·.Thus it·seems judicious 'to reduce the size of the bet. variation to a level that does not attract so much attention. The first edition of Beat the Dealer has made the casinos very cautious. Consequently Jbe large bets should be no more than three or four times the size of the smaller bets. Let us run through a simple calculation to see haw costly this is. ·

Suppose we are playing 100 han98 per hour head on with the uealer. Then, aceording to Table 4.4, there are · about 9.8 favorable sitUations at 34. per cent in favor ~. !he player and about 90.2 unfavorable ones at 0~2 pn- cent_ ucfavor of the house. If we are betting $1 and $500, we

4 . Winning Strategy 57 I~ o.oo2 x 90 x $x, or xu,. 011 the uofalozablo situatioua. and gaiD. o.o.34 x io x Ssoo, or $170. 011 the favorable situations, for a net profit of $16g.82. If iDstead we bet $125 in the Ullfavorable lituationli, we would lose 0.002

. x. 90 X $1~5 or $22.50 in theae instances bUt would again wm $170 m the favorable situations for a net pro& of $147-50. · It should be emphasized that these profit 1ig1m:s are rough avc:rage ~ounts for ~,very large J1UJDbe.r of .bands. In-any brief series of a few hundred hands, there will JikeJy be considerable deviations from these figmea. WZJ

We can now use these figUres to estimate the awnge hourly wage for the F.wes system. ~ that we. are playing ~ead on at the rate of xoo handS per hour. We saw preVIOusly that we average about $140 per hour when our bets range from $125 to Ssoo. 'I'herefcmJ wellioal(t make Ss.6o. per hoUr when we bet from $5 to $2o;. 1be ··"' player who only bets from so# to $2 will make a modeat s6f per hour. .

One sldlled player whom I know asserts that he caa#~ • · play 3SO hands per hour W,hen playing head on. Betting. •

from $x to $5oo, he wo1lld ave.tage $170 x 3·5 or about Ssgs per hour. It is. in the sy.stem player's beat iDfaest 10 be able !'> play rapidly; When more players ate ..,._.. the tnu:tion .of hands that. are favorable dwindles. Futthet­more, .BUlCe -1t takes longer to play out a 1'0UJld. each pJaJer getS kwer hands per hour. . ·

Cap~l Required, Extent of Risk, Rate of Profit We will now answer the questions:

, How mttCh capital is required to start? .llqw much risk is there? . ' .What is the average rate of profit?

. .• Yust you must 4ccicto. how much your iDitial capital Will.~ You must NEVER.Ni'VBa play with~ that it .. !lillhurt you to lose. Besides the usu.l.~ ~

58 BEAT THE DEALER

this, there is one more: playing with money that you can .. not afford to lose produces psychological disintegration, bad play, and cOnsequently a greater chance of defeat. Conversely, playing with money that means little to you lead~ to cool confidence and devastatingly accurate play.

Now you have cut your stake to a sensible level. Next you must decide how small to make the chance of ever losing your entire stake. For any stake, the player has many

I! choices. If he plays boldly, taking a large.chance of being I

l T t i

l

ruined, . he can make a comparatively large amount per hour. Instead the player may divide his stake into so many units that there is virtually no chance to "lose it all. But · the price paid for this is a considerable. reduction in aver .. age profit. .

As an indication of what to expect, suppose that smal~ .bets are 1 "unit," where a "unit'" is an amount determined by y(,u, the player. Suppose that big bets are 3 "units."'

1 Then with a stake of 150 units, the chances are less than 1\... :I £1 in. 10 that you will ever lose your stake. They are more 1- ~"(.!laD 9 in 10 that your capital will grow indefinitely. If . • jour stake is only 75 units, the chances are about 3 in 10

of eventually losing it and 7 in Io of its continuing to grow · as long as you care to play on.

Before we .get on with learning practical winning strategies, we take time out in the next chapter to describe

' the original casino test of the first of these strategies, the , j Jl()W famous Ten-count method. ;: •I

My Ideas Are Tested

in Nevada

I thought that the strategy based on counting Fives ~-. make an interesting paper at an upcoming Annual M~g: of the American Mathematical Society in Washington, D.C. I planned to fty down from the Massachusetts IDstitUte of Techllology, where I was then teaching and where I made my blackjack computer calcUlations. A few days before tho meeting. the society, as is customary, published abstracts of the two hundred or so talks that were to be givea. In-· eluded was my abstract describing the Fives strategy, .. For­tune's Formula: A Wmning Strategy for Blackjack" [67].

Two evenings before I left for the meeting. I was sur­prised by a call from Dick Stewart, of the Boston Gloh

· inquiring about the abstract. The paper sent a photographer­out to take my picture; meanwhile I explained the basic ideas of my system to Mr. Stewart over the telephone.

Tile next morning I was amazed to see a picture of myself with a story on the front page of the BoSUN& GloH

• st•

-,

~~: 60 BE.AT THE _DEAJ.E~;-[1. [4]. Within hours, the story and more pictures were released [' by the news services to their thousands of subscribing news-,, papers across the ·country (6. 27, 39, 43. 57. 7.8]. 1' . Following my paper in Washington, I wu. forced to \: give. a press conference. After this ·I was· televised by a

'7! major network and interviewed on a number of radio pro-j\ grams. When I returned to my office at the Massachusetts il Institute of Technology, my desk was heaped with mail h and phone messages. spurred by the continuing publicity !i . too. 6sJ. . . ·. · - During the next weeks hundreds of letters and long

distance~ phol)e calls rained in, the bulk of which were ri> ~- for copies of my paper and any . further available information. Interspersed amQng this correspondence were·

l a ~blc number of offers to back me in a casino test \1 of my system. ThO amoUnts proftered ranged from a few ~ thousand dollars to as much as $100,000! Together they l totaDed a quarter of a million dollars. i ' - CarcfuJJy, I screened the offers. I reject«:~. an off~ if lf.e ~or persons putting up the money could·not prove ! that they could dcml to Jose tbeir total investment. The rea·

11 sonfs, of coune, that there is some aman risk of a very bad 11 streak even with wiJming Slrategies. as we discussed earlier. !i . I was also worried about the poss1"bilities of be~ cheated. 1

· Since the $100,000 was the most attractive, I con-aldered itfirst. It was offered jointly by two New Yu-kmulti-

n millionain:s, whom I sbaU tefcr. to as Mr. X and lvfr. Y. \1 TheY are both large-scale gamblers. Mr. Yonce lost $Ioo,· ~ ooo in one of the casino games without being ~ly hurt ·r finandally. Mr. X's gambling actiyities involve hundreds of i! thousands and even millions of dollarS in profits; he has ·jl been famous for years in gambling ckc1es from M"aami to •' Las VegaS. I later 1eamed both that he was famUiar with

:;: thc'exptoits of "thO little' dark-haired guy" (Chapter 3) and \: that he had ~ade large sums himself at blackjack. Th~ he ·1; was thoroughly "sold" in ad~

-- ---~--~. ---~--·-·-----.---- ~··,··--···--·---~

My ltktu A.re Tuted iia Nevada · . 61 , l . . . ' l.

Prtparation8 . ' When I told Mr. X of my interest, he drove up frOm

New York One Sunday. He showed mo enough of his pJ&<> tical gambling knowledge and c:ard skill to persuade me that he could qu~ckly detect cheating. At tile invitation and ex­pense of"Mr. X and Mr. Y, I lew from Boston to N~ York several times to discuss the system and to plan a trip to NevadL

As the reader may have already learned from practiCe at home or in the casinos, the FIVes system wins "too slowly" because the good situations that it locates are fairly rare. Fortunately I was already working on a far more powerfUl system when I announced the Fives system. This was the

. ten-count method, described in detail in Chapter 8. It was this method that I piatmecl to use in the casino test. To mate the story easier to follow, I describe this method briefly.

The player keeps track of two kinds of cards, TeDI and non-Tens or "othen." 'Ibele are 16 Tens in each deck and 36 others. The player's edge is measlited by tho ratb' of others to Tens. For one fun deck, it is 36/16 or 2.2$. When. the ratio Js betow 2.25, the player has the edge.. Wlleft it is abOve, the casino has the edge. Roughly speak­ing, the farther the ratio gets below or above 2.25, tho gRAter is the elfect. .

The player has the edge half the time and his edge ranges up to IO. or IS ~ cent. The casino edge only ranges up to about 3 per cent! ·

There .were two main approache$ that we c:ouJd adopt for betting. ~ which I shall term "wild, "involve$ betting the. casino limit whenever the advantage to the player ex­ceat.· some sman figure, say 1 per eent. This method ~ dnCes, on tho average. the greatest gain in the shortest ~ However in a short ron of a few days the fluctuations in the· player's tc)tal capital· generally are violent. and a large . baDkroJ1 is required. Mr. X and Mr. Y said that they would

62 BBAT THB DBAJ.Blt " Myldeos A.r~ Tested in Nevada 63

back this approach to the extent of $Ioo,ooo and that the,. •·· lisluJieqt closed for three. hours beca~ of Good l1riday, would go farther if necessary. · · · · we returned to Reno. During the evemng we investigatecl ~

; .llUDlbel- of casinos to determine which had rules that were The $zo,ooo BankroU ... most ~voribJe. As the best spot for practicing, we selected

I was not in favor of the wild approach since there were ' a casmo. that dealt down to the last card and allowed the too many things I did not know about the gambling world. ; player_~ ~ouble down on any hand, split any pair, and m. l· also had no idea how I or my backers would rezct if I ; sure. This IS a more favorable set of rules thaJl is ordinarily. found. . . were to get behind, say, Sso,ooo. Furthermore, the purpose of the trip from my point of view was'· to test my system After a sumptuous dinner and a rest, I returned alono. rather. than to m~e big money for Mr. X and Mr. Y. So to the casino we had chosen. It was then about 10 P.M. Mr. I preferred being certain of a moderate win, rather than at• ·X did not accompany me because he is·wen known·to that lemJ>ting · a probable. but sotttewhat uncertain. big win. I casino's owner and we did not wish to attract attention. I therefore favored another approach, which I .shall call"~ began by alternately playing for fifteen or twemy minutes at servative" play. 'Ibis involves betting twice the ~aunt of a time and then resting for a few niinutes. Whenever.I ·the minimum bet when the advantage is ·x per c~t,.four . would sit down again I would always choose the table with times tho minimum wheri the advantage is 2 per cent. and the fewest players. My· behavior pattern-1 pauSed for finaDy leveling off at ten tim~ the minimum when the ad- thought and stated at all the cards played-made it appar-: vantage is s per cent or more in the player's favor: I deter- ent that I was using some "system." But systelli players aro mined that if my bets would range.from Sso to Ssoo.(~e •. frequent. if not common. in the casinos.IQ. fact, they &Je

· highest casino maximum generally available), then $6.opo welcoDie as long as they are losing, and gradually I feD .· or:$r;ooo would probably be adequate ~pital. To be safe, further behind until. by S A.M., I \vas down $100. · we took along $xo.ooo-a hundred one-hundred-dollar At this time, business fell off sharply and I was finally bills. . able to get a table completely to myself. My new dealer was

. , When theM. I. T. one-week spring recess came, Mr. X : 'ParticularlY unfriendly. When I asked to be dealt two~ ud I flew on_ a Thursday evening to Reno, where Mr. Y was . .she refused. saying that it was house policy that I must bet to join us later. We checked into one of the large Reno ' $2 per hand to play two hands. Since this change in the hotels at about 2 A.M. and immediately went to sleep. Early 1 • acale of bettinJ would confuse my records of the evening's . the next morning we began investigating casinos. . play. I refUsed. Besides, I was getting ~ and irritable. ·

. I pointed out to this dealer that at least eight ~ Thtt Warm-up dealers had let me play two hands without complaiut ad

Out. plan, titsisted upon by me, was to proceed with tllerefore it could hardly be a house policy. She said that the caution. We would -start "small," betting $I to $10, and reason was to keq» other players from being crowded out. would gradually increase the amount of the bets as I gained remarked dtat there were no other players at my table, experience. Eventually we planned to bet Sso to·Ss~ ber.reason did n~t seem.to apply. She became angry'at

Fm;t we drove to a casino outside of'tOWD. m·.an hour·- .and dealt anap1clly as she could. . or s0 of play I won i.few dollars, and theD when the estab- , :~::\'A few hands later, the ratio of others/Tens dropped to

f4 DBAT THB Dlh\'Lif My ldeasAre Tested In Nevada .65 2.0, a I per cent advantage for me. Being 'thoroughtY- ~- • -Mdl ·again visited the casino outSide town. Within minutes. . aoyed by. now, I broke my self-imposed discipline. I ado.~ playing the' $1o to $100 scalo, I won $200 or $30o. vanced to ~e $2 to $20 scale and ·bet $4- I won and the ,Mf', X joined me and we played for a couple of hours. Wo ratio advanced to 1.7, a 2 per cent advantage. I let my ~ .accumulated $650, aild the house began to shuft'le tbe deck ride and won again. The ratio obligingly diopped to 1.5, .a· .~eral cards before the end. Since the favorable situations 4 per ce11t advantage. I let my $16 ride and won again. I arise with greatest frequency at the end of the deck, shuf- · Jeft-$20 of this $32 on the table with the remark that it was :ling up can sharply reduce the rate of profit. Because we time for me to take a small profit. The ratio fluctuated boo ' cwere only practicing,. it seemed discreet for us to leave now tween 1.4 and ~.o and I continued to make $20 bets. By and hope that we could come back later fora few fuU-scale the time we came to the. end of th~ deck, I had recoupect llourS. _ my $100 Joss and had a few dollars' profit besides. Mr. X and I were still expecting Mr. Y in Reno. On

As I picked up my winnings_and left, .J noticed im ~ Saturday evening Mr. Y arrived. After dinner Mr. Y and mixture of auger and awe on the dealer's face. It. was as :1 set out to seek our fortune. We first visited the famous . though iho had peeked for a brief moment through,a famil- Harold's Club, ari enormous building in the center of down­iat door i!tto a familiar room and, maybe, she had. glimpsed :town Reno. We began to play at the Ssoo-maximum tables. aometbhig ·strange and impossible. (The maximum. generally ranges from $100 to Ssoo in

' This ·~g session bro'!lght mixed blessfhgs. I wouid Nevada, varying from casino to casino and frequently from -agree my rash behavior in a few days, for the casino's opera- table to table within a glven casino. With our capital, we tom:toet apeciaf notice of me. On the other hand, my attenio preferted the highest maximum possible.) In fifteen minutes Cion was drawn to the doubling-up betting pattern that I had }1{e won Ssoo, warming up at a .$25 to $250 scale. Our used in the last few minutes; it c:onsisted of betting I uni~ ,dealer decided to alert the management of the casiao. She winning and letting the 2 units ride, winning and letting the pressed a concealed button with her foot. Within minutes 4 units ride. etc. This pattern of play resembles· the well· H~ld Smith ~r. and Jr •• arrived. They eJtchanged plea&­known doubUng-up system, or Small Martingale, which is antnes and politenesses With· us, but they made their point: widely used-~ almost every gambling game. The pattern J ~ deck ~~d be shuflled as often as necessary to prevent used above JS not senst'ble for those gambHng games ill ··PI from ~g. • . wh~ the house has the advantage; but in blackjack, with. . Most casmo owners had learned, over the last decade. the player's use of counting methods, it is as profitable as ~ some players would wait until very special combina­any other way of putting·down money at favorab~ timeS. tions of cards arose, near the end of the deck, and that Forthern\Ol'O, since the system is so. widely and so unsucc~ ·thea they would sharply up their bet, sometimes going fully practiced, it makes an excellent disguise for the count1 from $I to Ssoo. These players were stopped by shuflHng ing player. Also, the casual touch of leaving your • -t» deck five or ten cards from the end. untouched between hands seems nice. Therefore, to be safe Harold Smith, Sr., instructed. our

•"'ll'~' to shuflle no later than I2 to 1' cards from ~e end.

• 66 B BAT. THE D.BAL,Blt. ·My ltktu A.a Tested In Nevada

Fortunately for them. they waited to see the results. We were ·whether to quit if I lost this one, I picked up my hand and not .planning any ulterior moves; we conti~ to use the foUnd· a pair of Bights. They bad to 1» split. I tlung three

. same Tens strategy that we had used all evenlng. This strat· $too billdrom my wallet onto the second Bight. On one of eg locates favorable situations after the first hand.has been tho Bigllts I was dealt a Three. I had to double down so I played. even if only four cards have been dealt. tlung three more $xoo·bills onto this hand. There was, now

A few minor yet favorable situations appeared and · $900 lying on the tab~the largest bet I had yet made. were exploited by us. Thereupon the deek was shuf:lled 25 · The dealer was showing a Six up an~ turned out to cards from the end. Some favorable situations still arose. have a Ten under. He promptly busted. Now I was only F'mally. the de8ler began shufDing 42 cards from the end, $800 down. This deck continued to be favorable and the that is, after only two hands bad been played! This fencing next went favo~le after the first band. In a few minutes went on for twenty minutes or so, and in that time a com· I wiped out all my losses and went ahead $255. With this bination of bad luck. the club's unfavorable rules, and the ·burst of good fortune, Mr. Y and I decided to quit for tho sbuftling allowed us to squeeze out only an additional $So. _evening. . It seemed useless to continue playing at this casino, so we Again the Tens system had shown a feature that woul4 stopped. appear .repeatedly: moderately heavy losing strea.ta. JDixcd

·We then visited a casino in one of the .large hotels. with "lucky streaks" of the most dazZling briUiance; We had been told that they used a "cheat" dealer on "big- Theaext aftetnoon Mr. X, Mr. Y, and I visi.ted tbe:ea- · ·

·money" players. After being cheated on the very first hand, sino outside tow1i a~. BefOre sitting down to play, I in-an incident descn"bed in detail in the chapter on cheating, . made a phone call. WbeD I c;_aDle back my friends told me we m~ on. the casino bad barred us from play but that it would be

only too happy to pick up our meal tab. I called over tho Nine Hululred Doll({rs Bet on a-Single Hand· tloor manager and asked him what this was aU about. He

Ia the 'next casino the maxin~um was only $300, but explained, In a very~ and courteO~ manner, dlat tbe this .limit was compensated for by excellent rules: the p~ayer staff had seen me play:ing the aay before~ that tbq. were coukUnsure, split any pair, ~ double down on any set of very puzzled by my steady whining at, a rate that 'Was large. cards. Wi( purchased ·$2,000 in chips from the cashier and for my bet sizes. He said also that they could not fi.gore out selected a table at which there were no other players. I lost . what was going on but that they had 1ina1ly decided, ia tho steadily, and at the end of four hours of play I was almost light of their previousexperi~ that a card-counting sy&-$r, 700 behind. I was quite discouraged. However !followed tem was involved. My ~ was becoming hanl to the pattern of countless hapless players before :me (with, I detect. hope, ~reason) ~d decided to wait for the' deck to bC- Evidently they were discouraged when .they esthtWecl.· come favorable "just once more" so l could recoup some the power of the system that faced them, for ~ 1I09f lball-of my losses. ager said that the owner had deliberated at length befoie

In a few minutes the deck obliged, suddenly producing . · ~g to bar us. The casino, he said, had fearlessly played aiatioofothersjTensof 1.4,a S percent advantage, which 'c ·~all cam CO~ he reeled off a series of caUed for the maximum bet of $300. Curiously, my rettl&illlo"'.& .' --~--· rbat JJJeailt JJOthing to me-and bad beaten 1hem ing chips amounted to preciiely $300. As I tried to ·. · '-al\ VitJl cme exception. He described the oaly piaF that

- _-. ..

68 BBAT T~H£ DB~LBa

had been previously barred as "'a little dark-Juiired guy from Southern CalifDm~" We have- already mentioned this . individual and wiD say more about him and other fam.ous early players later on. .

We returned to our hotel, and while my friends took . CJU'C, of business for a couple of hours, I passed the time

away by betting Ss to Sso at the blackjack tables. Despite · the annoying presence of a shUl, I won about Ssso. At this

point, the pit boss asked me to stop playing at the hotel and to tell the same to Messrs. X and Y and any other :,friends I might have. He did say, however, that we could . enjoy iuilimited drinks on the house. Immediately I had a Moscow mule and then went to tell my friends that they Jlad been banned from this casino without their ever having

_-played .there. ~

It was almost suppertime Sunday when the three of us revisited the casino at which I had made the $900 bet. I was warmly remembered as the rich playboy Qf the night be­J,~ who had. been down $1,7cio before wriggling off the book by some quirk of fate. We were invited ·to dine, courtesy of .the bouse, .as a prelude to the evening's gaming festivities. After two $4 entrees of baked oysters on the half ~1 and yarious supporting dishes. ca~ with wine, I set out ~ewhat unsteadily for the gaming tables: I was truly· a lamb readied for the slaughter. Within a few minutes. how­ever; I was at peak ale~ess. After four bouts' of betting $25 to $300, I was ahead $2,000. Since I was beginning to tire,

. with the utmost reluctance I decided to return to my hotel. . I remember that casino fondly: the courtesy eel hos­

pitality, the spacious, attractive modem dining room with 'its .fine cuisine, and the.. casino with its juicy little clusters of .blackjack tables, the favorable rules. and last but not· leas~ the free money.* ·

· •niS casino had its revenge. Nine months tater lrevisitecUr. A lki'UM cheat separated me from $6oo in tOn minutes (at $25 pei ~ Wore I realized "times had changed. • · -

My ld«u .u.· Tutetl fn N.,.g;, ·159 ·1 t

The Twenty--Five-Dollar Minimum Game ~

My friends and I were again ready for. action·(~ I both in the customaryse~ andd )in the

1 mMathdema~~ l!

of the .SJlDl total of all bets·ma e ear y on ay w.~~ ~ We drove to the south end (Stateline) of Lake Tahoe. ~ About 6 P.M. we arrived at Harrah's large, brightly lighted I gambling factozy~ It was jammed. I was barely able to get a seat at the blackjack tables.

A few··minutes after I plated on the table the $2,000 worth of chips I had purchased from the cashier, a pit bOStl rushed over to invite me to dinner and the show. I in tum requested (with success) that my two friends be included. I began a game and within a few minutes-as I began to win-Mr. X joined me. In forty minutes, I won $1,300, and ·. Mr. X. who was betting wildly. won $2,000. Then we .took ,:, time out tOr our free dinner. which featured filet mignOn ana .· champagne. Within hours, destiny would present us ·With a bill for our "free" dinner. The charge? Eleven thousand dollars! , ~

· After. dinner we strolled across to Harvets Wagoia Wheel. There were both the $500 limit and acceptable ruleS. As usual, I purchased $2,ooo in chips from the cashier and selected the least b\isy _table. From the beginni~ I Waa plagued by $r bettors who came and went, generally slOw­Ing down the game, who concealed cards so that they were hard to count. and who created many other small annoy­ances •

Whenever a small bettor arrived at the table I pointedly reduced my minimum bet~ $50 to $1. After a few~ utes the pit boss "got tho message" and asked me if I wouic1 ~ a private. table. When I said it Would "transport Die with.f:l:i&tUy.",he explained that, in general. tlie club did ·not .like ihe psychological effect of a private table qn the other ~. However, with -a trace of a smile. be said that a ~ gam~ could ~ arranged, and wondered if

70 BBA'I' TH.B nEA:LBB.

that .would be satisfactory • .J promptly agreed, and a sig.n to that effect was installed. which cleared th~ table of aU cus­tomers_ but. me. A small crowd gathered: quietly to watch their·. somewhat plumpish fellow lamb go to the slaughter.

Seventeen Thousand Dollars in Two Hours After I had won a few hundred dollars, the pit boss

was amazed and delighted to see another ''well-heeled" laml;).wandet up and sit down at my table; it was none other than my friend Mr. X, who thereupon "jumped in" the game. 1 then took the responsibility, for both of us, for keep­ing the count and calling the signals. Within thirty minutes we had emptied the table's money tray-the blackjack ver· lion of "breaking the bank." The once smiling pit boss trem­bled with fear.

- Otl1Cr- e~ployees began to panic. One of our -dealers bleated to -bee boyfriend higher-up, "Oh, help me. Please. Help me." The pit boss was trying to explain away our win to a -nervous knot of subordinates. While the money_ tray wu being restocked, the crowd swelled. They began to chc:er. on their David again the casino Goliath. ,

One bystander blurted out rather loudly that he had seea 1,1S off to a roaring start in Reno two nights earlier and wondered if we had done there what. we were now doing he~ As the pit boss listened attentively, we quickly hushed up the bystander with tales of woe.

. . In two hours we broke the. bank again. The great heaps of chips in front of us included more than $I 7 ,ooo in profits. I bad won about $6,ooo and Mr. X, betting wil~y. had won $n,ooo. I was tiring rapidly. The aftereffects of our-huge dinner, the increased effort in managing two hands, and the S!I'ain of the last few ~ys were telling. I began to ~.it •very difficult to count properly and saw that Mr. X ~ ·ecpaU,y·· far gon~ I insisted that we quit, and I_~ in my $6,ooo. As I did so, I was startled to find three or four prqtty girls wandering baf:k and forth across m1 patlutnft.. ing a1fectiQDately. · ·

My Ideas An Test.etl"Jn.Nevada 1i ·•· After wending my Ulyssian way back t9 ~e ta~ -I·

watched,· horror-stricken. ·Mr. X. 1laving refused tO.'stop · playing, was pouring back thousands. In the forty--A"* min- · utes that it tOok to pel'Sllade him to leave, it cost:tfte two of us about $n,ooo of our $17,000. Even so, when we_ returned to our hotel that evening we were ahead $13,000 so far on ~e trip. · . .

On Tuesday we paid ·a series of \'isits't9. .. a downtowJI club that had bad rules and shuftled five to ten cards from , 1 the end. We ~dually but steadily lost about $2.ooo, ptar- · • ~ ing Sso to $soo. The player could dou?le down on 1:0 mid · II only, cou1d not insure, and the ~ler bit soft 17. As will . ·' be seen in the chapter on niles variations, the player is whittled down at an average .rate of slightly less than 1 per cent while awaiting favorable situations. Although these sit­uations dO 8rise, they are reduced somewhat in botJ:l ~ queney and favorability. Playirig Sso to Ssoo, the Teas strategy p~duces. perhaps Ssoo • .per h?ur• ·with favorable ru.les and about $400 per hour Wlth typical rules. t With the· unfav9f'&ble rules- just described, the strategy probably· pro- . ·.· duces about $250 per hour, and the risk of bad fluctuatiOns rises sharply:t . ·

The new and powerful po~t-count strategy works qutte· <

wen on these unfavorable casinOs. That strategy, presented . in Chapters 6 and ;, allows the player to couht aU the card&. It i$ no more difficult than the Tens strategy. ·

My friends and I recalled that the club in .which I ~ practi~ed so lengthily had excellent rules and made a prac-" · tice of dealing down to the last card in the deck. We d~ . to pay it a Tetu.(n visit. Mr. Y and I purchased $1,ooo-.m : chips and begatl to pl~y. We immediately begatt to wm, but within minutes the owner_ was on the scene. In a panic. he savo_ the dealer and the . pit boss instructions. . . . 1?illli · aa amazing perfotmance began. Whenever I··

72 B B AT . T H B'. DB A L BIt'

changed my bet size, the dealer sbufBed. Whentwer l varied the p\lmber of bands I tOok (by this time I could play from -one to eight hands at one time and faster than tho best dealers could deal), the dealer shulllecl. The dealer against wbom'I had played last

1in my practice session was standing­

_in the background (had she .. fingered" me?), saying over and over in reverent tones how much I had advanced in akill since the other night Finally I happened to seratch my nose and the dealer shuflied! Incredulous! I asked her wb.ethc:r she would shuftlo each time I scratched my nose. She said she would. A few more scratches convinced me she m.eant. _what :sho said. I asked whether any change in my behaVIOr pattern, -no matter how minute, would cause her to lhuftle. She said it would. . , I was now playing merely even with the house, • as the

ihuflling destroyed nearly aU tny advantage (except that pined from seeing the burned card). But by chance I ~ ahead about $300. I then asked for some larger­denomination chips-$ so or $1oo-as aU I bad were twen­ties. The owner stepped forward and said that the house ymuld not sell them to us. He then had a new deck brought m ~d carefully spread, first face down, then face up. Curious,l asked why they spread them face down. Although the· practice is a common one in the casinos, seldom do they examine the backs of the cards for a couple of minutes, as these people were doing. The dealer explained that it was believed that I had unusually acute vision (1 wear glasses) and could distinguish tiny blemishes on the backs of the Q8fds. This, they surmised, is what enabled me to foretell what~ were going to be dealt. lsco1fed, bitt the house stiJl panicky as my wins continued, brought in foui ..; tfecks in fiVe minutes. - - . ·

My Ideas Are Tested in ]llevada · 11

A.fter. disposing . of that particular .. h~. theory., I . p~ them to tell us What they thought about my "secret."_ The dealer claimed then that I could count every card as it was played. an4 that therefore I knew exactly which cards bad not yet been played at each and every instant. Now, i~ is well known ·to students of mnemotechny (the science of memory training) that one can readily learn to memorize in proper order part or aU of a deck of cards as it is· dealt out. ~owever, · l am familiar enough wi~. the ~ ~ volved b41 to know that the informatioP, when so memor­ized, cannot be used quickly enough for play in blackjack. So l challenged the dealer by rashly claiming that no one in the world could watch 3S cards dealt quickly off a pac~ ~ then tell me quickly how many of each kind of card re-mained. . . '

She answered by claiming that the pit boss next to her could do just tbiit.l told them I ·would pay. Ss on the.tpot for a demonstration. They both looked down.~ishly and­would not answer. I made my offer Sso. They remained 'silent and ashamed. Then my friend Mr. Y .increased the offer. to Ssoo. There was no response from these "'sports­men." We left in disgust. . _

At the next club that Mr. Y. and I visited, the bllick­jack tables were packed, so we inquir~· about a privatO game. A balding, effeminate man scampered· out and ~ nervous high tones told us that he knew What we were up

· to and they were on to ~ aQd "No, thank you ... Ano~ sportsman!

Since I had proved the system and the D}illionaires W busiDess elsewhere, we agreed to terminate ®.f. little .gam• bling experiment. In thirty man-hours of medium-~ large­scale play, we had built $1o,obo intq $21,000. At. JJC)I·

did we have to go into our original capital-more than · _ expenses). Our experiment was a su~ and

pertonned ~ practice just as the theory on \tJlic~ preaicted lt would. . . , ·

_....._. ·- hour to kill before leaving for the airpoit.

BEAT 'tHE DBALEll

we visited a friend of Mr. X at the Primadonna, a casino the friend operated. I was in favor of having a Jastbig round of play, but Mr. X did not want his friend "hurt." Rapidly bored by the conversation, I wandered to the blackjack tables. I found three silver dollars in my pocket, inflicted on me as change by the last local merchant I had patronized. I decided to dispose of the silver dollars at the table. Soon a great sequence of favorable situations came along, and in five J,Dinutes my $3 became $35. Mr. X's friend never knew that a word from Mr. X had saved him more than $1,000

in that few minut~. My trip to Nevada gives an ironic twist to the words of

a casino operator Vf}lo was being interviewed on a national television program. When he was asked whether the cus­tomers in Nevada ever walked away winners, he said,

· "'When a lamb goes to the slaughter, the lamb might kill the butcher. But we always bet on the butcher." ·

. The day of the lamb had come.

6

The Simple Point-Count

Sys~em ________ , ...

firs edition of Beat the Detlln' were Many readers of the t · · " int.;coWlt" sys­able to discover for themselves vanou~ .po the .fidt edition tems. • These systems were n?t present no: been completed;. because the needed calculations had both be~ The point-count systems Will be .:;u:fu~ in dealing with

:!:x~~:n=~;?;~~ = ~untermeasures a~ · successful system ,riayers. This c~e~~= ~e S:

version of the potnt count. You · the Puerto of how one point-count player won Sso,ooo ln cllan&C Rican casinos and single-handedlY forced them to

their rules.

The Simple Point Counl When the deck is poor in high cardS' ( IO,A ), Table 4-'1

indicates that the casino has the advantage. ., =::rd:Ck is poor in low cards (2.3.4.5·6>• Tab~ 4 '

1

. . . c • It is permissi'ble also to caB (,, J, ~ s. 6, 7) low anc1 (9, zo • .C)

, ld.ll_. ~- 'lbo_ ·results are pradkallY the same. ~~ •fl• .

f/;

t , .• 76 .Jl~,#.T THB J)JAJ,-

iudie~-that the player has the advantage.t 1'bi& ~­a JYS_tem in which the player semehow ~~. 'the deck has. an excess of :mga cards (good) or low- cards­~). and bets accordingly. The simplest method is to COI1Qt low cards + I as they are seen, or "fall," al1li .to ~uutlligh~rds- 1 as tlleyfaU. Sevens, Eights, and Nines • RQt counted. Cards which arc not seen are not-counted. W'Jlh one. full deck ,and typical rules, the c:ount statts at

, zero. . - - . For example, if cards from a full ~k are .seen ill.

the order 4,2,J,6,g,s.4r?.· the point values are A(- I), '2(+.1), J(+t), 6(+ I), 9(G-), j(+ I)o 4(+·I), 7(0} • . ,.. to&al-~ of points at any given time-is the.1111Dlber we ~· For ~pie, with the cards in the order descn'bed ~---~be point tottt& are: before any cards are dealt, o; ilfter Jf, -- 1; after 2. o; after 3 +I; after 6, + 2'; after g. +a_; _after s. + 3;..after 4. + 4; after 7· + 4-, . ··,To p/lq._ the .simple point count, all you do is keep · ~--·of.JM t<>tlll polnta. seen. Plily your cards using the · basic 41trl(egy. If iM point-count Jotal is :.ero or minus

· when you mau your bet, bet only I. unit. If it is pluS, bet ~-lflllR1.un/ts_ u .. point-count total.

· ·., 111 the· ... above, if it were tim,e to bet just after )'011 had seen -~ 6, you should have bet 2 units. If you ,. tA 1110r0.receat ~ ot ~·•,potbapl ~a

· JIOiakount a,}'lt.ca.forhimsilf Js giveala tbe San F..,.. Clqofticff Oi lune l.f. 196J, A ~story -card Seereta foro sate'" tellS .of a· ._..who • Hid lo-••lleeomO Jateidesl-·.ia blackjack "pUreey by =tr_h.._eJa Ia ){arch of •..v't~-By ac:urious ~ lhe U/eatory t!)at ~1 ~But tilt Df4lw ~In MatCh of 1964 ~'111tllliUl olfCNII w..a,stem 1or ~a~o. &O san PraDciicui tor s~~ Huakl ead:l ~ ID dlo..deck fa aSsfped oae of four \'lllueat·--IO. -.s• 5.:.. or_. 10 (.of~ lit~-· , -z. ~~. J, or 2 WOaJcl Jaau_~ u -WBJ. • 'tlie. fti\i"H*IIt of sucli point -values (ono_ of leYeral pGISible ,._:: .,.. ... otltQa &1ft.~ -the ~ re.ulta) can reiCIIly- 1re derived fMtQ Tllble "''· It Ia: .c. -1~ s fa' lo; ~ .3. 4· 6, • , ~ .s;•~a.•tor-s. bot o fa:tialtern 9.111ill~«J are ~·no.~ .a tbe · Polllf--~ ~~U ~ a& ... 'IDI'rafale citJ*_. · ... CIIllill ~ •. .-111-.-.-.!Jfdlc~.~-.. -lelde\t' m ... w ~-···nat·-:_ z. ~a ·ew_p··· "· __ •· poilina11101 b'tlluardi.JiSJIJ&tU . . fil Ta\u · " ! ~ c· .

·· l'Jij: Shnp-le Pbtni-Cotint System 'fl'

ha4-·Jpsl seen the s. ,W ~ld have. bet ~ Utdo.lf "* half just seen- the· 4 -or tbe 7, )'<)U should have bet .4 units. otherwise you shoul(J have bet I ultit. . . · . ·

Notice tba:t when aU four Fives are. gonct, ·the point• COUDt total is (on the average)·+ 4· The poin~ cans for- a- 4-unit bet in these situati6nL So the poinfcount·and tbe·Fives system agree ~tit is good to have~ all tho Fives'_ · out. 'Die. itdvantage of the point count is that it ~eli m~ · more favorable situations than the Fives system. and· it fiJlds them -earlier. You win several times as fast. •

You should now ·stop reading and take tiJJiec Olit co­practice the simple point count. If possible,· have a. friend deal· to you. Play slowly enough so that you do not :maJaj niistak-es. Do this even if:you are ridiculoUsly slow at~: · ~ yOu· practice, y~-will gradually speed up. Pl!ly. . ·. _ ... :

chipS. This will add a touch of realiim, and ·it wm~4 give you·an idea Of how rapidly you 'will win· When' )'W are playing ~e simple point count accurately and at a normal speed and feel quite comfortablC With_ i~ ~ on.

R.,efinementl . - •, . After you are completely comfortable with ~ peiiir· •

count, there- arc several refinements which make it stroager. If the point count is, say,+ sand the deck·is aeailJ.

complete, the deck is not as rich in high car4s as it it_if. for example, the point count is + S .mel on_!y-m, ~ am left. 'Then they are an high. So the advantage or ~ tage to dte player ~ependS I10l only on the;point ~· fi.tat c

aJso.on how niany ~~-.re left. If ·we take this intQ accouat. we can play more preciSCly., The'c:ortectway tO do t1dS is to cn~-:the point total by the Jl\mlber of decks left. "J."he..auuJi. bet. We 8et is·a stiU •better- one ~ the ~ to~ !D' JISt • for ~pq~ For ~le,jf the poutt ~ 1fJ + 1 . ~~deck remai~ we-can bet I +(*) = 1 x(:a/~) i::·1~)1stead ~ .. OIJly·'t.-.ut MOte iUlportant. --cor-. ~-adjUstS th8' point coiQit lor playing ... casiDOJ

~;

l l

I j-j 1 !

' J

l l

78 B'EAT THB DE'ALJfit

using 2 or more deeb. In a two-cieclc game ·the point Cbuitt for the ahpost complete two decks should1>e + 4 f~ a 2-unit bet + 6 for a 3-unit bet, and s6 on~ However, once the two deCks have been played down to one deck. the betting· should be the same as for one deck. ·

· Later, in the advanced point count, or high~low, strat­egy (Chapter 7), we shall learn to keep a precise count of the remaining cards. This will allow us to compute our bets still more exactly. ·· . ' •

It is also true that the strategy changes as the pomt count changes. As an extreme example, if only low cards ~ Wt, the player sh9uld always draw on~d 15. • Since he eould draw at most a Six he could never bust, and draw­ing must itnptOve his Iiand.

· Roughly speaking, when the point count is plus, stand more bffeil~ double down more often, an4 split more often. ·When the point count iS minus, draw more often, double dOwn· less often, and split less often. The .details will be covered ill Chapter 7. With this background on the strategy chtiriges.- r can describe my Puerto Rican advcnmre and -the Salmon's" great Sso,ooo win.

1'"~ Simple Point-Count_ System 1~ \ S~s.ooo cluillcnp JD.atch with 8 wen-known Las V~ : casinO),. bankrQUed my play. -· . . ~ • :

· . 1 spent an enjoyable afternoon with Henry Mo~ Jll the New York offices of "I've Got 8 Secret," attenq>ting to teach him the point count. But his jokes ~re so m~ moro interestiftg. and he himself so uninterested in the dry rQUdno of "trainirig," that I settled for showing him the maill parts. of tho basic strategy. Hopefully, ho would manage to break even. (Profits would go to charity, of course.)· ..

I sta~ with Messrs. M and N in the luxurious. La Concha hot~L It is one of about ten in San Juan that bavo C&$inos. Henry _settled •where. Casino hours were gc:n­erallY from 8 P.M. to 4 A.M. N and'! arrived about~ nigh~ got settled. and played for ll couple of hours. ~ · get the feel of things. M had come ~ several days ~Uet and bad won about $1.000 so far. He showed us a~ and it soon became appar_ent that there were signiftc8Dt differences between the Puerto Rican game and the Nevada game. . -· First, the Puerto R1can casinos were government _re_g.-ulated. This meant that ofiicial printed rules of the-game~.

·' blackjack were posted on tho wall of each casino. in plaiD ! sight. The same was true of many procedu~;es. The cas~._

Henry Motgan and 1 visit Puerto Rico_ couldn't suddehly change the rules .on m~ as they had ~ After the first edition of '{Jeat the Dealer was out, I Nevada. You may recaJJ. for example, that in the test of the

apj)eared on ~'I've Got a Secret'' (April. 1964). My "secret" Ten-count system (Chapter S) Harold's Club ~ed. up_ was, of course. that I coUld rapidly and consistcrntly win at ; on me and another club refused to sell me Iarge-ssze chips. casino blackjack un<ler the· usual playing conditions. But The· two ot:11et games that were ~erally played, roulette Gary Moore, moderator of the pr~m, had a secret for and craps. also had their rulc;s and reguliltions posted. Tho the panelists too. Hi~ secret was . that Henry ,Morgan was rules for the games were unitotm (a f~ optionJ were at--to.try out the system in the Puerto .~can casillO$. on a bank- lowed) for the several 'casinos we visited. · roll . of $200. Henry was to report back to the program As lin added protection for both customer and casino. · the following week. .. . .

1 a govc.t'DDlCDt man was on duty ill the casino playing area _at

·x ·decided to go down, too .. and see. \;Vhat tlte,,.,~, · , aU 1imes. &ld. every ~lo etfort was made ~ infoml RiCitn_casinos we~ like. Messrs. M,ancJ N~-1:WO yollQg~: / ~=~ these.facts.'lbfs CODti;asts with _tho, situation_­YOt-k entrepreneurs (also the promoters of_ an;:~~ ·"! · ..,

.. •To ezpens: 1 ~ think..., ~"HfoA situ~Jii...,. \ {~,~i»~ ~-itself was dealt face up frQm two. lie shoukfDl, but to pvo them llcii WOII!d-O!Ily c~utWupa. ;~,!fl~~ J .. ~-~:cje"'··· - .· . · · ·

. . .. --'~"''''"~ _·:, '~"'->f;:;-;:c~::;>'

i-

j i l: J :_ i:

80 BBAT TBB ·D.BAL'Bll ... The Simple ·Point-Count System Jt. deeka. 'l'he. cards were dealt from a dealiDg box, or "shoe." . . The rules \VCI'C Jike the typical rules of (llaptet 2; with This makes the sleight-of-hand card ·lriCltely- 5o· oftett ._ one important exception. Doubling down Was restri~~ to· COPntere&Hn Nevada (but so seldom noticed; see Chapter totals of hard u only. When questiqned, tbe casbioe said 10 for clctails) much less likely. Don't think you can'( be that soft 21 could also be counted as 11 and doubled down cheated from a dealiDg box. You can, as we'll see ·shortly. (m (the printed rules seemed to imply it.also). They were :&ut the Wrlety of cheating methods are fewer and the quite aniused by the question and wondered w)lat f9ola chances of catching them are higher. So they deftnitely dis- , would ever double down on soft 21. C)Qurage cheating. · Here's an example where doubling down ou soft 21 fa

To further discourage cheating, two decks with dif. the best play! Suppose you are keeping track of totalpoints. ferent color backs (e.g., one red deck, one blue deck) were total remaining cards, and total rem~ng Aces. Suppose geaerally used. The top of the dealing box was tnoStly open you have just been dealt (A,Io) and the~ shows a.zo. ai1Ct you could seo the color of the back of the top card. If Suppose also that the point total is riow + 3, there are tbNe ~watched closely you could sometimes also see the.oolor cards left, and no Aces left. Then the three cards left unseea Of the back of the card dealt to you. The dealer's hand often i are all Tens. So the dealer's hole-ard is a Ten. He has 20;< oo'Wered the .card so wen during the instant before the card he can't draw, and you.will win one and a half times yo\11' was-turned face up that the color couldn't be determined. If : original bet from your natural, or bladcjack, if you stOp you saw a red card and got a blue one, you would know. you fl'. now. But if you double down, you will get another. 10 fQr wore cheated. · a total of ordinary 21. Your 21 beats the dealer"s .~ and

. . Another nice feature of the Puerto Rican casinos Is you walk away with twice your original bet, instead· of oa1y that no liquor is served in them. A favorite device of the l one and a half times iL .. Nevada casinos is to ply their customers with "f~" liquor. 1 Admittedly this is an expert· play. You would not bO This is often done deliberately, to impair the player's judg- ; expected to do it after learning only the simple point count. JDODt and to remove .his inhibitions. It works. I have seen However, once you have masJered Chapter 7, there is_-& 1'/ell-heeled -drunks who pay their employees one dollar an good chance you will be capable of such feats. The point _ hour fting thousauds across a blackjack table in a few l of the story here is that precision play of this sort was ·be­min~ And l have seen poor drunks Write checis ~ey t yond the imagination of the casino personnel we met. ~dn't cover. The last am perhaps tho saddest cases. The 1.. There was one other important rule in PuertQ ·Rico. Puerto Rican casinos willingly serve unlimited snacks- ! The betting limits were $1 to Sso. compared with $1 to sandWiChes. Cokes, etc~ 'lbo•1ltm0Sphere is f!uiet and re- $500 in most of the larger N~ada establi$hments. The )axed. Evenmg dress is the rule. - maximum win rate in 'Puerto Rico would .only be about

The blackjack rules allOWed the player to play as one tenthwhat it~ in Nevada. So M'.s wiA of.$1,000 in many initial hands {before pair splitting) as thete 111010 three ore fou! days would--correspond~ a win inN~ empty places. There were~~ so a,.__._ l (multiplyins .U bets by IP)-.of $1o,ooa. .

could play seven hands at a time. Ttle:patet. - • .. ~-.".-·-.. •.Ia' t~ . .After the·evenin .. g.'sw. arm-.. ~wetJI~m.about. S A.~J;F. of hands per hour, the ~ the ~ ~ 4ch,._ . One~ the grea~ stratns of thesoli~tle v~ IS teadjnstmg used. so we goD*II1y playe4'this ~ Witlf!JP-10 , ,the body~ sch~Qle toJbe rev.eraecUsourubat gamblers~ aeVes\hands. . - .. ·. ·. --~~,> t ·to keep. ..

''· -J. ,·:.

BBAT TB;B I)BALB&

Bnfer the Salmon .. Before we arrived, M had noticed a system pta~ ~o

.was having great success, playing e'!ery night and WUlDUll steadily. He and M ~~ acquamt~.. .

. The casino personnel nicknamed him the Salmon (pro-nounced Sal-moan, with the stress on the se<:ond syllable). lnspired by the first editi~n of Beat .the D_ealer, he had

.. begun to play about six months earlier, w1th a s~ of $200. I said in the book that this s~e ~o~d 81": the player a 99 per cent chance .to go on wmmng Jndefinitely. There was only a I per cent chance that an extreme run of bad luck would ever wipe out the stake plus acc.umulated wimWlgs. . . The Saltnon took me at my woro ~-got to .wor~ J1C found that the Tea-cQWlt was too labonons, so he dis­covered for himself the simple point count plus refinement&

··When we met him, his original $200 had~ to $20,000. · The Salmon put on a splendid and effecti_ve show. He

would come into a casino, spot an emp!J tabl~ ind purchase leveral-thousand dollars' worth of chips. which he stack~ in several great irregular. columns, up to a foot or two 1D

-< height. Th6 colu!Jlng of~~ like m~ suc:; kings in checke~ ~e scattered his great columns of c ps an over the layout,_like pieces Jn a game. ~ut he always lllanaged to scatter them so . that they effectively . bloclced anyone else from playing. While he played, the Salmon kept . :up a disarming and amusing line of patter with the nearby

casin~after I arrived; recent copies of Life ~ag­azine [491 tdached Puerto Rico. In them was a ~page story, with picture$. about me and Beat the· Detiler •. ~ book had alSo moved onto the NfiV York Times _non-,fiction best-seller list. I was. theJ1 ~ by the casmo petson· aeL After the casinoJ shUt down at 4 A.M .. M, N, nc:l ~ ~

· a snack with some Of the people from one of the .caamos. •

_The Slf!2ple Point-Cf:!unt System

We learned that the salmon had been winning regularlyfor several months, but oo one knew the trqe el(tent o~ his winnings. , .

t asked why he was called the. Salmon and was t9ld tliatit was because he was like a fish that swam up stream. "But we'll get him in the end," a casino boss said. ''We Call it a Ia larga [in the long run]," he said. "And we call it 'in the long run,' " I said. Salmon later told me that "Sabnop" was Puerto Rican slang for "jerk."

The casino view that the Salmon was a jerk was furth~ reinforced by the way he played sqme of his hands. At the end of the deck, ~n apparent recklessness or disgust, he

. would sometimes hit a blackjack again and again until it busted. Likewise with a pair of Teris or a pair 9f.Aees· Other times he would stand on a pair of deuces! Surely this was madness, the casino personnel would. say to _n'l.~ again and again. . . . . · ··

I coul(i only smile and ~y that it certainly was difficult (for them, that is) to understand how su~h playing oould lead to anything but disaster. I pointed out that my b~c Strategy, the Fives strategy, and the Tens strategy, all forbid · such plays. Was Salmon mad? Far from it.

'fhe Salmon was· using "end play," as. described in the . first edition (see also Chapter 8). Let's start with an ex~­ple. The Puerto Rican casinos dealt two decks ~11. the way to the end. The last card of the two decks was pulled back, unpllfyed, however. Suppose now that the simple point count is, say, -8 and there are (roughly) about l6 caJ~S left to be played. Recall now that the Salm~n has blcicked ~ff the table with great sta~ of chips. s_"mce there- are &eVen places, he can play from one toseven baruis on each deal. He now takes, say, four hands a~d bets $i on each hand (remember, the deck is btid). He and the dealer be­tween 1,hCm get ten carets on, the deal: . Suppose that the· Sa:Imon's" first hand is (IO,IO), his second hand is (A,IO) .a:ild .~ rest are 5D1all cards, He hits the (zo,IC)) until

-~ '

- ~-- :. , .. _-----84 . .BEAT TRE DEAI,.~.lt

it busts. Then be does the same with· the (A,IO) if pos­sible. Then· he draws to the small-card hands, witho1Jl busting any of them. When the deck is exhausted, the used cards are reshuffled. The (IO,IO) and (A,IO) cards are in this reshuffled group. The table is covered with small

cards wliich are not. The next deal comes from a shoe which is pOor in low

cards. The Salmon bets $50 per hand and ~as the advantage. On the average he will have_ the advantage dqwn to the end of the two decks. At that. time end play is again used to exert control over the composition of the next shoe.

The Salmon deliberately lost some $1 bets so he could Win some $50 bets. And the operators decided that he was

a jerk. . . We promptly adopted the Salmon's tactics. And for

sevetal evenings the casinos would see one, two, and some­times _three master players march up to the empty tables just after opening time. The "masters" would cover the tables with great irregular towers of chips and then begin to play one to seven hands at a time. The Puerto Rican dealers were very fast (on the average much faster than the Las

· Vegas dealers, in my opinion). Yet each of us could play still faster than the fastest dealer while we were supposedly counting myriad cards.·~ ment~lly computing at a great rate. ·

At the end of a shoe one evening, I had been losing ·for an hour or so. My dealer had a ten up. I had -seven hands with various totals. I was using the variation of the point count where 2,3,4.5.6,7 are +1, 8 is o, and 9,10,A are -I, when they fall. The deck had run out on the deal, and the point count was zero. Therefore the one unseen card, which was the dealer's hole card, was a "zero." So the dealer had an Eight in the hole for a total of 18. · . . · As the play of the hands developed after the reshuffle, I bad to hit several totals of hard 17. They all busted.

The dealer looked up scornfully, saying to me with a

· '!ire· ~~ple Poilu-Count <System · 8~ laugh, "So you count the ~ar~s. amigo. Why (ha,. ba), 'Iii bet you even know what I ve got under here." A couple of t~ other dealers grinned. So I said, "Why, you have. an Eight under there/' The dealer laughingly summoned sev­eral of the other dealers and the pit boss. He explained contemrtu.ousl~ that the Americano "expert" said ~at he

bad an Eight m the hole. A babble of uncomplimentary remarks in Spanish passed back and forth.

I w~s tired and about ready for a break. I had made an oce;as!onal counting error over the last hour.• There was a chance I would be wrong (~tter for me if I was, prob­a~ly). Then the dealer turned over his hole card. It was an. Eight. An.d the babble in Spanish raged anew. .~

, . We played on for five nights .. During this time our c.ap1tal fluctuat~ rather violently, considering the small bet SIZe: At o~e tim~ we were a couple of thousand .dollat$ behind. nus made us redouble our playing efforts. DUring these f~ days I was playing at my peak. I would cou~ Aces, pomts, and remaming cards, or Aces, Tens, and uon-

. ~~ns, and some~mes not just three but four or five q~ tities. I was making perhaps one or two counting error$ an evening! Yet I was having difficulty pulling abead. ·

I looked f~r cheating and found -only one incident. M and I we-,e playmg at the same table in one of the crowded clubs (not La Concha). Our dealer seemed extraordi­n~ri~y clumsy. Two cards kept trying to come through the sht m the side of the dealing box. \fhey would jam and he would fumble around. We finally got tired of this and moved to another table. Then that box was moved to our table! w_e moved a~ain and the box.moved again! Masked tO examme the deahng box, and we summoned the govern­ment man. Nothing appeared to be wrong . · But we knew shoes that deal seconds have been'tnade.

• •It is an important and interesting fact that errors in card count• aug; .if !,heY are "ran~om:' . i.e., have_ no intrinsic "patterns" or "ten­cleDcies; do eotnparativcly little harm to a system player!. ·

"

~' " 'i

).;

36 BE. AT. THE 1)_1) A L.~tt

They were commonly used, for _example, in faro as ~layed in the old West. A clever mechanism was concealed m ~he

. thin sf<;le panels. If this were such a shoe, one of the s~de . panels might be hollowed out. There was one long stde panel and two shorter end panels. We tapped both end panel~ with a chip. They sounded the same. Then we tapped the longer panel. It gave a higher tone. By the la'Ys of physics the tone should have been lowe:. We t~ted other ~parently ide:ntic_al dealing boxes. Thetr l~ng side panels

. gave out the expected lower tone. ConclusiOn: we would avoid this club from now on. ·. · Why didn't we ask that this box be confiscated and the ·matter be investigated? Mainly because we w~re ~nable to cpmmunicate with this government man. He dtdn t seem to understand. English and he dicfn't seem to know wh~t we were driving at. Of course, any effective action would have to be initiated there, on the spot, wjth the evidence at hand. Once it disappeared, so did the chance of makin& an effec-tive complaint. · .

· We ended our stay with a joint win ()f a li.ttle under $2,000, which barely covered the cost of a l~unous vaca­tion for four. This would correspond to a wm of almost $20 ooo in Nevada with its higher betting limits. :aut we sho~ld have won much more. Playing conditions were ideal, 'and dealing down to the last card enabled us to ~se end play very strongly. Also, in the same amount of time the Salmon piled up $7 ,ooo with play that was not as stro~g as ours. We found very little difference between our wm­nings when we used a point count and when we used a Ten

. count.

An Interesting Idea for End Play · Finally, the day before we left, 1 realized how we could

make a fortune in Puerto Rico as long as they dealt down to the end of the deck. My idea had begun to form a couple of evenings earlier. M was very impressed with the- courte­ous. friendly, relaxed atmosphere in the Puerto Rican

·~-.----.,..--~

. ''The· Simple Point-Count System

'casinos •. It was ·so different from his ~ences in Las Vegas and the other Nevada gambling meccas. He asked N to count Aces for him with chips. This worked fine. NeXt it was Tens and Aces. This worked too. F'mally, N would take turns fully relieving us of our counting chores by~­ing With the chips. Surpri$gly, what N was doing appeared to pass unnoticed.

This.made me realize that it wufeasible to count all the cards so that, every so often, we could know the deaier'8 precise bole card. The profits could be spectacular. We practiced with a miniature tally sheet in our rooms the next afternoon. My idea was that the player could be assisted by a helper who counted all the cards with a tally sheet. We Would start at the beginning of two fresh decks by ~

· &even hands at $1 each and playing them out with the basic strategy. This would be repeated, to use up most of the . , two decks quickly. Now, 102 cards are actually played out of the 104 cards in two decks, since the top one is burned and the bottom one is pulled back. About three cards are used per hand. So in a round of 5even bands plus the dealer's, about 24 cards are used. In three such rounds. about 72 cards are used, leaving about 30. .

Suppose that this in fact happens in an .actual game. We would next take some number of bands that guarantee. that the box will run out in the following round. Bufit must run out after the dealer gets his hole card so we can then tell from our count what that hole card is.

We take enough hands so there will be 16 or a little fewer cards left after the round. The reason for- the un- . certainty ·at this stage is ·that we can't control in advance how many cards, if any, the dealer will draw. In this case, five player hands plus two cards drawn or six player hands

·plus no cards drawn would Use up 12 cards. Since the dealer's hand uses up 2 more cards of the original 3~ this WOUld leave 16 cards less the number the dealer dra-ws to his }tote card. - .

SupJ>9SC it ·turns out that he draws two cards. Then

·88 B E AT· .·T H E I) B·A 1.:1~ ll

there will be fourteen cards left to play on the final deaf of the box (plus one card that will not be played). We now take six hands and bet the maximum of Sso on each of them. The fourteen cards are just enough to deal out aU our hole cards and the dealer's two cards. Our~ards are face up, we can see all of them, as well as the dealer's up card. We now tcy to hit the first hand, no matter what its value. The dealer pulls. out the last card, notices that it is the last card, and refuses to give it to us. Once we have seen and eounted this card, the only card left unseen is the dealer's hole card, and we now know from our count what that card is.

After the dealer shuftles the used cards, we get on with playing out our six hands. He asks us whether we still want to draw to our first hand. (If the casino rule is that we had to draw to this hand because we had requested a card

. before the shuffle, we would have countered by only betting $1 on this hand.)

l'lllying Strategy When the Dealer's Holl CtUd Is Knou'll

We play this hand and the five others with a new .. basic" strategy. The term "basic" means again that (for simplicity only) we are not considering information about any cards other than the dealer's two cards. and the hole cards of the hand being played.

Instead of a separate playing strategy for each dealer's up card, we now have a separate strategy for each pair of dealer's cards. There are fifty-five such pairs. The strategies were calculated in detail by Julian Braun and forwarded to me some months before the Puerto Rican trip. The results are given in Table 6.1.

Note that the strategies are yery much alike against dealer's cards of the same hard total.

Braun calculated that the player's advantage in a one­deck game·with typical rules, when he knows the deaier's hole card, is 9·9 per cent. (Just as a matter of general inter-

---- ----

'Phe~ Simple Point-Count System

' est, Braun found that if the dealer also won all ties in sueh a game, the player would still have an edge of 2.1 per cent. We omit the somewhat different player strategy for this situ!Uion.)

Value of Knowing the Dealer's Hole Card

We could have averaged $25 ($5.0 per hand x 5 hands :X to%) or so per box by using the complete count of all the cards to determine the dealer's hole card. We played at least six boxes an hour. At six hours per evening (rest ·two) this would have been $900 per evening-a "living wage"! But we were unable to become proficient enouoh

c ' as. a. ~eam, to get started \Jefore we .left And the Puerto . Ric~ rules were changed soon . afterwards, as a result of :the Salmon's exploits, so we never. had the opportunity to take advantage of the method. •

. In an ordinary game the player sometimes. finds out . what the dealei"'s hole card is. The first time this bappened to me was at one of the big strip hotels in Las Vegas. One ·man was playing alone at a $5-minimum · table. He was

' bettin.g $200. to $500 on each hand and winning :rapidly. ~hats !~'ore, he and the dealer seemed to be getting along JUS! ~~e. I sat down and bet $25 to $Ioo, figuring that my actlvlttes would go unnoticed alongside his bigoer bets ..

I soon noticed that when the dealer had 1; or more, · the player always hit hands of 16 or less. Once when the dealer had a 10 up and a 6in the hole, the player stood on. 12! Then I noticed· that ·the dealer was tipping off the

, p~ayer as to whether he had a "stiff" (I 2- I 6) or a probable . t1,fl (4-6 ), by looking unnecessarily long at his hand. If . e had a pat hand (17"'21) or a probable good hand (7-11

• There was little chance in any case that I penonaUy could have ~on. r_nuch ~oney in Puerto Rico. The intense public interest and

bc:ity which ~f!OUnded me at . the time made the casinos acutely ware of my activities •. As soon as my winnings became impressive the

. ~ W9ll~ have been changed. The Salmon's success deperide4 o~ tbe that he .concealed the ~ extent of his winnings for a tong time.

heil be "opened up." the CI!Sinos diil change the rules, as we shall see.

.·,

~·.

.!!

90 ·BEAT • t ll 8 D 8 A L F. R · .· The S.i(tlple Polnt:Count System 91

O.Oie • •

9,10 [19 19 9 10,10 20 20

up .card is a. Ten, the reverse happen~; Mw they wonder if . they ba~e an Ac~ •• If the bole card is large. they can see this· . very. quJckly~ If •t IS small, they must bend the bole card up qutte a lot before deciding it is not an Ace. Mr. F ($ee

. Chapter 12) gave me this tip in gratitude, after I bad won · $1,600 by standard me~ods for him and Mr. X at the Las ~egas <:Iub in a couple of hours one evening. Mr •. F and

. hts cromeS call this dealer giveaway the "telL"

. A La Largo: The Salmon Wins $5o,ooo

. .. We left ·Puerto Rico with fond memories. hoping to return on another plea~ure trip after school was out in the ~mmer. The Salmon was fearful that our visit and impend­mg return would ~peed the day when the Puerto Rican

· goose w~ul~ ~op laying golden eggs for him. He began to · . bet the hmtt tn all favorable situations. Within another two

months his winnings rose to Sso.ooo. The Salmon ha~ played an average of five or six nights

or soft 12 to soft t6), he looked at his band only for an , a. w~k. seven or eight hours a night, for .nine months. instant. He was telling the player what to do. Play for 17 FJgunng forty hours a week for forty weeks, that•s 1.600 (or more) if he looked only briefly at Iris -card. Stand on .hours. At 150 h~nds an hour, he played 240,000 hands. hard 12 if he looked a long time. And he won Sso,ooo without ever making a bet over Sso , Of cou~ the details only became· apparent after a per hand (except in the comparatively few instances when while. But I ~aught onto the "stiff'' or "pat .. part right . he. doubled down). This is the longest casino test of the away. In the next twenty minutes the player w.on an addi· ~mt count that I know of. It is a spectacular demonstra--tional $2,000. He tipped the dealer $300. I won about $50<1 , lion of our methods in action. ~ in the same time. Then the dealer went off duty and the , · In Las Vegas, with a $500 limit (and an honest Situation broke up. · ga~e!) t~e result .. would have been $500,000 instead! De-

When a dealer in Nevada bas an Ace or a Ten up. he : ~pite the 'modest amount that he wo~, the ~almon's win ~becks his bole card before giving additional cards to an) ,.,as one of the great marathon feats of gambl~og history. of the players. If he has a natural, the hand is settled the~ ,,A Ia larga~e long run-showed the Puerto Rican casinos and there. If the. dealer. is inexperienced, be may give awa, ,. the handwriting on the wall. some information abOut· his hole card. For example. if the ·: fh R l , up card is an Ace, and the hole card is. smaU, some in ;. e u es A~e . Changed" in Puerto .Rico experienced dealers see very quickly (from the lack ol '_:-, . . The casmos then s!opped their prac~ce of dealing to .. paint") that it is not a Ten, and pause only briefly. If the i;the-end of the box. Agamst good eiaye~ (the Salmon, M.

,.

92 BEAT THE DEALBll

and N, for example) they reshuftled as often as was nec-essary. .

· I learned the sad news when the Salmon called me to see what could be-done against this. 1 advised him to hunt for greener pastures. Try Panama, Curacao, Aruba, and (cautiou~ly) the Grand Bahamas. Stay out of Haiti and the Dominican Republic: it may take the U. S. Marines to get your winnings (if any) out. 1 told him that Las Vegas was a harsh, pitiless place. At that time there was a par­ticularly severe cheating problem. But he could win heavily there if he could dodge the cheating, or if he adoptecl the "paper-route" . technique described in Chapter 9· He tried Las Vegas and was sobn stripped of $2,500.

The last 1 heard from the Salmon was that he was waxing prosperous in balmy Puerto Rico. He was the proud owner of a fleet of taxicabs which paid him $100 a week each in rentals. He dallies on the beautiful warm beaches and in the night clubs, living the life of a playboy. Every onca in a while he hears of a "good" blackjack game in the. Can"bbean and rushes over to "kill" it. But he longs wistfully for the good old days when money grew on trees, and an be had to do was go out and pick it.

7

The Complete -

Point-Count System

Like the simple point count, the complete point count, or 'high-low system, had ~urred independently to many read-

. ·ers of Beat the Dealer. However, it was first announced-to the scientific public by Harvey Dubner, then of Simmons Precision Products Corporation in Tarrytown, N.Y. Dubner presented his results at a panel session of the 1963 Fall Joint Computer Conference, a national semiannual meet­ing of thousands of computer experts. This time the con-· ference was in Las Vegas, Nevada. The panel session was on the use of computers to study games of skill and chance.

The leading experts in the field talked of their work in analyzing such games as blackjack, baccarat, roulette, and go. Besides Dubner, the panelists were: Julian Braun of-the IDM Corporation (Braun's detailed blitckjack cal­culations, based on his. extensions and refinements· of my

. original computer prbgram, are the most accurate in ex­

. istence, and he has kindly atlowed them to be used through-• 93.

:­

ii .,

94 BEAT THi DEALEtl

out!his revised edition); Richard-E. Sprague, directot;of eomputer systems, Touche, Ross, Bailey and Smart (author of Electronic Business Systems); William B. Walden, then of tho Los Alamos Scientific Laboratory and now dii·ector of the Computing Center at the University of Omaha (he and 1 jointly developed a system for Nevada baccarat, and otlr winnings forced the casinos to remove certain formerly profitable bets from the layout); and Allan N. Wilson (author of The Casino Gamblers Guide, a recent coverage of the principal casino games; readers of this book will be interested in comparing Wilson's lengthy coverage of black­jack). Theoretical remarks were made by Robert E. Ka­laba, ·a mathematician from the RAND Corporation and an expert on gambling systems like the Kelly system (.see P· :23). '

I was the moderator of this panel session. · An enthusiastic Dubner made great claims for the

complete point-count method. His calculations supported his claims. And his play in the casinos (for low stakes)" during the computer conference was very successful. The interest of the other experts was aroused. Braun then made detailed calculations. (His computer techniques were based on the methods developed and used to work out the Ten­count strategy fof the first edition.) Although these showed that there were some inaccuracies in the details of Dub­ncr's results, Braun found that the complete point count was a powerful and effective winning blackjack strategy.

Exactly how much ·better or worse it is than- the Ten­count method is not known. But they are of com~rable power. In an era of casino countermeasures; tightening rilles. md dealers who are finally learning to count cards, the complete point count .is a wekome new weapon. The beginner should probably read through to the TeD-CO\J.Dt chapter and then choose either tho complete point count Of the Ten~count as the first powerful winning strategy to be mast~red. For those who are already proficient in the Ten

· «nul~ the COJD.pletepoint«JUnt is a valuable alternate 8tra'-eg and should also be learned, if possible. . ,

(;.~nting· the Ca~ds _In the complete point cOunt, ·we simply perfect the

~ents of the simple point count. The first sfep is t9 have an exact ·count, rather than a rough idea, of the numb.er of unplayed cards •. So we will need to r~member two numbers, total points, as before, and also tot~l unseen ards. The count of total unseen cards is quite simple. For one deck, start your count at·sz. Each time you see one of the cards used in play, .subtract I from the current totaL If a card is played and you do not see it for one reason or

.another, do not change your total of unseen cards. Adjust the total if you see the burned card or if the dealer flashes the bottom card.

When we were only. counting total· points, the cardS could be counted any time during or after the play of the hand. The only important thing was to have the total pomt count available in time to decide how much to bet ·on the

. next deal. You could even wait until a band was over and then count the cards all at once. We can still do this whca we are counting both total unseen cards and total points, provided we use the information merely to arrive at the proper bet size. We still play our hands with the basic strategy.

When we want to use our card-count information, to vary the strateg for playing the ·hand&-to improve on the basic s~tegy-we can still count this way. But it will be still better to adjust the count as soon as we see a card, wWiout waiting. This method of counting is called tht ~running count." It is something b'ke a full-court press in basketball: ·your count totals are ~ways ready. You don't Jet ~· except when the cards are completely reshuffied, or the ttea~ers change, br you change games. ~y the running·count is more tiring. If counting

r-c-_c··. c.~- ~'=o--~ f "96 ,j: is -easy for you, use it; it is best. If co1.i1lting iS bard .for

you, don't worry. Figure out the easiest w;q. to count and Use that. Your results will be almost as good. Perhaps they will even be better because you will be less likely to make mistakes.

I L

I d I !

Here is how we use the new count to decide on the bet·size. Divide the point total by the total of unseen cards. For example, in one-deck game, if 5.5,3.8 fell, the point total would be +3, 48 cards remain (unseen), so we get 3/48 or about o.o6. I find that the easiest thing to do is to estimate the nearest per cent. In this case· it would be 6 (per cent), because to change to per cent means to multiply

. by 100. I call this final number the high-low index. If the same cards had been seen from two complete

shuffled decks, we would have +3 points and 104-4 = 100

· unseen cards and a high-low index of 3· If there were 4 decks to start with and A, 10, 10, 9, 8, 8, 10, A; A fell, the high-low index would be -6/199 or -3 (per cent). .

The Bets Bet 1 unit for a high-low index: of 2 or less. For a

high-low index of 4, bet 2 units. In_ general, bet half the index in favorable situations; index 6, bet 3; index 8, bet 4; index 10 (or more!) bets. For index values between these, you can shade your bets up or down, at yolir pleasure. For example. if th~ index is 5, you can make the bet correspond­ing to an index el :either 4 or 6, that is, either 2 or 3 units~ The reason f91' llettin'g no more than s units, even when the index is over 10, is simply that the casinos get too excited if you do. -

Drawing and Standing

· The basic strategy is the best way to play against ·a ·

·The' Complete Point-Count System iyr that tlie game is being dealt from four well~sh~ffied decks and that the only cards left are Fours or smaller. Then tho player should draw on hard 17, no matter what th~ dealer's up card is.. This is in sharp contrast to the basic· strategy~ which recommends standing on hard 12 against an up card of 4, 5;or 6.

Advantace

15i

1091.

To use Table. 7.1, you look in the square correspond­. ing to the dealer's up card and your hard total. If it ·says

.. draw" or .. -stand," things are easy. Suppose instead that you find an index number. Then you stand if your index is

. - greater than the one in the table. But if your index is less · ·than or the same as the one in the square, you draw.

For. example, suppose the dealer shows a Ten and you have hard 13. The table'says d'raw, no matter what your index. Now suppose you get a Three. You now have hard 16. If your index is o~ or less, draw. If it is more than 02,

stand. (Warning: Minus indexes are less than plus indexes. Using < for ':'Jess than," we have in general • • • < -G3

. complete deck. But if some cards are missing and we know . something about them, we usually can HnJ»(OVe on tlte basiC · strategy. As a simple (and unrealistic) example, suppose

< -oo <...;.or < oo < 01 < oo ••• and·so on. So a large index with tl minus is less than a small index with tl minus. 1his may be news to readers who are not used to n~gativo aumbers.)

l ,j

.. II

= 1J

I 1 . I

~+:-:,,...;-j! 1-o;..,.J;;;.:.L...L.J A

c7 ll'!j

~ ~ .. • .. !!:

J· ~ ... J: f4J"' -.. -.. ~

~~ !a ~~ co

-"' -

r717 I ~ -:::~~

The Complete Polnt.;cozmt System 99 Table 7.2 gives the drawing and standing strategy for

soft hands. It is read in the same way. as Table 7.I. Notice that the soft strategy is the same as .in the basic strategy except for three ~.

J. With soft 17 against 7,. and an index greater that} 29, stand (the basic strategy says draw). ,

2. With soft IS against 10, and an- index greater thaD . 12, stand (the basic strategy says draw).

3· With soft I 8 against an Ace, and an index of -o6 or less, draw (the basic strategy· says stand) •

The first two exceptions are quite rare in actual play and may be neglected until you become an expert. Si~ce -o6 is fairly close to oo, we can simplify the third eX­ception by saying: bit soft I 8 against an Ace if the deck is

' moderately unfavorable. So the results of Table 7.2 can be simplified to: For drawing and standing with soft hands, play the basic strategy with just one exception. With soft 18 agafdst an Ace, hit when the deck is moderately un-

- favorable. ·

Doubling Down The doubling-down strategy for hard hands appears

in· Table 7·3· Note that doubling down with a total of hard 5 is sometimes best play! Such facts were undreamed of until only recently. Notice that the more favorable the deck, the more likely- we are to double down. Double down if ,your index is larger· than the table entry. Do not double do~ if your index is less than the table.

Table 7.4, which gives the strategy for soft doubling. is read in a similar ~y. Once again, thtt more favorable the deck, the more frequenJly we double down on Soft totals (and the greater the profits).

Po.ir Splitting Table 7·5 gives the 54ategy for pair splitting with the

high-low index. ,

RD1204 13664

r~oo­~~ \~ Insurance ;1

If the .index is greater than o8, insurance sh.ould be taken. Otherwise it. should not be taken. A more detailed discussion of insurance'will be found on pages 106-1: I.

Advantage and Frequency of Favorable Situations

Figure 7.1 illustrates bow the player's a4v~ntage .changes as the index changes. Notice the gain insurance produces. It is also interesting that the player gains more when the index is positive tl)~ he loses when it iS! negative. This is because the player ean vary his strategy. Therefore be can to some extent reduce the disadvantage he gets from poor decks. He can also increase the advantage he gets from good decks. ·

The figur~ gives the imp~ession that if the index is negative enough. ~ player regains the advajltage. This bnptessic;m is correct but the ·situati~ns are rare in play. Table 4.1 slu>ws some of these situations. For example, when Q(Io) = o, giving'an index of -16/36, or -44 per cent, for an otherwise f~.~:ll ~eck, we have a player edge

-of 1.62_per cent. However: the average situation with an index 9f -44 is still disadvantageous for the player, as indicated in Figure 7.1. When the index iS -Ioo, in which case the remainder of the deck consists of cards having values of Two through Six only, the player always has an advantage, averaging something like so per cent, and de­pending on the· precise cards remaining.

Table 7.6 gives an idea of how often various values of the index, and various advantages, arise in actual play. Notice that. the chances of various negative values of the index. ~tly balance the chances of the Corresponding­positive values. To illustrate the use of the table, note for example that after five cards have been dealt, the index is between 05 and IS, 9.S per cent of the time. It is between -sand os, 81 per cent of the time. It ~betw~n -xs and -os. 9·5 per cent of the time. ·

ThB Complete ftJW-Coant System

~ ..

00 ~ ;t ":"! ~ .. In -l":go,o ""0"' .-:Vl"'!O N '"<oi ~~V\ -- -~ S"'"'= "'0"' .,~::i~~ ..... ~

~ :.i • N -

.• 2 .5 ·f1s .... .,..o.,. "l ~ l'f.'OI; C! ""': .... a\ f'ol ..... ......... _,..f'ol

;; 01 -.§.g .., .. "'!oN :!S~ '01; .... 110~'01;"\"'!0

.!! -~ ~ .. r-: coi\ISV.CI\1"10\ ..... g --::! .. () ::> ..

"l oq . ttl "' ~ <'1 "! oq "'! ,. 0 =~0"1 "'0"' ~·5 o .. - Cl\.,...,...,...,.oo,..,...., 11_ .. ..,. Po ..,.C'IC'-.&N..,.C"'..,..,... _.>-

0'0! -5 5 ~C!\C!"l~t'!\1:!"'!~0. t-o:! ·~ ::; s:; ~"'O"" M'i' .. o 8 ..... ~'"'-0\ ..... \0\0t9''11101'00 a: .. 00\0C'I\Cf\lf\Ctf\ .........

!..S =· -8 -·c:l· .!a ~ ., u .. ii:"'o"' VlOI!Vll'fC!"'!f'!CII!f'! . -s s.."lo~ -.:s· ... 4JC"\ ....... ,3i .. 'i' o.:::~~~:!l~~~

r: 0 8' I I u .. .. :;V\ V\ "~ ~ "'!· .s "l "'t::i:l~~~~o ·&o- =~si - c a"' ~ --~~ "' 1- I > -.:s8, ...,

"l~:!~~::i <Q) ""': "'o"' ~ .. . ,s, 'i ..... '()£ -. .. ..... <;;;

~s:l "'O"' .-:.-:::i~::! !j= T .. 'i =.!! I I iS~ i.s1 "'o"' .-:Vl<'\0 ... e: '1' .. 1 .. -~

"' t-:ffl ., r-: "' "' '2 f) .... .2 ... .. "' ~ I .. ..

.R -0 .. ~

. ~o"' sa :l~Hl ~ ~ ~~

101.

3 I j j J. ... 1 t ~

lt <~r .;"' e8 " ... -ig Sa. .s• n §9 8'ii .... 3J ]! ... fli sf ~~ ~ .. t• .,.s .!11 h .. 8'lS

~~

l ! i·

_._' '-l'

8

A Winning Strategy Based.on Counting Tens

The strategy to be discussed in this chapter, the "'Ten-count" strategy, was. the "workhorse" winning system i.ntroduced in the first edition. It is comparable in power to the com· plete point count. The expert should know both .systems. The beginner should atten:tpt to master only one ~t firs~ Readers who. know the point count may either skip chapter or read it through qu1ckly.

A Winning Str.ategy ·Based on Counting Tens 103

. that the dealer has an Ace up~ Sometimes the play~r should hit bard 17; other times he should stand on hard 12!

One might wonder, in view of Table 4.1, how a strat· egy based on Tens could give greater advantages than one based on Fives. Card for card, Fives have more effect than 'Teris: four Tens add~d to the deck give the player an advantllge of I .89 per .cent, whereas four Fives removed from the deck give an advantage of 3.58 per cent. The solution is that there are I 6 Tens in the deck and -only four Fives Therefore much oreatet' deviations from the average can o~cur in the n~mb~r of Tens than in the number of Fives.

Effect on Player's Advantage as Proportion of Tens Varies .

The richer the deck is in Tens, the better off the player is, generally. We shall now think of the deck as divided into two kinds of cards, "Tens" and "others." During play, we shall keep track ot the number of others and the num­ber of Tens that have not yet been seen. Thus, with the

· Tens strategy, we take into account only the cards we see, as we see them. From these two numbers we shall deter• mine "Ten richness" by computing the ratio "othersftens" ·(others to tens). For example, suppose the complete deck is shuffled and prepared for play. For the complete deck, the "count" is 36 others and 16 Tens, or simply (36,16). the corresponding ratio is 36/I6 or 2.25. The approximate advantages for·several ratio values are given in Table 8.1 for. quick reference.

The adva~tages this strategy finds for the pl~yer gen· erally range from I tQ 10 per cent. The large adv~tages yield heavy winnings. The smaller advantages g1ve ~e player camouflage: it is natural in this . s~, as w1tb the point count methods, to vary·the bet StZe Wlth the ad· vantage in small steps all· the way fro_m _small waitin~ bets Learning to Count to a size several times these bets. TJus lS less· conspicuous Our first goal will be to learn to keep count of the lhan just two kinds of bets, "large" and "small."- others and Tens yet to be played while also playing with the

Another camouflage advantage of the detailed Ten- standard strategy. Here. is an exercise which should be done count ~ategy is that the player's ·decisions greaJ:ly depend as a preparation in learning to count. Take a shuffled ~om~ on the composition of the unused deck. Suppose for example . ·ptete deck, and turning cards over one at a time, "count"

• 102•

- 1 ~· ·BEAT THE n.·E~LE'R ~':Winning S~gy Based on Counting Tens lOS

'

TABLE 8.1 Appro~imate· Ptayer Advimtage In the Tin-Cbu'itt Sttategy. ·

Normal approximate /

advantage Others/Tens (in per cent)

J.OO -2.0 .. 2..25 +0.1 2.00 1.0 1.15 . . 2.0 1.63 3.0 1.50 4.0

/

1.35 5.0 l.i5 • 6.0 1.16 7.0 ·1.08 8.0 1.00 9.0

them and drop them face up onto a discard pile. For ex­ample, I have just picked up the deck on my writiqg desk. I count, "(36,16); ·3 of Spades (35,16); 5 of Clubs (34, I.6); 3 of Hearts (33,I6); 4 of Diamonds (32,I6)-the ratio is now 2.00 and the hands. dealt now give the player

·a I per cent advantage (Table 8.1); 3 of Diamonds (31, 16); 6 of Spades (30,16); Queen of Diamonds (30,15), etc." A few cards from the end of the deck, stop and record. your count. Then see if the remaining cards agree with your count of them. In the example above. when I stopped, my count was ( 2, I) and the last three cards were the deuce of Clubs, the Nine of clubs, and the King of clubs, in agree­ment with the count.

The first few times it may take you two minutes or more to go through one deck without any mistakes. How­ever, you should be able to drop your time, in six or eight fifteen-minute practice sessions, to between twenty-five and fifty seconds. Fifty second& is more than adequate and twenty-five seconds is excellent. I had trouble getting below twenty-five seconds until I found that it takes me twenty

to twenty~five ·seconds to ;.tum over the cards whether .1 count them or not. For th98e who want to push this exercise to the limit,. there is a way to break this twenty-five-second barrier. Remove a few unknown cards and spread the reo maining cards face up in a row, with enough of each card showing so that it may easily be identified. Then ~ount. by reading fr.om left to right or from right toJeft. You should

· learn to be equally at ease reading in either direction. Your count should check against the unknown cards that WefO removed.

Shortly after I had practiced spreading the cards for rapid counting, there was an opportunity to use this skill •. I was examining a certain casino to see if it cheated and began, ·:naturally, by watching the table_ where the ~~t money was being risked. After the shuffie, 1t was the casmo s practice to have the dealer place a joker face up on the oottom of the deck to separate the used cards from the unused ones. At the end of one deal the joker had vanished! The amazed players asked to examine the deck. The dealer spread it in the standard fashion and then scooped it up ·again in about four seconds. Even with the rapid count I coUld only count the first 12 cards.

· . The players demanded a closer look at the deck. This time the dealer gave them ten or fifteen seconds. When I reached 38 cards (28,10), the dealer began to scoop them up again slowly. I quickly counted the number of cards remaining, without regard for denomination. There were 20 left: the deck had 58 cards! Of course, the dealer still had not allowt(d the players enough time to examine the deek. They called for a new deck and requested that the pit boss examine the old one. He counted the old deck off to one side, holding it in such a way that no one else eould count ajon:g with him. · · · When he finished counting, an odd expression flickered across his face. Then, without offering a word of ex.plana­

. tion to tht;· players about the recent puzzling events, he

;

••

•U)6 BRJ\T 'rHE -I)EAL'BJl

left.· ~king the old deck with-him. The trusting playerscon­ti,nued the game and -soon forgot the matter. The pit boss had accurately judged their narvete.

While you are increasing your counting speed with this exercise, you should also practice maintaining a count while $0meone deals to you and you play the basic strategy. Have them deal slowly enough so that you can count easily. Play with chips and start with 200 units. Before each han~ use your count to estimate the ratio. Theq vary your bets : according to the scheme given in Table 8.2.

·; . Roughly speaking, whenever the ratio is between 2 and

:'l"ABLB 8.2. A Conservative Betting· Scheme for the Ten.Counl Strategy.

Ratio

above2.00 2.00-1.75 1.75-1.65

bdow 1.65

Bet (in units)

1 (minim11111) . 2 4 s

1.65, we-are betting about twice as much, in units, as OU!'

adva.ntage i$ in per cent. We level off below 1.65 at 5 'unit$

so the variations in our bet size will not unduly alarm the casino. You do not need to perfonn division in your head to figure out the ratio exactly. Rough guesses, say to within o. I or even 0.2, ~e very satisfactory.

Insurance There is one important change from. the standard strat•

egy. that you should take into accoun~ at once. ·Whenever the ratio is Jess than 2.00, take insurance if .the ~pportunity (dealer's up card an Ace) presents itself. If the ratic;» is 2.00 or· more, do not insure. This is reasonable. If the deck. is Ten~rich and the dealer ,sllows an A~e. lie is more likely th~ usual to, have· tilackjack. You are allowed to check y.our hole cardsc (and you. may have.been able to aee other

;ft~lnnlng Slnltegy · BaS«l on Counting Tens 107

: ~yer's ·bole cards- too) before' insuring.. You :have a1sO • seen the dealer's up card. All this can be taken into account,

if you wish, before deciding whether to insure. · . We can c~lculate either the player or the bouse ad­

vantage from the ·insurance bet wheMVer we know tho , number of Tens and non-Tens. We illust{ate this calculation · ~ forthe.case in which bands are being. dealt from one com­

plete deck, a situation which represents the average bouse advantage. In this instance, the dealer's up card is an Ace. Since thv dealer's Aet; is visible, there are 51 possibilities for his hole card (assuming at the moment, for simplicity, that we do not use our )Cnowledge. of O_!Jr own two bole cards as well), 16 of which are Tens'. On the average, ~ player wins twice the amount of his insuranCe wager 16 times out of 51, or 314 per cent of the time. The bet is lost 35 times out of 51. The average house edge· is 3SfSl ' - 2 ~ 16/51, which is 3/SI or 5·9 ·per cent.

If you wish to take into account your hole cards, thero . are three~ to consider. If your hole cards are (Io,Io), the house edge is 35/49 - 2 X 14/49, which is 7 I 49 or 14.3 per cent. If they are (Io,x), where x represeilts a·nOn. .. Ten, the house edge iS 34/49-2 x 15/49. which is 4/49' . or 8.2 per cent. If you hold (X,X) the house edge is 33/49 -2 XI6/49. which is only I/49. or 2.0 per cent.

Insurance was originally introduced by the casinos as just one more way of fleecing the player. It is ironic that a bet providing such an average advantage for the c~in~ can be turned against the bouse. The trick, of course, is simply more of what we have been 9oing all along. The average hoUse adVantage is 5·9 per cent, but there are times when the advantage is in favor of the player. At these times we

· insure and otherwise we do. not. For example, wlien the count before the deal is (1o,1o} the-player's avetage IJ!Ofit on an insurance bet is 2 X I0/19 ~ 9/I'):. which is 11/19, a healthy 58 per cent of tim amount of the insurance.

. Once when I played in a large club in Reno. I nQtited

108 BEAT THE DEALER

that it had no insurance bet. Since one of the ·owners was standing at my elbow (for I had begun to win rapidly and they had rushed up to stop me), I asked why there was no insurance. He said that because it hurt the players, it was taken out for their benefit. As a .large'? bettor (large bettors generally are humored and given little privileges), I asked that I be allowed to insure, explaining that it gave me a feeling of security when I made large bets (when the count was (IO,IO) for example!). My request was refused with· ~ an explanation. I later learned that one player, using end ·play (discussed later), the insurance rule, and card counting, had taken at least $40,000 from this casino before be·was stopped. .

Many dealers and players alike share two widespread misconceptions about the insurance bet. They often become . annoyingly insistent in their attempts to "correct" ~ player who does not share their view. The first misconception is that a player should always take insurance if possible when dealt a natural. The argument is that if the _dealer has a· . natu~al also, the hands themselves will tie but the insurance bet wins one unit. If the dealer has no natUral, the player's natural wins 1.5 units and his insurance bet of o.s unit is lost. Again the win is I.o unit. In either case the player has a ~ure profit of one unit. Why not take it?

First, I will show you that there is a case in which the msur.a_nce bet is wasteful: Suppose that you ate eounting Tens and non;. Tens, and that after you see your hole cards and before you decide whether or not to take insurance. you find that aU Tens have been played. In this evenr the dealer can_not have a natural. If you 'insure your natural, you have a sure profit of exactly 1 ;o liJlit. -1lS discussed above. However, you know that your natural is a winner, so if you do not insure, you have a sure profit of J;S units. In this instance, to jnsure is to-throw away o.s unit.

Now, suppose the ~k had only one Ten and, say, eight non~ Tens left. Should you insure? No, ~lfSe even . .

c::;j ~~

=~~ Cll::l~

~

·~ .,;~ jh ~~ ~~

~"": "'"' . .., ... ..... "'"' .,;.,; ..;...4..-i

j- ., ... :1~ ~...;~ criui

."'t :: «!::1 -~3 "'o~ ~:s ltl..; N"l ""' "'"l "' .,;c-~ .....

~~ ::lCII .. h ~!l

c"! ""!~~

"' ~ ...... ~:::1 ~

.. "' '~ -~ ..

. .

I~::~ ~ .: I:~"!~,., ... ...

·10

=•· ·l'i . .

N~ ·~ . ~~

-~1 ~~;

fl

:;}

~

:~ h

... ~ --'!~

-~ 19 10: q<'i .... ....

!: "! ... !! .... "' .. ~

I"':! :J~~ ...... "!

"!"

·. ~ N ~1 f'Ci.C <!oi:C !~

. ·. ~~

"' ..

I <

. -~ Wfiiiilng. Stii,zlfgj BtiSed on ·Counting T~ 1 U

though your insu«tnce bet may win, yon are probably throwing it away. On the other hand, if all the remaining

.t:! cards were Tens, the dealer would be certain to have a Ji~ natural also, and insurance wins x.o unit for you. If in

·~ [· ....,~ ::l I! t:~ imagination we let the deck get richer and richer in Tens, •, ~~·~1<'1~4-1-1 j fc_ "".e:j•_~ there ought to be some critical point beyond which in-

.... .. ... 5 - e _. s~rance is profitable and below which it is u~profi:table. It 1: !i: j f !:i is the ·same point we recommended above. When the ratio l J! i! ta.l!ll falls below 2.0, insure. When it is above 2.0, do not insure. c! .. 1--ll!!l+-14+-liiil.. j J !~ ~ When it is exactly 2.0, there is, in the long run, no gain

• ·• ~r'.l j~-f or loss on the average, so yo~ can do as you please. "'~!1--J.....j....!.-w.i ... l'.:.j;;l ~ Lfl -~ i it _Actually, in this one instance, to insure a natural will re-:.

t--""~H-14-P"!~ l<'lH"' '"'Jll · duce the fluctuations in your capital. Thus if you havo ~J "'!!! ~~~~ ~~ f. ~·b . limited capital there is a minor advantage in doing so for

a ratio of exactly 2.0.

The same arguments apply, with greater force, to the second misconception, which is ·~nsure a good hand and do not insure a poor orie.''

· Strategy Tables There is a complication in giving the Ten-count strat·

egy. For the best possible play, we must vary our strategy as the ratio varies. For each ratio there is a corresponding strategy. Fortunately, all these separate strategies can be

· cqmbined into a single chart, given in Table 8.4 or, better,. jp Table 8.3. The player using Table 8.3 keeps a "runmpg ~·;that is, he keeps track of the cards as they ap~.

· \Vitll.his up, to-the-second information, he is able to play his hands. with great precision. Many readers of the first edition became experts with Table 8.3 (Table 5.j· in that edition). Tables 8.3 and 8.4 have our usual format, Witll one exception: some of the squares, instead of simply beo ~g ·shaded, have numbers. In the case of doubling down .and pait splitting, these numbers are to .be interpreted as foll~ If the ratio is equal to or less than the number in the square, consider the square shaded; thaf is, split the

,, ,;,-

iH DEiT·!f"HE 'UBA:tB'iL

pair or double down. If the ratio exceeds the number in the .)'Oll. most of the theoretic.al advantage. However, if you. square, consider the ilquare blank; that is, do not split the ,wish to add other parts ()f the strategy, the order of im­pair or- double down. There are two numberS· marked with portance is the .same as in the standard strategy: firsHe<Ull an asterisk(*). These numbers have an opposite interpre- hard doubling. then pair splitting, and =>Oft doubling last. tation. H the ratio is greater than such a number, ~ider Memorizing Table 8.3 in its entirety seems like too the square shaded. Otherwise, consider it blank. . much work. When I began play in the casinos with the

Referring to Table 8.3, read the minimum standing Ten-count strategy, I only knew approximately the standing number c~rt as follows. The soft standing numbers are . number and doubling down parts of Table 8.3. For soft the same as for the basic strategy except when the dealer•s doubling. I knew even .less. I played the standard strategy up· card is an Ace. In that case, it is 18 as usual for ratios until the ratio fell to 1.4 and then I doubled down on all of 2.2 or less. It is 19 for ratios above 2.2. The hard soft totals from 13 to 20 against cards 2 through 6. Tune standing number against an Ace is 17, as usual, if the ratio ·and time again dealers have smiled as though I were iJt. is less than or equal to 3.1 (but greater than 1.4). For sane when I doubled down on soft 2o and turned it into ratios above 3.1 it is 18. The hard standing number for a poor total. J;lut their smiles have disappeared when they up cards of 2 through to and for an Ace when the ratio is bust themselves. 1.4 or less, is read, as follows, from the chart. For a given The nmning count sometimes yields gains at the end ratio, ·shade all squares having numbers greater than or . · of the deck because the player can figure out frotn. it what · equal to that ratio. The lo\;VeSt shaded square is the correct the dealer's hole card is. I played a hand in which this made •tanding number. Alternately, one might think of all the a difference of $250.,I had bet '$125, I had glimpsed the squares. with numbers greater than or equal to the cqrrent other players' hole carets, and so when my turn came (I was ratio as the goal. Draw until your total equals or exceeds the the last to- pl~y), I knew twO cards remained, both Tens. totals represented by these squares. For example, if the dealer The dealer's. hOle card was therefore a Ten and only one shows a 4; the standing numbers are: 12 for ratios of 2.2 card, a Ten, remained unplayed. HI were to draw, it·woold or less; 13 for ratios above 2.2 but less than or equal to be offered, then the dealer would see it was the last c:.ro. 2 •. 6; 14 for ratios above 2.6 but less than or equal to 3·3· and as was customary he would pull it back and shumt.

Notice that the hard standing numbers against cards The dealer had a Ten up, thus he had 20. I haCl hard: 18; 2 through 6 have all dropped to 12 by the time the ratio If I did nothing I was sure to lose. drops to 2.0, which is the point at which we begin increasing I attempted to draw a card. As predicted, there. was our bet size. You may recall that the hard standing numbers only one more card, a Ten, so the dealer pulled it back and for cards 2 through 6 were also aU equal to 12 for the ahufiled. I attempted to draw again. When I saw the card Five-count strategy. As the ratio becomes larger, which cor- I almost fell oii my chair. It was a 3 for a total of 21 and a JeSp<>nds to a shortage of Tens, these hard standing numbers win. When the dealer turned up ~y cards he was astounded ~- They reach 17 against a 2 when .the ratio is above by my draw on hard 1-8. I was. barely able to explain it 3-9 and against .a 3 when is above s.o. . away by seeming dazed and saying that I had added my

The Ulost important part of this strategy to inC()rporate cards to IS. When the same situation ~rose half an ·hour into your plan is the standing numbers chart. This gives later, except that I held: ha:rd 1_9 against a sure 20, I did'

I~ ,. II4 BEAT T.HB DJL\1.ltlt J:j;~4¥ilirilng'Sttatigy'Baied on Counting Tens Its not dare draw seeking an Ace (to tie)~or a Deuc.e (to win).

When there are other players at the table and-you have not been able to see and count some of their cards by the

·time it is your turn to play, it is profitable, particularly ~ the end of the deck and when big bets are involved. to make inferences as to what these cards are and to use these inferences to modify your behavior. For example, suppose the count is (9,6) after you have seen the dealer's up card and your own cards, you are the fourth of four players, and the firSt three players stood {without hesitation) on their hole cards. Then the inference is quite strong that they each hold one and perhaps two Tens. Estimating their hole cards to be four Tens and two non-Tens, the true count, for pur­poses Of your draw, is perhaps {7,2). Hence, if the dealer is shpwing an Ace and you hold hard 14, IS, or 16, you should draw, rather than stand. It is even likely that you should draw on hard 17 in this instance!

Learning the Strategy Tables

During the writing of this book, I taught the system to several people ,with diverse backgrounds and interests. One· purpose was to see whether the card-counting methods and _charts of this chap~er, the key chapter in ~-first edition and the. most. difficult one, could readily be learned. Witbout exception, everyone was discouraged when he first saw Tables 8.3 and 8.4 and when he was told he would have to count the cards. Nearly everyone was surprised at the speed with which he learned. A couple of one-hour practice sessions, with someone else dealing, was generally enough to fix the basic strategy. Two more one-hour .ses­sions were enough to teach the simplified Five-count system. By then almost everyone gets impatient with waiting for the relatively infrequent favorable (Fiveless) situations. Two to five additional one-hour practice sessions, plus some counting practice, are usually enough to teach the players to c<>unt the Tens and non-Tens~ and to vary their bets accord-

. mgly (see· Table. 8.z), while they still play the basic · strategy. 'thesingle·remaining problem is memorizin& one

of the tables. We illustrate a method of memorization that has been worked out by experience. · .• . .

. The first logical step is to learn the parts of Tabltf~.3 that have to do with your big bets, namely th~se parts m­volving ratios of 2.25 or less (favorable decks). This in· formation is given in Table 8.4, an approximation to Table 8.3 in which the precise Ten-count strategy is played for big bets and the basic strategy is played otherwise. One might think of Table 8.4 as the basic strategy plus .some modifications.

Learn Table 8.4 in several stages. Begin with the standing-numbe:s table~ it i.s the m~

important. Like most tables in thlS book, th1~ t~ble 1S

more easily learned if one looks for patterns m 1t. ·For instance, against 9, the change in ratio between squares: is three tenths. Against 10, the changes are not very reg-

~ ular; in tenths, they are two, three, three, four. Note, however, that the standing number drops from 17 to 16 if the ratio drops to 2.2. A very slight excess of Tens is enough to change the standing number. In the Appendix we shall see· that in the basic strategy the hard standing number against Ten is generally 16 !or hands of 3 or more cards and barely 17 otfierwise. ' ' . .

The table next in importance is hard doubling d~ - · There are not many pair-splitting squares to learil.

However, the Ten splits are quite important. A player is ordinarily dealt a pair of Tens on about I hand in II. 'Jlte frequency. is even higher when the ratio drops. Note that Ten splitting (and also Four splitting) occurs only against the favorable dealer's up cards of 2 through 6. Once ag~ there is a sharp division betweep. 6 and 7· . · · The soft-doubling-down table is the least important. It

ean. be omitted or learned only approximately,. if desired. . .W'Jten most {or all) of Table 8.4 has been learned, th~.

· next sta&~:.is to get an approxinuite idea of the hard stand-

i t II j! .•

f~

'

n6 B E AT THE i> J! ALE 1\

ing numbers against 2 through 6 when the ratio is high. You probably already will have been ••leaning" in these situa­tions.

Further proficiency in the details of Table 8.3 are for the expert.

Rate of Profit

By this time the reader should be wondering whether the Ten-count strategy will win enough faster than the Five­count strategy to justify the extra work involved in learning

. and playing it. Table 8.5 illustrates the rate at which the Ten­count system wins. The ratios were calculated and recorded by a computer for Ioo,ooo hands, and ihe results are typical of what ~rises in actual play.

Table 8.5 is read as follows. If 25 cards are dealt off the top of a well-shuffled pack (and counted by the player), the player will find that a ratio of I .o or less will occur ·only o.I per cent of the time. A ratio of I.7 or less, but above 1.6, will not occur at all. A ratio of 1.8 or less, but above 1.7, will occur 14.5 per cent of the. time. A ratio of 1.8 or less· will occur··24.0 per cent of the time. (This last figure is obtained by adding all figures in the 25-card column up to and including the ratio for 1.7 to 1.8, namely, O.I + 0.5 + 2.0 + 6.9 + 14.5'= 24.0.)

Note· from Table 8.5 that a player counting Tens has an advantage of I per cent or more (i.e., a ratio of 2.0 or less) about a t~ird of the time. It turns out that the advan· tage ranges between I per cent in favor of the player and I per cent in favor of the house about a third of the time,

.. and· that the house has an advantage of I per cent or more about a third of the time.

· When more than one player is at the table, the Tens strategy loses somewhat in efficiency, but the decline is not nearly as sharp as it is in the Fives strategy.

Study has shown that a scheme for betting that gives very good protection against ruin while also providing a

• k Winmng Strategy ·Based on Counting Tens 117

'i --:"l 00

-00 00

_.., o...:

~ -"' 0 ..,.. N tf"a . ...,. .n \0 I""" co 0\ C ~ f"! ff'!

~ c;;...:...:_:..,:_;..;..,.:....;_;_.:rtNNr-1

'tn 0 ..- N """ V "" \0 .r-- ~ 0\ <:::! """': f"! "'! Q...:..;_;....;...;.-...;....;..,....;NNC"o\N

--------·~~- ~--~-

' II8 B E A:'l' T'li E DEAL B 1l

large yield is to risk a percentage -of your initial capital. equal to your percentage advantage. For example, with $:zoo and a 3 per cent advantage, bet $6; witlr a 10 per cent advantage, bet $2.o; with a I per cent advantage bet $:z; with situations less favorable than I per cent, bet a minimum $1.

In the test of this system which I described i_n Chapter s, modified proportional betting was used: I-unit minimum when my advantage was below I per cent, 2 units with a l per cent advantage, 4 units with a 2 per cent advantage, and so on, up to I o units with a s per cent advantage. Above S per cent all bets were leveled off at 10 units to reduce the possibility of frightening the casino. This pre­caution turned out to be insufficient in several of the casinos.

, In the present era, with the widespread successful use of our methods in the casinos, bets should be limited to ratios of I to 5 or I to 3, or even kept constantt

, There is a variation on the proportional betting schemes of the foregoing discussion that is mathematically sl:lperior to them but involves. a little extra mental work. In it, the player bets an amount approximately propor-

l!l.:.l tiona! to his current capital. The amount bet should be

equal to the player's percentage advantage. For example, suppose a playeistarts with $200. In a IO per cent·situation

,f he bets $20. Suppose he later builds up to $300. He would 1: - n_cjw bet $30 in a to per cent situation. [l · It is not necessary to bet the precise amount called t! for in any of our betting schedules. The results do not vary f~ significantly,. even Vtith considerable deviations from the

suggested amounts.

· Including Aces in the Count

Your results improve further if you adjust your bet size for an excess or shortage of Aces. When all the Aces · are gone, subtract 4 per cent from your estimated ad van-. tage. When the deck has twice as many Aces as normal,· add -4 per cent to your es.ted advantage.

A Winning- Strotegy Based on Counting Tent 119·

We usc some extreme cases to illusttate how Al;es can. --!feet your advantage. Suppose the unused cards consist exclusively of Aces and Tens. (I have seen the last eight' cards happen to be all Aces and Tens.) How much shoultl you bet? If possible. bet one half of your capital; sa~ tBe, other half for pair splitting or insurance. If the dealer has

, an Ace up, you can insure and prevent him from winning in case he has olackjack.

If you lost your insurance, the dealer must have a pair of Aces and must bust when he draws; therefore you will

· win your main bet. If he has a Ten up, he may have an Aco · under and, consequently, have blackjack. A part Of tho

time you, too, will hold blackjack and the game will be a stand-off. The rest of the time you will lose, but this is ~ -

. dnly time you lose. If the dealer has no blackjack, he haS a Ten under also. Then with (A,Io) you win; with (Io,Io)

. you can tie or, if any Aces remain; you can split your Tens · with the chance of a net gain. With (A.A) you can split · for a sure win if there are cards remaining. If the deck is · exhausted, splitting (A,A) ag~inst the dealer's (Io,Io) keeps-· your average loss fairly small.

The foregoing discussion is included to make it seem reasonable that w®n hands are dealt only from a collection · of Aces and Tens, the player is greatly favored. A detailed mathematical analysis confirms this.

As you become comfortable with the Ten-count strat-. egy, you can begin to keep tra,ck of Aces. When the c;leck has an excess of Aces, increase your bet somewhat over that· which is called for in the straight Tens strategy. On the· other hand, bets should be reduced when the deck· has a scarcity of Aces (is Ace-poor.)

· .You must be more careful than ever, when counting Aces and T~ns, to avoid letting the casino know that. you are keeping track of the cards. The story of Junior illustrateS _ what not to do. He was countmg both Tens and Aces and

. betting heavily. After. a while, he made a maximum (for ··. :him) bet of $200, since his count showed a. very favorable

11 r

l20 B BAT T lt E DB 'A'i:'l! ll sitUation. He was dealt a pair of Tens;; There was still orie Ace unseen. The dealer bad a Ten showing. but did not have a natural. · .

,.4. WiiJning Strategy Based on Counting Tens UI

· ~~sed cards. A negatiye figure means the deck i$ Ac~ ' pO<>r· and the player's advantage will be .reduced (perhaps

even eliminated altogether) by the negative correction. It is difficult to make these additional calculations while

playing the Tens strategy. I recommend that the ~eader who is countina Aces as well as Tens merely "lean "In the indi~ted direction" rather than make a precise calculation.

Junior bad seen the burned card and knew that it was not an Ace. Since there was only one unused card left ~ the pack, it had to be the remaining Ace. Furthermore, thts casino was at that time dealing the last card (the customai?' practice now is to withhold the last card and to place It The Remarkable Gain /rom Proper End Play · with the used cards and shuffie). Now, placed in a situation A few years ago, a now legendary figure, sometimes such as this, knowing you would get the last card, an. Ace, described as "the little dark-haired guy from Southern Cal· if you requested it, what would you do? Draw? Spht the ifornia'' (we purposely avoid giving his name), approached pair of Tens? - . . . a large and famous casino in Reno. The story goes that he - Junior asked to double down on }Jts $200 bet. Ptty-: explained he would like to play for large stakes--the J:ouso ingly the dealer attempted to explain to this "foolish free limit or more,- if possible-and that he wanted a pnvate spender" that he must have wanted to split his Tens. They game without publicity because he had tax problems. He argued until finally the pit boss was called over to settle the . · &et down carefully stipulated playing conditions that prob.­confusion. Now both the dealer and the pit boss pleaded aoly did not deviate from the spirit of the game. As a with him, in an attempt to "save him from himself." By bachelor "steadily earning five figures," he had accumulated this time a. crowd. of e~ployees as well as spectators had . appreciable capital and w~s able, no doubt, _to convince the gathered. Fmally, mfunated and exasperated from~: long_ .. casino that he had constderably more still. The house,. haggle, Junior yelled, "Give me the g- d--d. Ace~ ~e · thinking it had its usual advantage, was probably more card was dealt. It was ~n Ace. The amazed ptt boss patd . than happy to accept these conditions: .. the $400 and then escorted our ~ero to th~ door. Of course - · _ Although I do not know the detatls of the proposttiott, he was barred from further play m that casmo. · it is not hard to make a reasonable guess as to what they

The effect of Aces can be taken int~ account. rather were. From what I have learned through the grapevine, it precisely. The idea is to estimate the ~elative ~ce nchness · seems likely that what I call end play (to be described · of the deck and then to add a correction to the advantage below) was the main ingredient of this particular coup. If that is computed. from the ratio of oth~rs. t? Tens. For sc)~ the conditions for the game would be as follows. ~he example, suppose there are 26 cards rematrun?, all four casino's usual rules, as to drawing and standing, doubhng Aces flmong them. The average number of Aces IS two. The down, splitting, and insurance, are to be in force. In addi­average number can be computed from (26/52) X~· Thus tion, from deal to deal, the player may vary at will both the in this case the number is double the average so you mcrease number of hands he takes and the amounts he.bets. Further­your estimated advantage by 4 per cent. The general for-. more; the casino will deal down to the last card before ,mula for the correction to your per cent edge, due t~ Ace shuffling. At first sight, this set of conditions seems pretty richness or Ace poorness, is: [I3A/N-I]X 4 where AtS:the" harmless. But before we see what happened at the casino number of still unseen Aces and N is the total number of

-122 BEAT ,"'f Jf:B D·B·A'&Jill· ~5 WmningStrottgy BaSed on CoUIIting Teill U3

in question, let us examine play under these C91lditions We now. return to the story of what happened. at tb.e more closely. . · · . casino. The little dark-haired guy is said to have played for

Imagine, first, that seven cards. all Aces and Tens, re- several successive nights. The first njght he won. ten or . main to be played. What happens if you decide to take fifteen thousand dollars. Then, on successive nights he lost exactly three hand.s? Then when you pick up your three and won similar amounts. When the casino became ao-

. hands you find each one consists of either (A,A), (A,IO), customed to these. large surges and when it was clear tb.at or (Io;Io). The dealer, however, receives only one Ace or they were primed to hang on even though they were 'Well one Ten, and since the deck runs out, he must shuffle before behind, he began playing to win. Hour after hour the money getting his next card. You now have thre_e powerful hands piled up. It is said that somewhere between $40,000 and facing him, and besides, he must draw h1s next card from $86,ooo,the:casino "snapped" and called off the game. Th~ a deck that is poor in Aces and Tens. Generally, all three latter figure is_supposed to be the authentic one, but there of your bands win. The advantage ~requently is IO to. IOO ~re varying reports, probably because there were only four per cent in these situations. Money 1s won at a truly dizzy· witnesses to the game-the player and three casino people. ing rate. . , . His idea of no publicity paid handsomely. During the next

, Here is an alternate variation. Suppose there are five two years, the little dark-haired guy sold his proposition to cards left, mostly Aces and Tens, and that you decide to other Nevada casinos. He was finally barred throughout take five hands. Then you get all five of the cards in this the state after he had won· more than $250,000 •

. favored group, and the dealer gets none of them, for he Of course, since nearly every casino in Nevada now runs out of cards and must shuffle befor~ dealing the first ... refuses to allow end play, the method is nearly dead. Many card ·to himself. If you now get a Ten as your first card ·. casinos are so intimidated that they will not set up private

. this gives you a 15 to 20 per cent advantage; starting with games; But keep it in mind. It paid off in the Puerto Rican an Ace. gives you a 35 to 40 per cent advantage. casinos for several alert readers of the first edition.

If it happens instead that the end of the deck is very poor in Aces and Tens, this too is to your advantage. Suppose there are twelve small cards left. Take five hands and- place very small bets on each. All. twelve cards are used up in dealing the hands, and since mostly small cards are involved, some cards will be drawn, forcing a shuffle. When the deck is shuffled, twelve small cards are -missing and therefore the new cards will be dealt from a residue whose ratio is 24/16 or 1.5. A few cards will be drawn, but, although the ratio fluctuates in individual situationS, on the average it will also be 1.5 at the end of the round. Thus by taking five bands to keep1 the small cards on the table during the shuffle, the player has created a series of highly favorable situations.

l

, -,-, '''-Beathig the Canno Countermeasures

9

Beating the Casino

Countermeasures

There is an abundance and variety of casino counter­measures. We cannot discuss all those in use at the time of this writing. Furthermore, as the players become more re­sourceful, so do the casinos, changing their countermeasures to meet new situations. The discussion of the principal countermeasures should make the player versatile enough to cope with new developments. •

Slwffle up When I wrote the first edition I had the mistaken im­

pression that the casinos could stop a system player by frequently reshuflling the deck. Let's consider the extreme case in which the game is played with one deck and that one

- cleck is reshuffied at the end of each and every deal. Suppose

- • An interesting book in connection with casino countermeasures Is Inside Nevada GambUng-Adventures of a Winning System Player by Glenn L. Fraikin, Exposition Press, New York. $4.00. It tells of Fraikin's being barred, harrassed, forcibly ejected and embarrassed. You can generally avoid these problems if you are careful.

•t24.

~, a player is alone at the table, playiug one hand at.a time. ·The best you can do, it may seem, is to use the basic strat­

egy. Against the typical rules of Chapter 2, you will have an edge of about 0.13 per cent. For practical purposes, the game is even. ·

But suppose now that you take six hands, playing the whole table at once. On your first hand (if the cards are dealt down), you have seen only your hole cards and the dealer's up card. The situation is the same as before. But when it is time to play the second hand you have already seen the cards for the first hand. If you use the playing strategy for the complete point count or for the Ten count, you are able to play this hand better than if you stuck to

_ the basic strategy. Naturally your advantage increases. You will win at an average rate that is greater than 0.13 per cent.

Things are better still on the third hand, and they are very good indeed, by the time you get to the sixth and last band. You. ·now have seen at least 13 and probably 18 or zo cards. If you split some of the earlier hands, the figut:e could run much higher. Your average advantage, due to improvements in playing strategy, is roughly 1 per cent. This is about the same as the casino's edge in baccarat. You can make a big bet on the sixth hand of every deal. In

. fact; you could make a small bet on the first hand, a little ·larger bet on the second, and so on. The largest .bet would be on the last hand.

Sitting in the last seat at a crowded table, you can make a .big bet" all the time if you generally see most of the cards played ahead of you. With cards dealt face up, as they are in Puerto Rico and in a number of the Nevada games. you see plenty of cards. There, in· fact, from every seat you can see at least the players' hole cards, for an edge of 0.5 per cent or more.

In a casino where the cards are dealt face down, it may not be easy to see most of the cards before your turn

U6 ll BAT 'l H·E D E AL. II'& ., E0' ;Beating fbe ·(!dn() COUntermeasure# ··127

· to play. In fact,. some casinos now use shills and have them '· , · • know .yet that this first bet of yours is going to turn out conceal their cards. The dealer rakes in their cards without , to be "large." ever showing them. You can prevent the shills from play­ing, and make sure you see everyone's cards, by filling the table with your cooperative friends first. (They can play the basic strategy, have fun, and break even, if nothing else.)

.Dealers Who Count

After Beat the Dealer became a national best seller ~d several hundred thousand people read it, some casinos had their employees read it. Then some of the dealers. were taught to count. There were two ideas behind this. First, these dealers could help spot count players (more on this later). Secondly, these dealers could shuffle up when the deck went good but keep dealing if it went bad.

Suppose a dealer does this. What do you do? The only solution that I have found, besides looking for another and better game, is ~o sit in the last seat. Bet heavily on the

. 6r$t deal and hope you win with your I per cent edge. Bet lightly otherwise. If ·you pull ahead (and the odds favor you, but not tremendously), your dealer may give up his tactics.

Strike When the Deck Is Hot

We have just seen how dealers can shuffle away the good cards and )Feep playing with bad cards. You can do very much the same thing. Suppose you are about to enter .a game. Stop. Don't sit down. Get your count first. And be sure you have some casino chips in your hand. Now wait until the count is good. Then pull up a seat and tOss your large bet down. This way you always start your game with a good ~eck. You are striking when the deck is hot.

This is a very nice way to pick up extra change when you are on your way through your hotel's gaming areas. The dealer generally doesn't reshuffle on you: he doesn't

Punishing Fakes .Many dealets reshuffle when you increase your bet. A

few of these are legitimate Ten counters and know that tho deck has gone "good." :aut most of them are not. They just reshuffle to be safe against system players. And a good· many of the ories I've played against are out-and-out fakes. ·That is, they pretend (either to the players or to their pit boss) that they are counting Tens and non-Tens. When a big bet goes out, they reshuffle. But actually they have no idea what the count is.

When you find a fake, stay with him. He is "money in the bank." Here is what you do. Start by making a big bet (3 to s units). When the deck goes bad, increase your bet, and when he reshuffles, pull back the extra amount. · When the deck stays neutral or goes good, keep your bot the same. The result is that the dealer shuftles away tho lr.id decks. You always play against av.erage or good decks.

·You have a consistent advantage and probably wiD.

Multiple Decks There'has been some increase in tho number of black•

jack ~~·played with either two or four decks. Some casinos believe that it is much more djjicult to count several de<:ks. You will find· it a little more work. But if you aro lairly~ with one deck, you shouldn't have much trouble.

'l'hC most serious drawback to two or four decks is that the· advantage for various values of the high-low index or ·the Ten-count ratio are not quite as large. They· are_ roughly 0.4 per cent less for two decks and o.s per cent less for four decks •. Also, the deviations from normal deck com­position are smaller. So there are .somewhat fewer favor. able situations, and the ones that occur are less favorable. If the casino were to reshuftle two Qr even four deCks after

B.~ AT T H B DB Al. Ell .·~ r~ ~tlng th~ Cdno Countermeasures

every deal, even sitting in the sixth seat wouldn't help. But the casinos won't do this. The game would Slow down to such a crawl that bored players would keep drifting away.

Multiple decks have one big advantage for the player who strikes when the decks are hot. When two decks, or better fou.r decks, go good, they stay good much longer. So when you sit down to a good situation with a big bet,. things are likely to stay good for a while. This partly camouflages your actions.

Rules Changes

The gambling world was electrified on April I, 1964-The Las Vegas Resort Hotel Association announced that the rules of blackjack were being c;hanged [34]. This was the first time in history that the rules of a major casino gambling game had been sigpificantly altered. The rules changes were made to combat the winning blackjack sys­tems (primarily the Ten-caunt) introduced in the first edition of Beat the Dealer. In the words of Gabriel Vogli;­atti, a spokesman for the Las Vegas Resort Hotel Associa­tion [45]: "In the last 15 years there hasn't 'been one plane ~t landed without at least one person in possession of a system. This guy [Thorp] is the first in Las Vegas history to have a system that works." .

The specific rules changes forbade the splitting of Aces and restricted doubling down to totals of hard I I only. As you see from Table 9.I, the effect is to reduce the basic· strategy player's advantage by roughly I per cent. The re­duction is somewhat larger in favorable situations and somewhat smaller in unfavorable ones. Thus a situation which was a I o per cent advantage, for example, is reduced to a littl~ le~ than 9 per cent. Clearly the point-count and Ten-courit methods will still uncover favorable situations. But there won't be quite as many. and they won't be quite asgood. . .

. How did the rules changes affect the system players?

. th:e good players went right back the next day arid COn-· tiriued w.inhing. True, their rate of winning was cut some­what. But it wasn't enough to cheer up the c~inos.

. Against casinOs with unfavorable rules, the betting scale should be adjusted downward. Perhaps the simplest method is to decrease all large bets by one unit. Alternately, if you •Use the point-count system without. strategy. refine­ments, you can simply decrease your initial count by one point per deck in use. For example, in a two-deck game with poor rules (such as is played in Puerto Rico an_d in some spots in Nevada), start with a point-count total of -:a instead of o. Then play precisely as before.

. TABLE 9.1. The EOect of the (Temporary} Las Vegu ·Rules Chtmgu.

Basic strategy Basic strategy old rules new Las Vegas rules.

Player's overall .0013 -.0089

advantage•

Player's A# -.0847 -.0853 a!fvantage A• -.3603 -.3607 when 2 .1011 .0888

· dealer's 3 .1376 .1219 up 4 .1844 .1626 card s .2369 .2073 is 6 .242S .2121

7 .1464 .1374 8 .0547 .0490 9 -.0437 -.0471

to• -.1706 -.1717 tO# -.1032 -.1044

# Excluding the possibility tb&t dealer has a natural. • Including the possibllily that dealer has a natural.

The casino operators had read as far as Chapter S (Ten count), but they had not read as far as ChapterS (Rules Variations), which explained how to counter their rule5 changes. Further, I predicted when I wrote the ~ edition two years earlier that rules changes would be trted and· that they would not be effective. But the casinos hadn't

130 . B EAT T H ~ DE A L 'B~ ~ ~Hflli.~ the Cll8liltJ C.ounler'metiSW"t!6 .. 131 .

read that far yet. It took them two or three weeks to throw . c·option·ofsurrender. At any time, unless the dealer is 'Show­in ~ sponge and go bac~ to th~ old .rules •. What was ~ . big an Ace; the player ma'y .... surrender" his band, retaining trouble? A sharp young JOUrnalist named Jude Wanniski · one ball of the anioUnt of his bet alld losing the other half. explained it very clearly in a by-lined article in the Na- Braun estimates that with best play surrender gains· about donal Observer: 0~15 per cent for the player. This is more than offset by the

Overnight, play at the Las Vegas blackjack tables player's .~ of about o.S. per cent ~?ecause of the restrictions 1fen off. In fact, play at all the gaming tables declined on dou61ing down. as the ftow of tourists into the city diminished. Casino TABLa 9.2. Approxlrnllle E6m of Cmnmon Ruin YariDtiou employees, whose income depends in large part on the tm Play~;·, Advantage Wilen tile Basic Strategy I• Used.

number of tips they receive, began screaming that the Player•• Joss or pia new blackjack rules were a bane to the industry. Rules variations (in per cent)

First one casino, then another, quietly scuttled · forbid donbling down on

the new rules. By last week, Las Vegas gamblers threw bard II

in the towel. They admitted they'd rather have all their ::: !0

business back, even if it meant putting up with the liard s system players. all soft totals

all totals after pair splitting

Rule$ Jlariations allow doubling down on any three cards allow doubling.down on any number of cards fOil!' decb

; two decks dealer draws to soft 17 dealer draws only to soft 17 with Ar:t: up dealer drawing to soft 17 Is op&ional further splitting of. pairs

all pairs, one cleck all pairs, two decks .all pairs. four decks

-o.S9 -o.s6 -o.l4 -o.oo -o.t4

. -().13 0.19• 0.20•

-o.$1 -o.JS -{).20 -o.23

-(0.23+)

0.053 .08

0.11 (est.)

In fight of the attempted Nevada rules changes and the considerable variations in blackjack as .played through· out the world, the reader should be prepared to estimate the effect of any deviations from the typical rules of Chapter 2. You can do this by consultil)g Table 9.2. To the basic· st.rategy edge of 0.12 per cerit for the. player, add or sub­tract the correction indicated by the table. for each ~les variation being used. The final result is the overall player advantage (if positive) or casino advantage (if negative).

At various points in the book we discussed tlie varia­tions in England, Puerto Rico, and Nevada. One rule, which is apparently peculiar to the Far East, has not :been con­sidered. It is called "surrender."

all psirl. except Aces. two decks all paq, one deck. and unlimited draw to split Aces all pairS. two decks, and unlimited draw to split Aces

and double on hard 11 only all pairs except Aces

.04

.037

.06

.OS

• In the Far East, particularly in Macao (a Portuguese colony that is a short hydrofoil ride fro~ Hong Kong)· and Manila, the blackjack rules are like those of Chapter 2 ex­cept that doubling down is permitted only with a two-card total of u. But the player is offered the additional-strategy

· drawing any number of cards to split Ac:Cs no splitting of Aces · DO splittiaJ of Aces aad DO doubling down on ioft 12 forbid pair splitting two-to-One pay-off for blackjack Puerto Ricaa rules, one deck

two decb Surrender (Ma~o, Manila)

0.024 0.14

-o.t6 -o.l6 -1).46

2.32 -o.71 -1.()4 o.ts (est.)

..._ :~- ~~~~-~ -

Dll .~:t ~~~. '!#!tlllg tlr8 Cilsino Countermeasures 13~-~,, ·. · impossible to get a reasonable game. As a last resort. I grew

a ~rd for the summer and got used to contact lenses. Then fapent four days in Las Vegas, Reno, and Lake Tahoe. In ~the beginning I wore the full beard. My usual glasses wero ~aced by conta~t lenses and s\veep-around sungl~~· After I had won for two days in Las Vegas, players w1th beards began to receive most unfriendly treatment. Twa COI.Jlpanions and I went on to Reno and Lake Tahoe. We walked into a crowded casino on the north shore of Lake 'fahoe (~ed for gangland connections). T4ere were no sea,ts at the blackjack tables. Then a boss looked up ~d .,aw the bearded apparition. His jaw dropped. He called 111_ a dealer and opened up a table. I sat down. Two toughs trotted up and plopped down on either side of ~e. .

My companions (I NEVER travel alone m Neva~) thought I was about to be hurled from the casino ••

Camouflage . The casinos hav.e become painfully aware of tlie thou-.

sands of basic~strategy players who do not lose money. Even. worse, they are infested with hundreds of good Ten-~unt players who carry off money. These good players, particu­larly, have a problem. When they become known to the casinos, they find shuffie up, or the dealer hiding cards~ Sometimes they are barred (asked to leave the casino). Or they run into heavy cheating (sometimes undisguised!).

Clearly, if you want to be left alone to play a good game of blackjack, you must disguise your pta~. First, do not start at the beginning of every fresh deck With a small bet. With one-deck games and typical rules (so I have a .. slight edge off the top), I get best results by betting large off the top of the deck almost half the time. Of course, if you glimpSe a burned or bottom card, you can choose these times more effectively. Dealers think, "If he's going to bet , big off the top of the deck, shuffie up is a waste of time.",

· In casinos with two or more dealers or less favorable rules, you ought to bet big off the top much less ~requently.

The. size of your bets is also important. I have played against dealers who thought it was natural to bet a $5 chip or a $25 chip but not natural to carefully bet$_?, $10, $15,, $20, or $25, depending on the deal. So I bet $5 except when things were moderately good. Then I bet $25. In ~erto Rico I bet $1 while waiting and $50 in all favorable situa­tions, since nothing bothered them! You mtist learn what _is best in your situation. A ratio of 1 to 5 or even I to 3 m bet sizes might be a good beginning while you e~plore the sitU;&tion. A ratio of 1 to 2 is acceptable. Against: one deck and typical rules, a ratio of I to I is acceptable (all bets the same!) if you sit in fifth or sixth seat at a full table and can see most of the cards before your tumr

Disguis~s

As a result of the intense publicity and the wide distribution of my picture, I have lately found it ·nearly

. The two toughs, the dealer, and I _played on 1n com· plete silence for ten minutes or so. When I was absolutely sure the dealer was a cheat, I walked away. The two toughs immediately trotted off. The dealer closed u~ the table and_ went off to wait for the next problem. A mmute later, an was. as before. The throngs of happy, fun-loving tourists went on with their merrymaking. Not one of them had been the least bit aware of the little drama that had taken place before them.

It looked as if bearded players were through. But it had taken me four weeks to grow the beard and I was determined to use it one more time. After further adventures we-arrived in Reno.

At about 3 A.M. I began to play at a club in downtown Reno (about 50 or 6o miles from the scene in Tahoe). The club was well known for its fair rules and for dealing down. to the last card. I always collect a few hundred dollars from this club whenever I am in Reno. whatever befalls me else­where. This club doesn't seem to be "in" with the other ~rs. In particular, I thoJJght they would not have heard in adv~ of the bearded threat.

'~134 BEAT TRB ~BAL~;. T~"!r th6 .Cflsbro CountermetlSIIIW • 135 · ~8foiui~ sunglasses and wore only my contact lenses. I ~cOmbed my hair son1ewhat dilrerently. I was ready for tho ·.:My table was· !Ull. There was ~ opp<)rtunity tor enc!

phty. :But I kilew fr.otn past experience that this casino wouldn't allow that anyhow. I played on, b~tting Ss to $25, tbe maximum size and spread of bets that I felt would be tolerated here. I won steadily. The casino pU~ me h"berally with strong drinks. I gave the casino people the impression that l'was .careless, woozy, and a bad player. "Doubling down on Five:...:...ridiculous; and winning, too. What can you do about luck like that? But it will run·out. It's bound to."

But the cards ran well and instead of Winning an ex· . pected $25 to $75 in an hour, I piled up $300. So the casino personnel paraded by, one after another. They· SC{Utinized me, watched my play, and examined my face. At the end of an hour they had all examined me. They had bad enough. A hostile brunet pit boss (the same one who

. saw me practice for eight boors in the same club when I ·first played in Reno; see Chapter S) now took over as .dealer. She smiled pleasantly. She dealt_ the first hand: t;lackjack to- herself. l was .watc~ng for the second deal (see Chapter 10) and caught it, and ~. got up. Purring Cattily. she invited me to stay on and play. .

During th~ previous hour of playing, I had a single dealer. (This casino leaves the dealers on at one table interminably, but this firstone couldn't cheat). She was a youno blond, lonely, emotionally troubled, and in search of a ma~. When she glimpsed the stack of $100 bills in my waliet, she became exceedingly friendly. She was disap­pQinted and upset to be so abruptly parted from her fish.

The blond told me (wonderful fellow that I was) that she. bated beards and "everything.would be all right if I .shaved it off." So it occurred to me that I could later make a 'perfect test-of my disguise.

The next evening I shaved off the ragged full-Jength beard. The only clue to my having bad, it was an unusual paleness about certain parts ·of my face. Instead of careless

. casual clothes, I now wore a suit. I removed the sweep-

. :test. . - I called for my companions, who had nevet seen me ·sans beard. When they answered the door, they did not cfticognize me. Instead Of an aging unkempt JOan of forty· 'five. I was a springy, crew-cut young executive of twenty­five. out for an evening on the town.

When 3 A.M. approached, I left the game I was in an,d went to the caslno where the blond dealt. There she was, at the same table, and there was one vacant seat. I sat down and began to play. There was no glimmer of recogni·

"tion on her face. Again I bet Ss to $25, with chips I had been careful

to purchase in advance froxn the cashier, with a few care­tully preserve<f small bills. There was no evidence of the previous evening's stack of hundreds to give ·my dealer a due. I did not talk, I only gestured (not too unusual a behavior paltem). When the. waitresses- came to ply me with· drinks, I whispered •milk." The dealer could not hear . my ~ce. After two milks, the waitresses didn't bother me.

So far, my disguise seemed to be worlOng perf.ectly. But chance soon put it to a real test. The cards ran weU again and I began to pile up. winnings at almost the same rate as the evening before. Next, the player on my left (I was in fifth seat, my usual choice, since the casinos expect "S}'stem players to be in sixtb seat) turned out to be a cheat!· He would pile his bet carelessly ~n the layout. Then he would check his hole cards. Then he would try to sneak some extra silver dollars onto the layout if he thought he had a good hand. If he thought he had a poor hand he would try to remove some of his bet. The dealer didn't know what to do. Then the same parade of casino person­nel filed by to examine the cheat.

· · The ·bosses watched him play for a few minutes. He was such a bad player that, even though he was cheating,

-- .. ' .

-........ ------~~---~~--·· ~ ·~~· .' ."13v BJfAT TBB bBALB&. --~ !'! ~.~ · ·:- •·

he was barely breaking.·e\'en. They ga\re ordera to Ieaw . . ·A~O~TIC BLACKJACK~ .. hiiii 8Jonel ' ~ · · · '· ·. ~~ ~ · Is a gam.ng machine manufa~red in the sfclte of:

. Both the cheat and I had been carefuUy scrutinized bJ Nevad~. the same. g~g that had studied me the previous Dight. ~o '"AUTOMATIC BLACKJACK" is electronically oper- ·

· one recognized me. But after an hour, and another $300, oted us1ng a fuU simulated fifty-two-card deck-. their patience wore thin. As on the night before, the~chea'" a_!Jtomatically shuffled and ·dealt. Play is initiated ~g dealer eame in. I left. . upon the deposit of a coin, or coins, in any or all

~ Disguises do work. They are a lot of trouble, but they denominations of quarters, halves or silv. er dol. Iars can be fun, too. ~

I'M Automatic Blackjack Mochlnu One of the most interesting casino countermeasures is

the recent introduction of automatic (electronic)~ blackjack machines to replace the dealers. The machines have been

• or are being tried at several casinos. The text of an adver­tisement on pages 137-8 gives the details.

Let's use our. meth~ to analyze the form of automatic: bJaC?kjack presented in ·the ad. . . Notice iirst that the machine deals a one-deck game

and reshuftles after every deaL This goes a long way toward aegating the gains from card counting. Our fiist impulse is

·.to sit in the last seat and use the.~ we see to get an edge (as explained on pages 124-126). But there are only four IC&ts. So the advantage gained will be considerably less. probably about 0.33 to o.s per· cent.

. The next thing to notice is how the rules differ from our typical rules. First, doubling down is restricted to hard totals of 10 or n. Secondly, pair splitting is not allowed. ~Both these restrictions on the player increase the house ·edge: By Table 9.1, the lOSS due to DO pair splitting is. 0-46 per . cent, no doubting do~ on hard 9• hard 8, and the soft totals, ~ costso.14 + o.oo + 1.1401'o.a8percent, and there is ao.zo .per cent loss because the deal4r hits soft 17. Thus the usual basic strategy player advantage of 0.1 J per cent seems to be reduced by 0-46 + o;z8 + o.zo or 0.94 per cent, for a ·net casino edge of o.Sr per cent.

••• to a limitation of five coins of each denomtnation.

The sequence of the game is in accord with all stand· ard 118LACKJACK" or 1121" games.

The player has the option of "hitting or stailding" on the hand and score dealt. All SCORES ARE NUMERICALLY INDICATED IMMEDIAmY.

The "dealer'' will continue to draw cards autOmati­cally until. it has a score of 17 or more-at this tim(. the score of the player is compared to the "deOlet"' and payoff is automatically made CICQ)rding to the score and the amount of the beJ' made •. If the player P~akes a "BLACKJACK" he receives double his ori­ginal bet ~nstead of the usual one and a half payoff..

"AUTOMATIC BLACKJACK" is fully approved by the Nevada Gaming Commission. All card dealing is ab$olutely uncontrolled and based upon chance.

Any player, after receiving two or more cards show­ing a score of 10 or 11 may elect to "DOUBLE DOWN." Simply press the yellow flashing light but­ton, increase your bet in the same denomination up to double your original bet, press the ·"HI'P" buHon,

• and you will receive one card only to complete your hand.

111! AT T11 B b B A"L lit

• lnsert coins on 11MAKE BET.11

• ''HIT or STAND'' when buttons light. (Automatic stand after 1 0 second period).

• Deafer stands an 17 or over (must hit soft 17);.

· · • Aces count eleven or one.

• "BLACKJACK'' pays double, "PUSH" pays money ' back.

~ Flashing light indicates 1 0 or 11-press button and immediately increase your bet, one card only will be dealt to your hand.

• Nu~erical score supersedes card display. 0

0

• All coins rejected when game is in progress.

NEVADA ELECTRONICS INC. hno, Nevada

Notice also that a player sitting in the fourth seal-would ordinarily have an· occasional profitable insurance bet, pro-

' vided he knew the hole cards of at least one of the other · . three people at the tabl~ (The player must see at least five cards of a complete deck before there is a possibility of a ··. ratio below 2.00 3Dd a profitable insurance bet. Two play-

. ers' hole cards and the dealer's up card are sufficient.) But this. little advantage does not apply here, for insurance is aot offered.

. There is one more rules change. If the player ~ · blackj~k and the dealer does not (the ad doesn't say this but presumably it means it) he wins twice his original bet

· instead of the usual one and a half ~es it. Cleatly this helps the player. How much? The chance (in a one-deck game) that a specified player will get blackjack tmd the dealer willoot is 4.649 per cent. But the player now :wias an extra so per cent each time this happens. So the •

, gain to the player, due to this rule, is SO% X 4-649%, qt 2.32 per cent.· So the player's gain with the- basic atrate&Y.

· is increased from~-o.8• per cent to+ 1. 51 per cent. This is a tidy rate of profit for each and every hand. So

a basic-strategy player should make a steady profit. One drawback is the fact that the machine will accept bets only up to a total of $8.7s per hand. But at 100 hands per hour one should expect an average profit of $8.7S X IOO·X •·s•% or over $13 per,hour!

The ad reproduced here. was picked up in Reno and in Lake Tahoe. Recent information from Las Vegas is ~ the machines there all pay I .s to I for blackjack. Further. this is rounded off to the nearest quarter, against the player. For example, a 25¢. bet reCeives only 2S; for a natural, not l7~ ;._But a so; bet·receives 7S¢ for-a natural. Cle8rly

- the player should.only bet in multiples of so; against suCh a machine. ENen so, the edge is now an uufavorable o.81 per cent.

Warning: MachUles ·wear out or becOme defective. Also, an unscrupulous peison. could set an automatic black•

140 · 11 E AT T BE J) E ALB Jl·

Jack machine to cheat. Before 1 would play such a machine, I would tally 1000 or so dealt cards to see if appropriate proportions of each type of card were being dealt. I would further keep track of the percentag~ gain that a pla!cr experiences. in xooo hands or so. You are better off domg

. this by watching than playing.

The Paper Rouie Technique

The questions I have been asked most often is, .. Can a player using your system still win in Nevada, after all the publicity and reader successes?" and "What is the best way tO do this?" Y cs, you can go on winning in Nevada. ~ere is a technique which is very effective. Before you begm a session of play, set aside from your stake an ~ount equal to about 5 large bets-perhaps 20 or 25 units. Play untn you either lose this wount or win this amount. Do not show the 20 or 25 units at once, but instead buy chips from it as .required. If you play for an hour, stop anyhow. / . The point is this. If you play no more than an hour, you are less likely . to attract attention or be remembered. If you lose no more than 20 or 25 units in a session, no one ·cheating dealer can ruin you. If you win no more than 20 or 25 units, you are less likely to be acted against'as .a po,ssjble threat because of your wi~nings. "~e ~ just a little lucky this time but we'll get hlm next t1me.

Remember your dealers. Return to the dealers you do well against. Avoid the dealers you lose 20 or 25 units t9. This way you never get cheated . twice by the same dealet, You may lose legitimately and then avoid an honest dealer. This can't be helped. .

The method is oversimplified to Ulustrate principles. Make appropriate modifications for your situation. After a session, you should go to another casino. When you are . using the method, you stroll from casino to casino, collec­ting money, much as a newsboy for a paper route goes from ~r fo door.

~10

How to Spot Cheating

Blackjack dealers in casinos are often fantastically skiltM with cards. Before I became seriously interested in black· jack, I and everyone I knew ·believed that although dealCta could cheat if they so desired, they did not cheat. The well­_pubUcized argu~nt is- that the casinos enjoy a natural advantage in the game and wU1 win anyhow. Why should a casino risk possible exposure and the resliltant bad pub­Deity, loss ot customers, and perhaps even loss of their gaming license? Alte~atively, why would a.crooked dealer who works for an honest casino risk losing his job in order to line his pockets?

We might answer this with another quesdon: "lsn' there~ corruption in political life and in the busf.. ness world? And isn't it usually for the same stakes (money) and with comparable risks (loss of position or of ~ to operate, bad publicity, etc.)? Why then should legalized gambling be JllOre immune from dishonesty than 'Jegalizecr business?" ' ·

In recent years it has become public knowledge~ .141.

-~ BEAT TBR, J>E~L~··'·· · -ent gan~er groups (such as the. Mafi~ and~heCosa. Nostra) are behirid a number of the casmos, u~cblding so~e of the largest ones. Pulitzer-prize-winning writer Ed Re~ and coauthor Ovid Demaris detail the grisly story in The Qreen Felt Jungle. Mob control in Nevada reaches every· where, even to the highest levels. 'The book is "~ust" read­Ing for anyone who visits Nevada. Other details can be found in [17~ 24, 28, 33, 46, 51].

Reid and :Qemaris tell how casino mobsters, in vari·· ous falling-outs, blew each other apart. Not only do t~e anQbsters fauen their tills with "legitimate" profits from theU" cashios; they also take'money "off the top." That is •• they habitually report a ·smaller gross than is actually rece1ved. An account of these practices was given by Wallace Tur· aer in a series of articles bi the New York Times, Novem-ber 18-22, 1964- .

. Now let me ask you again, are men of the Cosa Nostra. who bribe publit officials, who steal money off the top~ w~ help to tinance their ·rackets (dope, prostitutio~ and smuggling) with their casino profits, who comnut murder to settle their differences-are these men going to atop ·short, at a little cheating at cards? Perhaps they are good sports. Perhaps they don't want the extra P,rofits that «nne frooi cheating, especially when the "suckers" won't have a chance in a million of spotting the cheating-and who, if they did spot i~ wouldn't be able to c:to · anything about it.

Before we continue, let me try to set things in perspec­live. In the great majority of blackjack· games there is no Cheating. But there is enough cheating (I estimate the aver­age player wiD face a cheat perhaps s or . 10 ~r cent of the time) to make it a serious problem. It can mean the difference between winning ~ losing~ So we must under,;. st8nd how to minimize our losses from. cheating ..

I was originally naive enough to swallow the argument that blackjack as played in the casinos is generally honest.

ti took painful ~nat experience to convince me ~­wise. The first such experienee was not long in coming.· ..

The Knockout Dealer. A Stubbom Expert Wastes $2o,ooo in a Single Night ·

One afternoon during the test of the system in Neva~ Mr • .X went off alone to play the Tens strategy. Early tho next morning he told me he had played steadily for eight to ten hours at one of the large hotels. He made house:.limit bets-of Ssoo in sufficiently favorable situations, and at the end of a few hours he had won $13,000. At this point the hotel brought in its "knockout" dealer-a cheat emp!Qyed specifically to dispose of big winners. ·

Her method of cheating was to peek at the top catd when it came time for her to draw to her own hand. If she · liked the top card, she dealt it (honestly) to herself. If~. did not like it, she dealt herself the card just below the top card, ~y referred to as the "second." Even tbotlgh she did liotknow what the second was, it was a better risk tban the top c.ard about half the time. . .

Mr. X stubbornly played on, hoping he could beat tho cheat anyhow. The cheat faced him for forty minutes at· a time. Then she rested for twenty minutes during which time the game was honest again. Mr. X hoped to win more iD the twenty-minute sessions that he would be cheated out of Jn the forty-minute sessions. But he made a fatal error. He continued to bet on a large scale against the knockout dealer, ratbei' than reducing his bets to a few doltars B!ld .waiting until she had to be relieved. Thus he lost too heavily against her. After a few hours he had lost back $:l.O~ cancelling his $13,000 lead and putting him $7,000 in tho

. hole. When Mr. X complained to the owner, a person whO was t:esponst"ble for the· operation of several latge ca&~ he explained that a (mythical?) Texan had won $17,000 the day before and the casino could not afford further losses. . .

v· -~-----X44-

The Queen of Heart. · ·- Anxious to learn to protect myself from being Cheated

I visited this particular casino the next morning, accrim· panied by Mr. Y. Mr. X had described the knockout dea!er to us: a thinnish, grim-faced woman of about forty, Wlth black hair that was beginning to gray. . ·. .

I purchased $1,000 Worth of chips from the cashier aDd seated myself at 'the nearest table. I bet $30 and thea thO dealer dealt a card to me and one to herself. As she dealt the second card to me, the pit boss rushed over, stopped her, took· the cards, and called over a dealer ·10 replace her. The new dealer was a thinnish, grim-faced WQDWl of about forty whose black bair was beginniitg to turn gray. .

I received a pair of Eights and the dealer bad a Three showing. I split my Eights and got totals of 20 and 18. The. dealer's bole card was a Ten. Mr. Y and I watched as the dealer, holding the deck edge. up, bent the top card back slightly to .see what it was. We saw it too: the Queen of HiaftS. That \vould have busted her so she dealt the second card to herself. It was an Eight. giving her 21, and she · raked m our $6o. Angrily Mr. Y spelled out for ber what she bad don~. She reddened and looked down. She said DOtbing. and pretended not to bear our loud, angry pro~ts. When the pit boss came over, he also sllowed no r~~n. There was nothing we could do-"it was our word againSt

- hers." We left, poorer but wiser. j. · After that experience and before writing this chapter, I

went on several exploratory trips With the purpose of in.. . vemgating cheating. I played at .most of the major casinos in. Las Vegas and Reno for periods ranging from a few minutes (cheating) to several hours, With bets ranging from $I to $125. I was cheated frequently enougb to learn to c~ and spot a dozen or ~-cu~ techniques. Thera, was cheating at large plush casmos. as wen as at·~ out-of-the-way places. There was chea~g at. all· betting levels, even for 25¢!! In many additional instances, ~cards

·.· ·· fiM·"tij Sjiot ChMttnt r4~ behaved so strangely that I suspected cheating although I could not actually see seconds being dealt (with a good cteaJer it is eXtremely difticult to see). . · . . .

On both the Reno-Tahoe trip and the first Las Vegas trip (taken foUr months after the Reno-Tahoe trip), I bad the good fortune to be accompanied by individuals Who were able to play the Teil-count .strategy, who were expert bOth at demonstrating and spotting dealer cheating and who patiently instructed me in the ways in which the dealer. can cheat. Furthermore, since each of these indi'viduills bad money invested in the play, .they were always at my side, watching attentively. . · · · I emphasize that the cheating incidents described are

.my personal experiences. I do not wish to imply that they are necessarily representative. The average amount of ~heat• mg is certainly less than the amount encountered bY, a Well-identified "prime target" like me. In addition, it ~ ably varies with such things as changes in government, . casino management, and ownership, and with the Size of bets, the time of day, and the individual dealer. ·

In some cases an honest casino may unknowingly ~ · a dishonest dealer. This dealer could cheat ~ house. by letting a friend win .heavily •. If the house checks teceipts regularly, they might notice that the given dealer frequently bas unusually poor shifts. In order to prevent detection, a logical cover-up by the dealer ~to cheat other players to make up the deficit. Some people descn'be this behavior of ·the dealer • a Robin Hood· function.

I emphatically dq not wish to imply that one part of Nevada is· more. or less free from cheating than any other part. I also be.lieve ~t the cheating problem is not so greitt that people should completely refrain from playing. However, anyone who does play blackjack ~hould Jearn without delay SOme of the elementary ways in Which his

, opponent can cheat. (Sad to say, this advice applies equally • to' the other card games, both casino and private.)

l have been told by a reliable source that in the

~- 146 BBA'l' Tllll Dlt~t.Jill '~~::~: ":~ SJIDI etwat~ng· '141·

1 first ·fiye ~,ears· of tho Nevada Gaming Control-Board's · , 0a one occasioti I was beUiDg from $2 to $20 whit'

~ 7

opera!i~ ~ "!osed down ~ than ~-casinos· f(Jr· , . a card expert-stood by to protect me agaiust possi~ cheatiag. Little if any publiCity (exceptiOn: (71}) was . log. A shOrt while after I began tq play. the deck was ~

!

gen~ly given to these proceedings. and the casinos usuall)' out and a new one was brought in. I requeste4 the QJd Bet fCOJ?OD premptly und~ new w.anagement. . · as a souvenir. I wanted it in order to C?heck f9r markings.

The amount of cheating 'VariCJ from almost- to Even though I iDSisted on that particular deck. the casino over 90per cent, depmding on which area of the~ ' tefused to give it to me and instead, after much hunting are playing· in. There is a similar variation iD the amount of ancl digging, produced another deck. 'I'he Jatter was iD con-help the authorities will give to a. player who has ~ sic:lerably better_ condition than the one they had J;efusecl to cheated. varying from no help at all to a great deal. You give up. Suspicious. l continued to pJay against the JlCW

may wish to investigate the conditions in your area ·beforo cfeek. and a I was winning moderatel~ .J'l8S a troo you play. . mendous streak of favorable situatioDI-my suspiciolla ~

· · There· are dozens of ways to cheat at b1ackj4ck and at lulled. Afrer about thirty minutes I stopped and my friend card· games in· general. All we Qll do here is. sketch some told me tbatlWas playing against a marked deck. Ho said of the more popular ones. based on my experience with the that both of the dealers agaiust whom I played dealt seconds casinoS. · · · whenever called for if the bets were $10 or more, and not

CJtberwise.lmmediately an odd incident came· to mind. Once a card stuck. in the deck, held only by its corner. It did not come Jooso undl the dealer fticked his wrist sbarply. Tho

Mlll'ked Cards _ ·. · Qne main technique in cheating is for the dealer to ..

Identify the top card in order to deal a second, if it is advantageous, at some appropriate point in the game. Tho . simplest way of identifying a card ftom its back is to mark

t on the back, in some kind of code, just what the card is. A ·marked deck is called "paper." Millions of decks of marked ·cards are produced annually and are readily purchased by man from supply houses specializing iD crooked gambling equipment. They are,&OOavailable in most .. magic" stores.

· card must have been a second, for it was hanging there and was held both above and below by other cards in the deck.

· My friend said ho did not pun me out of the game because J .,.. winniDg ·anyhow. Although I did win 80Jllo­

thing. my winnings were only a sman part of what dley might have been with such tremendously favorable situa• tiona as had OCCQrred.

All the most widely used standard brands are avail­able. There is no safety in the fact that a pack may have been IP8Dufactured by a reputable company; someone else

~ can easily "mark" the deck. For example. &rJyone ean t puldlase at nominal cost special inks and _brushes for

tbia purpose. For the details of how cards am marked. pictures of marked cards, . and some of the styles of lilark· mg. the reader is referred·to (IS, 21, 22, 36, 53-58, 66]. Photographs illustrating secOnd dealing (disc:~.betow)',,

Frequently dealers of seconds have the habit of ·~· ing" the wP&t of the,haDd that is holding the deck as part of. their motion of dealing. This ~ prevent cards from getting stuck and hanging as in the above incident. Thus whca you soc a dealer who has this Ofhel:w\se unn.ocessary modoJl, you should suspect very sttonJly that he is able,to

also appear in some of these referen~ · · , · .

. deal·aecond$. One of the cheat dealers, who had worlc:ed for the e.a­

.0. on 24-hout gdJ, Jhowed me a novel card-marking ~·He.took.lsisJhumbnail and pressed the backs. of Aces Qd Tens on their top edge. He did not scratch them

j:

~ h

li ~ !i !'

I '

. ·l

J

148 BBAT:.TBB DB-4& .. , " ttt·8pot~g: 149 but rather merely rounded them slightly; when this was done.'', : &Uri by1he exposed .part of its face but tWlt this portioa the cards seemed no different from· the othets in the ·dec:kt · Ottlle card is not visible to someone sitting across the table. as several friends and I learned when we htmted for the How; certainly no one is going to reach across with hi&othc!t' lliarkillgs. However, when the deck was held at a certabt · · !iaad and bend down the card. But a skilled dea1el ~ ~le to the light, the ec!ges of the marked cards gleamed J.ccomplish this swiftly and invisibly with the same hand Just enough so- that a trained eye could pic~ them out. with which he holds the deck. Fortunately many if not most Because of the angles involved in light reflection, when the 'Of the cheats seem to be much less practiced at peeking dealer can see the gleam, no one else can. -_ . than they are at second dealing, so if you are watchful, you _ This dealer claimed the Gaming Control Board had at can ~ften catch them as they peek. different times confiscated several decks which he had so -.Jf you ar& suspicious, there is one method which wiJl m~ked and that images of them had been projected greatly frequently catch the peeking or put a stop to it If tz. enlarged on a w.aU without the markings ever having been dealer peeks, his eyes must rest upon the card at which .he detected. is peeking unless he uses a "shiner." While one person plays, · ' Some people fall asleep ·at night thinking about their tDQther person stands (this has advantages over sitting, as Jobs. or their stockf, or their families. Mathematicians faD we shall see when we discuss dealing seconds) behind the asleep thinking out problemS in higher mathematics. Some , player and stares at the dealer's eyes. Whenever the delller'a people just count sheep. But this dealer confided that bo eyes took at the deck, the watcher's eyes ftick down to the and his friends fall asleep thinking up "new ways to cheat · ~ to see if a peek was poSS1"ble and then ftick back to •em.• , the dealer's eyes. The watcher .should also have his eyes

P-eeking ~ on the deck whenever a card is being dealt.

·. Marked cards have the disadvantage of being concrete · e\lidence that can be used in a court of law. A more coni· mon method of identifying the top card, which has the· fUrther advantage that it may be used with any deck ·what­soever, is for the dealer to actually look ·at the face of the top card. This is called peeking. · A skilled dealer can peek at the top card in plain view of a tableful of players with aJmost no risk of being caught Suppo_se that a pla~r busts. The dealer collects his chips _and his cards. He wtll often use both hands for this. If he · holds the deck in the left hand, as that hand reaches out, · it~- natural to tum it over so the deck is upside down. Try this and- freeze your left hand in this partially extended P;OOtion. Now reach out with your right hand and bend the : nght rear <:orner down slightly. Notice that you can identify

I have found this technique extremely suc:cessfol. Some c;heat dealers becom~ so tense and nervous about being caught that tbey become clumsy and.are caught all thtnnore­I{Uickly. OtherS stop cheating altogether under tho preSSUre.

A "shiner" is a little concealed mirror with which. tho dealer sees the faces of the cards, before or as they are dealt. It might be put in such places as the face of a ring, the inside of •pipe_bowl, or the polished edge of the money ;ray Is31·

~- S~ple Home Experiment ~~ that ,the dealer peeks and deals seconds at

wl1L Here- is a .simple home experiment to show you- the cmormous average advantage he gains whenever he decides tddo this. : . Deal out one hand to a $ingle player (imaginary, if

~"";:~~, .... 5-o--~---~-~.~.i:;-~~~~lfa AL · uecessary) and one to yourself, as dealer. Ha-ve thep'"'. YGG·<·il P,~ aboutto deal. Now_ rearrange the deck as follows. uSe the basic strategy. Each time, Wore you give a .1\C l~-front comer should be nestled in the "elbOw" or '?the player or yourself, peek at the top card. (It is ~ ~nd joint of your index finger. Tbe tip of ibis. finger

. aunply to deal from a face-up pack, with an cards left up- ~ould be. slightly above the top front edge of the deCk (it on the table.) If you prefer not to deal that card, hold if · ~ keep_ the third card from being pulled forward by -the: ~d d~ 9le seco~d card. A certain amount of judgment: ac;tion of the second). The left-rear corner of the deck JS ~ liere. When you feel your judgment is g~ s11ould be firmly seated in the palm. Tbe second, third, and make p initial bet of one chip on each hand. Record t1te·· fOurth fingers should go under the deck and around. the results for one hundred hands. Shuffte the deck well eaclr tight side. Their tips should 'also be slightly al>ove the top time you need to reshuffie. I tried 100 hands against a of the deck. · player using the basic strategy. The player won 9 units and . No-w, with the thumb, which should be lying com-the ~ won I 10. Tbe net gain for the dealer was: a · b1ably on top of the deck toward the front, pull the top whopp10g +101 per cent. Compare with the results ex·· ca.rd to the left, say half an inch. This half-an-inch figure is pected for 100 honest hands, given in Table 3.6. only for illustration. An expert would pull the top card

to the left (or down, for an important alternate variation) Dealing Seconds only a very small amount. If your grlp is proper, the rest

Dealing seconds fs the principle weapon of literally of the deck should have been undisturbed by this motion. millions of card cheats ·throughout the world. When com· The .c~er of the second card is now expo~. By using, petently done, it is almost invisible, even to experts. Sleight• ~r.:t~ght thumb on that comer much as in ordinary ~al~ of ~d and ~anipulation of cards were already well devel-.. ing._ slide the second forward and to the right. When it is oped m the SIXteenth century. With reference to this, see · -pal1 way out, grasp it with the thumb and forefinger just the report by Ge~lamo Catdano [50, pages 132_134], pe~ • in ordinary dealing. At about the same time, with the haps ~e most skilled gambler of his day, in which there are·· -left tbuml> slide the_ first card back into its original position. stories of the miraculous skill and tricks of Dalmagus (or :If your grip was proper, only the top.and second cards were Dalmautus) and of Francesco Soma. disturbed during the entire process. Proper height of the

One ~ethod of detection-listening to the sound of tips of the second, third, and fourth fingers will prevent the th~ cards. being dealt-is generally useless in the usual cards below the. second from moving when the second card DOtS)' casmo atmosphere. It is based on. the fact that the Is pulled out. ·Thus when the deal of th~ second is compl~ second rubs other cards on both its" surfaces when dealt,. the ~eck_ ~- normal. This is not an expert .technique. while the top card rubs only on its lower surface.' Thus· a' but 1t sliould gtve you some idea of how second dealing dea~ wh!ch mc:tudes-a second generally sounds something works. . like. swish, swish, scrape (second), swish. Of oourse these . If you used a deck with borders, you may have noticed sounds and the difference between them are slight; thus it is that, as the second is dealt, the right border* of the top card

pneraUy necessary to have quiet to detect this. - • I assume the dealer is rlght-lianded throughout ibe boot. If the . To get a very .rough idea of the technique of d~' c1ealer II. left-handed, the wordr "right.. and "left" should be inter-seconds, place a deck of cards in your left hand as though.. c:l!anpi_ by the reader throughOut many of these discussiona.

I .

1$2 JiB AT_ 'I' IJ ~ ~ Jt~ L:- "' ttY Spot CWi:tg -~

is almost lltationary. However, if a top card_!&- dealt, tbj: jltne. The rules were pretty standard except tbat ~e gapte right border: of the second is partially or ~~ -~ncealeJ ._. played with four decks all shufiled together. Further. until tlie- top card has cleared the deck. ThuS one way to 1Jie cards were dealt from a "shoe." This was a black plastic spot a second being dealt from a deck with borders is to . t;0x that was open at the top. The four decks were shuftled stare from above the deck at the right border (left bordet atlcl th~ placed in the box with their long edges down. from the player's side of the table)) to see whether or not · ·There was a slit in the end of the box at the bottom and a that border moves much as the card is dealt. sman oval hole running up from the slit for a short distance.

To counter this. many second dealers use decks with- 111e backs of the cards showed through the oval. The dealer o~ borders. However,. so do a number of honest casinos; placed his right thumb through the oval, and to deal the thus the mere use of a borderless deck by no means in~ ' cards, 'he drew them down and otit through the slit, one by· cates cheating. To make it still harder for the player to..see a · one. second being dealt, the dealer generally tips up the front of· This seemed like an ideal game with which to use the the deck slightly so that the player views the_cards edge ont:. - Ten-count strategy because, with so many cards in play, · In.this instance the presence or absence of borders is imma· the fluctuationS in favorability from hand to hand (which terial since the. backs of the cards are totally invisible at the teSUits from cards played on the p~vious hand) would be instant a card is dealt. much smaller than in the one-deck game. Thus when large

Dealers often tilt the forward end of the deck so far up ~were being placed, there would be much less fluctuation and back toward their chests that kibitzers who are stand• · ~P and down in the bet size. Half an hotll''S play at the iDg cannot see the back of the top card. In this position an~' table, waiting for favorable situations, confirmed this. When one can deal a second without being caught. If you try this a run of .favorable situations finally came, I changed from ~ yourself, simply ~de the top card down a short distance, waiting ~t 9f $1 to the $5 to $25 range. There now was deal the second by pulling it forward and up; and restore a long steady run of favorable situations before the foli~ the top card to its initial position. · · decks finally ran out. I won about $8o. During the next · Perhaps the inost popular type of borde~less cards now.· two or three- hours of play there were _comparable runs -in use in the casinos are the famous Bee No. 6j. The pattern with similar results. Altogether, I accumulated about $16o. on the backs consists of solid diamonds separated by brokeD . A card-expert friend who was standing by thought tho game diagonal white stripes. This pattern seems to dazzle or blind_ was so--Bafe that he wandered away. Wo were soon to get_ the untrained eye; irs use seems to increase the difficultiea ._ an expensive education. · in detecting seconds. When the wrist flick is employed, this · . Shortly afterward, a dealer against whom I had DOt pattern is even more effective in blinding the eye to tho previously played came to my table. After a few minutes.. deal~g of a second. · the four decks became. favorable. Only this time, I lost

Deck-Stacking: The High-Low Pkkup ·

Durmg my exploration of casinos to s~dy the Cheating methods actll!illy ~ployed, I came across a novel 5011 of

nearly every hand. Before the run ended I had lost $250. Startled and suspicious, I then watched tho dealer intently.

It was conceivable ihat ~e could deal· seconds out of , tl&e shoe. BuJ how could he identify the top card? Peeking

. seemed impossible; hence shiners were ruled out. . What

-f :>: -~ ""~ ~ c

154 BEAT TBE _J)~A_L,-~ ~1iiiW fj,$r}ot-ChMiinr !5.5

- about marked cards? I watched the dealer.'s eyes,. but be_-i ~. ' - The casino had dealt with my earlier run of "luck" never glanced at the part of the- back of the-card which : . rather promptly and brutally. I decided to find out whether was exposed in the oval hole;. • this was casino policy. I observed··that this dealer generally

· The four decks were about average durmg the next ten -played at the table that was getting the. most action. Furpter, .runs so I dropped back to $I waiting bets. However, I was most of the other dealers were less skilled. It seemed likely losing on nearly every hand! I then began to count, and.; that fbey were not "in the know." Sure enough, when I in :l6 hanQs, I lost 24, tied one, and won-one! The odds made $I bets at their tables I was not cheated, and tho against losing at least 24 of 26 hands by chance_ alone are · deck stacker busied himself elsewhere. I struck up conversa-about 2,000,000 to I! • I could not believe it. Could I be dons with a number of these dealers to check my $pecUia-so tired that I could not count? To be safe I counted the - tions. I asked whether there had been any winners lately. next group of wins and losses by putting chips in little piles. . ·A few dealers remembered only one such person in the last · 1 also was obvious about it so that the dealer would be sure ,week (the others remembered none). ThiS person was su~ _to notiCe. I wanted to see whether he wo~ld alter his be- posed to have made Ssoo. I guessed that he must have bavior. He did not. On the next I4 hands I lost 12, tied one, played only a short time and bet fairly large. Sure enou~ and won one. The odds against losing at least 36 out of 40 the dealers said he bet $25 and Sso ($Ioo on douliling hands by chance alone are about 25o,ooo,ooo to one! I' down and pair splitting) and played only a short time. Thus was baffted.....;.how did he do it? his win corresponded to $20 betting $I to $2: he was

And then I noticed an odd thing. The dealer, in pick- lucky and besides he probably quit before the big guns fng up his winning pair of Tens, slipped a small card from could reach him.

- my hand between them. An accident? I soon found that the. · . ·· I later found the high-low pickup used in many other -used e!ards were being stacked ~low-high, low-high." Then · casinos. Here is a test that will give you an idea of the 1 watched them come out on the deal. Six of us picked up power of the high-lo~ pickup. Let Aces, IO's, and g't -be our hands and we ali had {IO,j). {10,2), (10~6), (9,5), high. Let .2 through 7 be low. Let two B's be high and two etc. I quit playing now and watched the fellow ply his trade. be low. Stack the deck low-high. low-high. Now deal hands

. He picked up the cards so smoothly that his interlacing of to an imaginary player and to yourself as dealer. Deal so high and low was almost unnoticeable. He. preserved the that the player gets low-low and you get high-high. If the ordc:r of the cards through the shuffle (a false shuffle that . order of the deck is low-high. this is automatic. If it is high­looks like a real shuffle -but does not · affect the cards is · row, you can deal seconds. You should win virtu~y every Standard equipment with card sha~s). When he dealt, ~e· ·· hand. When a stacked deck is used, get out of that game never needed to peek or use marked cards. He knew wher, immediately. the cards were: just where he had stacked them.

• Readers who attempt to calculate this figure by assuming that the probabilities of winniDg and losing are each about o.s will get a figure of a Uttle-over 200,000 to r. However, the proba~ of a tie il aearty 0.1 and should be ~ into account. When this JS done and we ask, What is the chance· of at least 24 losses in 26 hands? with the probability of each loss about 0-45 and the probability of CllCh nonlosa (equals tie or win) at?out o.ss. we F' the figure of ~,ooo to 1.

Deck Stacking: The Seven-Card Step-up

One afternoon I wa! playitlg head on and bettitig·from $Io to $Ioo. After I had played .duee or four decb._l

· 'noticed. that the dealer bad started each deck with two . han~ one of which was 21 and the other 20. More amaz-ing. his 21 was composed each time of the Ace of Spades

r~- --~~~-.~--------~--- --~ --- . ... .. - - ._, , _;',i<

t IS6 B BAT TBB DBA L--B.

I. l

aDd. the same Ten~value ~ the Queen of Clubs. At_ my elbow employed in a private captlcltJ, was Mr. Michael MacDougall. one of. the world's ~utstanding detectors of card cheats and a special investigator for the Nevada Gam­~ Control Board. • (Tho rest of this chapter is ~ . from our experiences on a trip which followed by nme months the trip deScribed in Chapters. We spent twelve hours a day for eight days playing dozens of the prinCipal Nevada-casinos.~ Mr. Macl>Qugall told me. that the dealer had.set up a card sequence of ?.B,g,zo,IQ,I,Q,K,A. The sequence was preserved during the shufile. It is a stan~ tecbDique of card .cheats to shuftle . a deck repeatedly m such a way that some preselected clump of cards is left un­-disturbed. The sequence was "bridged." meaning that it was bent lengthwise down the center, .so that when placed face down on a table, the center -did not touch the table. _

The dealer had left the sequence near the middle of~ deck when he offered it to me to cut. An Ul'laware player ~ genefany cut the deck so that the bridged cards become the top cards. Try it yourself and see.

_ !At us return to our story. After I had cut the 7 to ~ top. the cards dealt werei 8 to me. 9 to the dealer, IO tome, 10 to the dealer. He held 19, I held 1_8. On the next hand the c:ards went I to me, Q to the dealer, K to me, A to the dealer. His natural beat my twenty. ·

As he p~ up theSe cards he stacked ~ ~ck into the san1e sequ~ce. With the next deck the Sttuation repeated-itself. The reader; can easily convince. himse~ that the same card stack is effective against two or three .players.

At this particular casino when the dealers went off duty they put the decks_ in their pockets. When . the new

• MacDougall Wrote about the cheating we encountered 'in hill syndicated column £35, 37]. H$ mentioned it OD "'pen End" when we &ppearecl as guests oa April 7, 1963- Ho exposes cheating iD ~ to clubs, dinner groups. and -the like, thro~Jshont tho country. By a Strange coincidence, in ·Nevada he doesn't seem to be · iD such peat

.· demand as an investigator any morL

.~~ 'fo.!V- ~ Spot. Cheating . ._57 . -~ eameon.·th~ took decks out of their pockets. we

fOun4 out that they were gaing off an4 setting the "~liy' mto the deck. · · - -r .-. It is said that a ~aler in Newport, Kentucky, was tht Inventor of this step-Up and . that he was so proud of his achievement that he considered demanding royalties from the casinos using it.

A closely related_ and much more innocent-appearing lttcldng idea (not currently fashionable) is to let the'-~ begin each time with a sequence of Tens. Then evetyOne holds 20 and ties the dealer on the first round.. However, aucceeding rounds are played from a Ten-poor deck. The effect is the same as if seYel'81 Tens were removed from the

· deck before play. The shufDing- technique used to preserve the step-up

cin be used to convince an onlooker that the deck ia·llOt being.false.shufiled. The dealer can even square it up after each shuflle. A high:low sequence of considerable length can be preserved this way.

One greedy blond dealer against whom I played brlely was preserving a high-low clump twenty cards long. When I called her attention to the clump, she refused to shuflle to destroy it.

Often one sees a blackjack table standing empf}', attended by a dealer and awaiting customers. It is common practice at such tables to leave the deck spread out face up. Presumably this is so the player will see that all the c:ards are there. If the deck is newly opened and the cards are in their original order, it is easily possible to tell whether or not any cards are missing. It is not easy to tell this when the cards have been mixed up. And as soon a8 you sit down the deck is scooped up, shuftled, and brought into play; therefore you have no time to check the deck. Mr. Michael MacDougall obsetved that ·if we stood back from such 111 anpty table and examined the ·deck we could find the -step-ups ·all laid out and waiting. We went to a suspect

of;-..;'

· .ts8 B BAT T II 8 D 8 ~ L ~~ '· - ::.,row to Spot CHeating Is!. .··casino. At the first table we approached. we saw the step-up. Ohto tlie dealer's handS and eyes. He did not do Ill~

W.e bet $1. The dealer otfered1ts the deck to cut, WithoUt Suspicious. Puzzled, \Ve·watched and play~. In a fe~_min· even bothering to shuffie. As expectea. we cut the step-up . ates an individual who would have otherwiSe seemed m.ccm,. to the top and· the dealer received 20, followed by 21. We M>ictious came steaming down the· aisle behind our row commented on how we had been cheated by a step.up ancf: of·blackjack tables. He was passing our table at high speed the Gealer laughed happily, proud of his handiwork and · when ~ pit boss· whistled. He made a sharp turn on his proud that someone finally appreciated it £371· . heel and plunked down in the seat to my right. I cut roy AnchorMen

One large hotel on the Las Vegas strip seemed to be completely free from cheating. After I had won a couple of huridred dollars in half an hour, pl~ying $5 to Sso, one of . the pit bosses jokingly 8$ked how the system was working. MacDougall told him "up and-down.. like an elevator." Sb1ce the pit boss was friendly enough and we were run· mng out of places in which to play, we adopted ~e follow­iPg policy with this hotel. Bet $s to Sso and stop playing ~-we had won $200 or played forty.five minutes, which· eVer was sooner. Forty-five minutes was short enough so that ill a few minutes I coUld fully recover ~m the strains Of counting, refreshed and relaxed. If we stopped at $200, our win was sniall enough, compared to ordinary chance ·ftuetuatioris, to seem to be just a little luck. .

· (The reader may be puzzled at the contrast between this timid low·scale betting arid the earlier test reported in Chapter s. Conditions ill Nevada seemed to have changed dnlstically between these two m.es. On this trip we learned . that if we won more than $250 to Ssoo or placed a single. bef above Sso to $too, we ~id so at our peril.) ~

This caSino wClcomed us back the next few times we came. We proceeded to beat them. eight times in a rqw. We were bUsily at work on a ninth win when the pit boss called over our dealer and· told. him ·something. My frlen~ Mr. MacDougall overheard ·tho dealer say, .. AU right, ru giv.e it to. him." When the dealer returned we asked him what he was going. to give me. He only &roiled. Out eyes ,locked

bets and awaited developments. . We immediately noticed that the dealer now might be

peeking, but he did not deal any seconds. The new player watched the dealer's eyes before he decided whether to draw or stand. I tried to see his cards when the bets WCl'C

being settled in order to tell if he was followii)g a consiste~t strategy. But either he threw his cards in face do~ or the

, dealer picked them up so that I saw only their backs. Final-ly, I got two glances at this fellow's cards. · - ·

He stood on hard 8 once and drew on hard 19 the Second time! Further scrutiny confirmed the fact that the dealer was peeking. If he wanted me to get the top card, he -signaled the new player (termed an "anchor man") to stand. If he wanted to keep the top card from me (for example, suppose I doubled down and he saw it was a Nine or Ten), he signaled the anchor man to draw. An anchor man makes the dealing of seconds unnecessary. Wi~ marked cards even the peek would becbme unnec-e§a~ .

The dealer ·could easily infer from my behavior .. · formed through long habit, whether I was going to double

down, draw; stand, or split a pair. It is thus a useful art in . casino blackjack to be able to play with a poker face. An ~sier way to combat an anchor-man is to wait until yOur turn' to even .look at your cards. In this Wa.y, you carinOt ~possibly help the dealer who uses. an anchor man. . ·

· We moved to another table, the one far1;hest i'rom ~ere W!' were. I sat so that a player already at that table was on my right, occupying the anchor man's future seat.

,"l1!F . BE AT_ T ~E.~ J) E ALE~> .. lt{)W to :Spot Cheatirfg.

The word was passed to the dealer and the anchor ~an waited patiently. In a few minutes the player on my nght got up and the anchor man plopped down. We left. The fun . was over in that casino [35].

Alternatively, the anchor man may be seated on the dealer's right. He then stands_or draws according to sign~ from the dealer. This enables the dealer, whose turn lS

next, to draw more desirable cards for himself. The entire table loses to the dealer's superior cards, in contrast to the first use of the anchor man, in which only one player seems singled out for hard luck.

· ·We had several additional incidents with anchor men in other casinos. They suggest to me that Nevada, a state without seacoast, has the biggest "navy" of any state in the union.

The Gratuitous Peek

When th,e d~ler has a Ten up, he immediately checks his hole card to see whether he has a natural. When he has an Ace up, he flrst asks the players if they want insurance (assuming the insurance rule is being used) and only after this is decided does he check his hole c~. When he has any other card up, there is no reason for him to check his hole card before his tum to play. I have on a number of· Occasions seen dealers check their hole card when an Ace

· was up before they offered the players insurance. They thea tiied-by their actions and manner to influence the player. If they had a natural either they would not offer insurance or they would rush the player. U they had no natural, t,bey

. gave the player plenty of time and even hinted with their facial expressions that insurance was ~ Once in a while I have seen dealers look at their hole card immediately when they had neither a Ten nor an Ace up. 1bey then continue to peek until. they find a card they need, which they hold for themselveS by· dealing seconds.

Miclumics on Call

· · A (card) mechanic is a skilled cheat who 8CCOJlloo

plishes his ends by sleight of hand. Some larger casinos have their own·mechanics working as dealers on the regular shifts. Since mechanics are paid much better than ordinary dealers, a casino which cheats will, for reasons of economy, hire only as many as pay for themselves by the extra revenue. Often there is only one mechanic per shift. One morning I was betting $ro to $roo and winning •. After a whiiC, the dealer went off duty and his relief came on. Sud• ·denly the pit boss rushed over, scolded the relief, sent him to another table, and made our dealer stay on. A few minutes later another dealer the pit boss had summoned relieved our dealer. The new dealer was, according to the expert who was protecting me, a mechanic. We could not see any definite· cheating, only suspicious moves. But good mechanics can often conceal their move from the most skilled eyes, leaving only a few suspicious side effectS. W~ immediately lost nearly every bet. At the $IO level we .. lost a couple of hundred dollars of our winnings in a few minutes. ·

We walked across the street to a smaller casino, one Periodically these boxes are brought into the_ counting $5 to $50, we were well ahead after twenty minutes. The pit boss was grimly hostile. Then he made a phone; calf. MOCc time pilssed. Then we looked at our watches and ~aw that it-was dme for our dealer to go off duty, yet he was not. in."a city without clocks," always cany a watch to see if your dealers are being changed after·what you observe to be the regular length of time for a shift, and at no other time. A break in the pattern indicates trouble.

· Our dealer had been on duty thirty-seven minutes {thirty \\l&s normal in this casino) when a man dreSsed ~ .a white shirt and black pants like the other dealers, but wtth­out the apron of that casino, hastened through the front

F~ ~

---~--~~-----~"lEII~!IJ':T'.c'~~...,.......--------ilfiillll!=illa!!!l!l~~

162 B B AT · T 11 B D :B A L~B a ::~ ii¥.1JI"nt 'ID SpDl Cheotbtg t6l ~ ctoor and up to our table. 'He immediately began to deal. 8tiher Jose )'OUr $20 or.double if. Then quit. If you an. Wo spotted him as a mechanic. W"rth a smile he ~purred. ~ playing after an llour, quit anyhow. Go on to ano.thet MWould you gentlemen.~ a drink?" Tb! pit~ became cass~ Return to casinos and dealers you do well agaiust. relaxed and in good sptnts. A ~ave o~ friend!~ seemed Avoid those you do poorly against. This way, no one cheat ·to pervade the grim. barren, midmormng desolation of the. lmrts you mudl. And you nev~ play a cheat twice. place. We left.

Miscellaneous Methods There are so many ways for the dealer to cheat that we

can only give a brief introduction to the subject here. The interested reader can study it in much greater detail in the lll1Ul)' references that are available [IS, 22, 36, 53, 58, 66]. Many interesting facts were also presented during a Senate iavestigation of gambling in the fall of 1961 [30, 42].

Despite all our warnings and the frequency with which dealers cheat, there are a number of casinos that are scrupulously honest. Our purpose in this chapter is to make tho reader acutely aware of the darigers of cheating and to give him enough knowledge so that, in most cases, he Will

/

' , be able to spot it and change casinos before suffering a serious loss. It should now be evident that 1t is suicide tq play in a place that cheats; you have virtually no hope of . winning.

A voiding the Cheats As a practical matter, it is not likely that-you will be

able to actually catch much of the cheating that you meet. The better cheats can only be spotted by experts, and sometimes not even then. How can you ayoid serious losses from cheating? ,

·The best me~od I know pf (the paper-route tech· nique ~ earlier) ~ now being quite widely and successfully used: Here again ls·the idea. Divide your stake into ~ ot twentY' equal parts. For example, starting with $200 you might divide it into t~ parts of $20 each. When you sit down to playt. buy $20 worth of chips. Play until you

------~-----·--· ~- - ··- .... t:--•,

II

Can the Cheating Be Stopped?

LIFE Breaks the Cheating Story Life magazine ran a feature article about the author

. and his methods in which Life's staff writer Paul O'Neil broke the long-concealed story of Cheating in Nevada black· jack [49]. "'Thorp has been victimized by crooked dealers. ••• He has been backed off (thrown out) by pit bosses, be has been harassed by shills, plied endlessly with booze,

-eyed significantly by plug-uglies and, on two occasions, rendered spectacularly rubber-legged and goggled-eyed by knockout drops, courtesy of the house ... • ·

In preparation for his article, O'Neil read .my work. Then be spent a week in Las Cruces studying me and my ideas about blackjack. Next he spent four days with me in ~Vegas (I was there for the 1963 Fall Joint Computer

•This was the house's last Attem})t to stop my five-night wbming llreak at baccarat. After my team IUid I pushed our W!n streak to teVCD alghts. the rules were changed. Tbe side bet we exploited was removed from tho layout. See [5g. 70} for details.

·•64·

Conf~e ). He watched me pile up $420 in winnings in the four ~d a half hours I could spare from my wotk· to play blackjack. (My bets ranged from $1 to $25, depend­ing on circumstances, and averaged about Ss.) During the twenty minutes that I was posing for pictures at the Tropi-

. cana (untecognized by the management), the tables kindly rewarded me with $62.50.

The Life article was painstakingly accurate and thor­oughly researched by Q'NeU. · Even !be authors of The Green Felt Jungle were unsure-:-probably due to insufficient information-about the cheating situation. The dust jacket of ·the hardback painted the most sordid picture of the other phases of Nevada gambling and then concluded, "Ironically, the gambling equipment is possibly the only thing in Las Vegas that is really oia the leveL"

Replies from Nevatla

The Life story could nt)f be ignored. There Were cries of protest from out West. For example, Edward ~ Olsen. chairman of the Nevada Gaming Control Board in Carson City, wrote in a letter to Life [48. p. 27),

Sirs: Neither your writer nor Dr. Thorp has evidence

to substantiate such statements as: "Thorp has·been victimized: by crooked dealers in almost all of the .

·major casinos in Nevada!' . Your write_r's statement that Dr. Edward Thorp

•tisuaUy knows in hi~ ne~e ends just when he is being double-crossed" appears more worthy of scientific in­jestigatioa than. his system of counting the cards in a 2.1 game. I think Life hQ libeled ~te of Nevada.

.EDWAltD. A. OLSEN, ChalrtnQII. State Gaming.Control Board· .

i66 B'-BAT TBB -DB ALB* On April 3. shortly iftcr Lite-s article appeared.

Mr. Olsen closed the Sllver Slipper. a Las Vegas casino, on charges of cheating. Meanwhile, several caSinos · haVe made changes in. their blackjack rules to thwart . the Thorp system.-BD.

But the right hand evidently did not know what the Jeft was doing. For at about the time the Olsen letter was sent, a Nevada newspaper announced [25]:

STlUP CASINO CLOSED BY STATE. SILVER SLIPPER

·" R#.IDED. "CROOKED DICE" CHARGED BY QAMlNG -91!• fiCIALS.

Tho story went on to tell how five dice were picked up on a routine check of the crap tables. All five -dice were later found to be percentage dice. '"The official statement of facts was signed by Control Board Chairman Edward A. 01· sen. • • • " An earlier incident of cheating at blackjack was reported as a second charge:

The Silver Slipper had no seriouS enforcement .problems before ·last year. It appeared then iil com­plaints filed by the Ganiing Control Board With the gaming commission after undercover agents found a dealer cheating at 21. The casino's management was notified, and the dealer was fired. ·

· No action was taken at that time, but the incident is BOW listed as a second cause of action in the current complaint.

How Card Cheating Can B~ StrJppetl

In the first edition I gave a detailed and simple pro­cedure which could \Je -used to stop virtually all the blaCk· jack cheating methOds described in this book. These &Ug­ptions seem to. have fallea on deaf ears in N~da.

What can be done by priVate citizens or-by the federal

f'QII, !~ CheatJng B~ ~topped? 167 goverpmeot _to eradicate the. cheating? Many people have asked JDe thiS and I have thoUBbt long and hard about thci ·answer. · ·

. The ~at· Stlm_lblin~ block is. how can you· get evi- . dence that wtll stand up m court about something so intan­gible ~ sleight-of-hand card cheating? What is needed is

, ~me~ow to get photographs, or better, moyies of the cheats m action. Movies would be ideal. for the cheating move could be followed frame by frame. Photos should show tho cheatin8 move, ~e cheat's face (for identification), and the club emblem on the layout. · • Discou~aging fu~thei' cheating would then be _quite

sunple. Publish t9e ptctures and an account of the· action in a nationally distributed book or magazine. . I am told that the most practical (though still· difti­c~lt? way of obtaining the pictures would be with a·sub­Dllma~ Japanese video tube ~d to transmit the pictures by radi? to ~ remotely located recorder. There might bo legal difficulties, however. I understand that a Nevada stat~ l~w prohibits taking pictures inSide a -casino without tho casino's permission. . . . . ~

How the U. S. Treasury Can Recover Tax Money Lost ~rom Stealing off the Top

In the summer of 1962 I~ visited by an under­cover agent of the U. S. Treasury Department. He was one ?f a h_alf-dozen or so sucb agents who were part of a larger mvestigation of tax evasion by certain casinos in Nevada. He told llle that certain casinos were taking large spms of money :•off the top." In blackjack, for example,. when yoa buy chips at the table, yo~r money. is generally pushed $'ough a-slot in the table Jnta a locked box underneath. ~erlodicslly these· boxes are ~roug~ into the counting l'Q<)ItlS where they are opened, and the contents are recorded !"' part of the casino's gross income. Accormng to &hiS

- : _,.,.;' ' ·-· · Can the c;hellting Be Stopped?. 168 BEAT TBB DBALBJl

agent, a common practice Is to list the contentS ·on 8n addlns-machine tape, recording the number of hundreds. fifties, twenties, and change. ·

He asserted that 101118 casinos keep two adding-ma­chine tapes. one with the 'true figures and the other with patly reduced· figu~ The smaller figures are reported to tho government. Tho difference between the true gross and tho~ gross escapes taxes.

J asked the agent ~ he knew about all this. He said that he and other undercover agents were posing as

. big.~ and gambling conspicuously. They were able to inptiato themselves enough to be taken to the counting fOODIS as part of a guided tour.

He said that the c,tperiencos of the agents suggested that· about one third of the winnings was being taken o1f the top at that time. If the gross declared casino gambling income in Nevada was $220,000,000 for 1961 and if $x8o,ooo,ooo we~·fpr expenses. the taxes would be about $10,000,000 on ure $40,000,000 net, leaving a profit ,of about $JO,ooo,ooo. But ·• SUSpec:ted $no,ooo,ooo more was. being taken off the top. This would mean a true profit of SI40,ooo,ooo! Taxes escaping the government would exceed S:~s.ooo~ooo.

Tho ageat bad to come to see me for two reasons. Fust; he and the others had learned to play my winning system and how to spot cheating. The government "couldn't afford .. to have them lose huge sums at the tables just to get .fOUlS of the counting rooms. He wanted additional pointers.

Second. the agent wondered whether there was a way of proving, statisticaDy perhaps. that the theft of money ·off the top was taking placo. We conferred for a couple of dayr. after which I suggested tho following scheme. Here is the original text (except for minor changes and clarifica· lions. mostly indicated by brackets) of my suggestion tO the · federal government.

Qperation "Money-PuTIJp ..

· ' ·· ]UM, 1¢2 INTRODUCTION. There is strong reason to believe that · the Nevada casinos, when counting the "drop" fn)m

the gaming tables (i.e., the money that accumulates in the money boxes under the tables between collec­tions), take money "off the top." This means that in the counting rooms they set aside money from the total so that the reported gross total is· less than the

. true total. Tlie evidence is of two kinds. First, an agent of the federal government has several times been an eyewitness to this in casino counting 1'0QQlS. Secondly, the casinos expend vast sums which are not covered by entries in their books, for items like complimentary girls for customers, mechanics who cheat the custom-· ers at the tables (they get a fraction of the take and a higher daily wage; rve talked at length with. one of them) and perhaps. for the financing of nation-wide rackets. Where does this [extra} money [that the casinos spend but don't report] come from? Logic sug- . gests it comes "off the top .. of the [casinQ] operation it is being used to support.

.m 1961 the Nevada casino gross gambling rev­enu~ was around $221,000,000 and a net taxable in­come of around $4o,ooo,ooo ·was reported. If the· casinos are taking a mere one sixth off the top, they have a hidden additional potentially taxable income of $44.~000 and are more than doubling their sup­posed net income (the reported $4o,ooo,ooo taxes..

down to about $Jo,ooo,ooo)! On the basis of the evidence cited earlier, it seems likely that the amowrts. taken olf considerably exceed one sixth of the total.

There was a famous muckraker. Lincoln Steffen-. · who in the period 1900.1910 exposed in a sensational series of magazine articles the then emerging pattern

170 BB.A1' 'rBB DB•Al.B:it •

of corruptioa ia · our bJg cities and in state • g0\1emoo . ments. In his autobiography he sum,marized his ex­perieoc:es ia a maxim! -wherever a graft exists -tqr the taking. people wiU arise to take it. .. The casinos can take money off the top with impunity: no preventiVe

, machinery now existalr Eiuther, if a certain casino is taking eff, say, a hunc&.,o thousand, why not take off

· a million instead? Since if by some accident they're caught the penalties are the same, why not take off all the traffic will bear?

We propose a test which will either .clear the casinos of this charge, or will produce court evidence of such strength that the responsible casino bigwigs can he convicted in'criminal court. On this latter event, which seems likely for some of the casinos, the effect

•of a few convictions should lead the remaining of.; fenders in the casino set to the i>ath of righteousness.• The potential gain in income tax should each year pay for the project scores of times over! .

TBB PLAN. We propose to gamble at selected tables of selected casinos. We would buy chips with hundred· c:k>Dar bills whose serial numbers will have been pre­recorded (this. record is not essential to our·main ob­ject but does have an additional application that will be discussed in a later J"Cpo~). _We would play con­tinuously from one collectip~ to another, recording the total number of hundred-dollar bills that go into the money box and noting the number which were ours.. A current practice is for. the casino to record the Dumber of bills of various denominations and to label the record with the table and shift number. Thus, at the end of the casino tax year, certain of th~ records can ~ checked to see if money 'flas taken off the top ot not. Our evldence is that it js currently ta,ken in the form of hundred-dollar bills. in .a propOrtion that m.

c.n ·the f;h«~Jinr Be s~ 171 CJeaSeS as the 1lUIIlbcr of lmndted-dollar bilts increase~.. Thus our test is apeclfic for the problem •

Now, it is widely believed that the casinos necet­sarily have a certain percentage advantage_ over the player at each of the games. At blackjack, for ~pie. a typical player seems to lose in the long run from 3 to s per cent of the total amount of his bets. Of course, he usually rebets his money several times. Thus he gen­erally loses several times this percentage of his ori~ in~ bankroll. This is measured by the casino when it determines how much of the drop it "holds," i.e., what fraction of the amount the player invests in chips is won by the casino. In blackjack it is estinlated.that the casino holds 20 to 30 per cent of the drop, i.e., the player loses from his investment an amount equal to from about four to ten times the casino's edge over him. This is due, as we've mentioned before, to the fact. that he gone...Uy rebets his money several times and thus the house extracts its percentage several times over.

'It might seem from this that to pump $120,000 into several blackjack tables (perhaps Ss,ooo per table per shift would be adequate, thus allowing 6o samples). some $30.000 of playing capital would be

· required and consumed .. However, it has been provea (both in theory and at the gaming tables) (67;68,69) that there is a winning (!) strategy for the player at blackjack.. This makes it possible· to pump in really large amounts of money-in the millions if desired­with the aid of the same $30,000 bankroll. We propose · a $30,000 bankroll for playing capital. The hundreds of samples thus attainable offer statistical and legal advantages over the· smaller number of 6o.

We propose that a team of perhaps twelve agents. soine ntale and some fema~ learn this strategy (two or three days training is sufficient!) and ~stematic8Uy

f l l •'·

r I '

. ~ 172 BB~T THB DBALBJt

but as inconspieuously as possible pump in very large sums in various casinos. At the end of a tax year many hundreds of table-shift record cards· could then be checked to see if money was taken off the top. With such a sample size, if the money were being taken it would certainly show up. Further, since the people who count also sign the record cards and have been there since the money· box was unlocked, they are absolutely ·liable for taking money off the top. Now the evidence would probably be stronger from the standpoint of convincing a jury if the same in­dividuals were found guilty of several violations rather than only one. Our sample size is designed to make this [more] likely. ·

As far as I know, •Operation Moneypump" is filed away in a dusty Federal drawer, certain casinos are still iaking money off the top, and our national tax collection ~till .doesn't cover our expenditures.

·~-·-=-

12

' ~

ScienE::e Versus Chance

The appearance of numerous system players will ultimately necessitate important changes in the game of twenty-one u it is now played. We can gain some first insights into theso changes by discussing the means through which the Nevada

. casinos have ·.dealt with the handful of successful players (whom \ve refer to as .. count" players because they coulitecl carcls) that has appeared in that state during the last dee­ade. Most of the stories surrounding these. early players are

· aot a matter of public record or even known beyond a ~ IDlall circle of· acquaintances. No part of the legend be­came known to me until sometime after I had completed the winning strategy outlined ~ this book and had arrived in Nevada to test it in actual play.

Early· Winning· Players

The first of tl.te successful system players, a much dif. , ferent personality frOm the others in the group and in· no :i

way representative, was a colorful individual known a.i .173.

~- ~p

~~14 B BAT T H B DB~!.~& .38cfence VdtrU (:fumce 'J1S

~:~ John.. Large and obese.. he acquired his name y consistent and the people who told them were direct~ . :::~=~'his habit of coining to the casinQ with a largo bag of . pants. Therefore, 1 believe the story is substantially"~ ~ · .y greasy fried chictcn. ~ played for as long as twen!J. · rate. · · · · . ·.·. homs at a stretch, never leavmg tho table. The casino sup- To my knowledge, the firSt person who employed a -~ ~ ,the ~nks, and innunlerable meals of vmying sizes -count" system. in successful casino play was Benjamin F.

·. could always be drawn from tho huge bag of chlc:ken. It ( .. System Smitty") Smith, a viell-known figure ln tho ~ · ·eopn became ~pp~t that -oreas,·totui'" wanted to play Vegas casinos [I]. A~cordlng to Mr. Z. a mutual acquamt-- alone. As crowded is the casinos ..-e. once he became a ance who has seen Smitty's voluminous notebooks, Smitt;r

,-~ bmlliar tace he did not !lave much trouble keeping other apent several yq.rs playing out Ioo,ooo hands ln an effort , ,~ ,players away. His profanity and drinking drove o~ all but to determine the proper standing numbers when a Ten-. tho hardiest of women players and finally the casmos fol'o count was employed. The system, as described to me by Mr.

bade all womeQ to play at the same table with him. z. gave a fair approximiltion of the totals to stand on for Since 3reasy John's hands were generally dripping 'ftrious values of the ratio Tens/others. However there were

with clucken fat. the cards soon became too oiJy to ~ certain moderate errors, which resulted, at least partly, from comfortab!y. Even though decks were changed frequently, the nature of the system. . the grea5e was sufficient to drive away the men players. In addition to the moderate errors in standing numben

.Greasy John played fOr long hours day aftC;f day, and that were part of System Smitty's meth~ there was ~ In a few months he became wealthy enough tO retire. He detailed strcltegy for doubling- down and pair sp~tting. These

_ IWfered a heart attack and died shortly afterwards. Wo factors, ln toto, probably cut2 or 3 per cent off the player's haw no knowledge of the system that Cirea9 John used. advantage, not to mention the increased rate of.attrition of lt ~ probable from surviving details that he employed the small "'waiting" ·bets. Since the bulk of the favorable

_ end play. As noted in an earlier chapter, end play wiB bets are ln the o to 3 per cent range, the player's rate of Win produce astronomical gains in a short time in spite of the is greatly diminished. The only alternative for the player fact that the player's )?asic playing strategy is poor. Further- who wants to make a big win is to overbet his capital (in mote, end play is a Very natural idea, easy to ;erify em- terms of the ~eory of proportional or "fixed fraction" bet-

. piriCally, and it probably has occurred to a great many ing [23,70,76]), greatly increasing the chance of ruin. and players. : . hope for the best. . · .

For those players who follow -system Smitty,• we SmiU, ·probably did precisely this, for he has had have ln most cases omitted the details of the colorful back· many spectacular win-loss sequences. Mr. Z said he was grounds of the individuals involved, and the parts of their present one night when Smitty won $xo8,ooo at the black-adventures that might serve to identify them. We also have jack tables (that is a considerable sum with a Ssoo limit) regretfJllly omitted .the human-interest portions of the and lost it an· back- by the next morning. He did not even legend, involving areas such as sex, vice, "con men." and have the price of breakfast left. · ·· the mob. We must further emphasize that the legend is a . · Smitty's system, which was first used in the mid-fifties, c:Omposite of many separate stories told to .m.e by different !believe, seems to have spread to a sman group of pta~ -peop~ However, for !}1e most part the stones are mu~ lncludmg' a certain old-~e gambler whom we shall Call

v~6 BJfAT.TB1!· i)l!ltL'Blt

Mr. F, Mr.·Fs mistress,. the Mra X ofChapter 5t Mr. Z, .lho. little dark-haired f}lY mentioned • previously, aud .a )'OUJJg player commonly known both as Junior and a8 . ~y."

This group of players pumped large sums of money out of the blackjack tables within the next few years. There is no way to determine exactly how large tbe·sums were. For what it is worth, the "grapevine" credits Mr. F with $so,ooo gross winnings, Mr. Z with $56,ooo gross (after-

. warcJs.divided with his bankrollers), and Mr. X with $too,• '()()O to $150,000 gross. The little dark-haired guy is sup­pQSed to have cleared $250,000.

. In any case, the nrembers of the group won large tgq()unts ·in short times in only a feVI( casinos, and as a consequence the casinos, which had initially been skeptical of the possibility that the game could be beaten, finally

.. barred each of the members of the group from play at the

. twenty-one tables and spread the warning.

. Omno Couniermeasures Against Count Players

During and after this period a number of casino re­JPOnses and countermeasures to count players either' came into existence or developed further. ·

i · ·· Cheating. · Cheating has already been discussed. Barring. A casino can exclude a small class of

playerS without difficulty. However, this solution does not ICeDl feasible on a large scale. With the early system players, photographs could be distributed to all ·local casinos, but for thousands of players this idea is simply impractical. Along these same lines, even though a given casino's employees may remember a particular individual and bar him from further play, barring is not a defense for the casinos as a whole because it is possible that the ·m- _ dividual may work his way through the hundreds of exist­ing casinos and allay suspicion- by winning opty a· few hundred dollars at each one.

.. , $clence Vme Chtnu:tt ·- -··171

It is obvious· that casino- employees me trained to ro-. member people. Junior (also caUe4 "SoDDy") toJ4·blo 1t1at after ho _. universaUy barred in the casinoB;· he weal to the make-up departm$1t of one of the Hollywood movie .. ~ atudi01. He paid Ssoo for a complete disguise. On: tho basis of his-facial structure, color, and build they decided to dis-

. guise him as middJH.ged Chinese 1be disguise even Ja. 1· eluded a carapace to be fitted over bis torso• He tried Out t ' his nice new outfit one evening in a casino in which thoro -~ were &ix employees on c:luty·who knew him. Five of them ~~­paid no atteution to-him. Shortly after ho began to play, the .~ sixth employee wandered over from tho bar. spotted him at 1. once, aad eKCiaimed, "'Hey, look everybody. There's SoDDy 't

all dressed up like a Chinese." Junior still keeps his beloved t Chinese outfit stored away somewhere, buric<l unc1er years t of accumulated dust. [.

Shuf/16 up. Shuffle up is another casiDo strategy tbat J; Is effective against a amaJ1 class of players bUt has a serious i:_· drawback when system pJayen become numerous. It costs ·r the casinos money by slowing down the game,.aad"it also 1_ ·.

· alienates some customors. Farther, as we haw 'already .J. pointed out in Chapter 9, shuflle up is fraught with diftl· ' culties for the casino. How does a dealer know when-be is facing a system player and when ho is not?1be best COUDt ·~ players can play faster than any dealer can deal, and r 11111oother and more effortlessly than most players. Thus they f have ample freedom to adopt a guise. And t11c= ate many ·~~ IUbt1o ways to camouftage varying one's bet size (when nee- ~ essary tho variation can be mado quite small, ~ to 1 or evea ~-· ~~ ~

Further DINlopmenta As we noted in Chapter 9. the casinos tried ~

changes and gave them up. Tho outcome of the exporimenta with automa;tic blackjack macbiDes remains to be seeu. ·

r-•. ~-~.,~~----~-__...~-~-B-;·~'l'.-THJ1 &BA!.IUl.\ -~VIf81#·~

I Low-cost miniature devices have been manufactured ·bJ . players to do most of the work of counting .cards and playin'g CIJrnputers Versus Casinos the various strategies. · The next step is obvious. A .computer oan be ill'<

119 -~

. . The ultimate player would be a higb.:speed computer. structed quite easily always to make ·the best possible play. A first step in this directipn was taken several years ago. by The player simply tells the -machine what the casino rules. Robert Bamford of Jet Propulsion Laboratory. Bamford de- are and then informs it of the cards he bas seen and of the signed a "black. box" to play blackjack. It is an electric shuftles, Then, a split second after ihe player tells the ma-analog device that performs an approximate computation chine his cards, it would tell the player the best play. for an arbitrary subset of cards. It tells the player his ap- It is technically f~ble :to link a casino blackjack proximate correct strategy and approximate advantage. player by radio to a remotely located giant machine which · Cards are. read in by turning knobs attached to gang- does the actual playfng: The. player would win at several switches. Information output is from a meter that is inter· times the rat~ of the best buii\BR players. Furthermore. rogated by push buttons. The analog calculation is based special situations such as the Puerto Rican end play of on a matrix approach similar in spirit to the matrix formu· Chapter 6 could be. exploited easily and precisely. . lation of the iz$Ute-deck baccarat calculations of [70}. The If -a large computer were used, it could play $eYel'al devico I saw Consisted of two portable-radio-sized boxes games at once. In industry computers are now commonly •. wired together. Miniaturization to book-size or smaller used on such a time-sharing basis: while the machine is WO\Ild be quite f~'ble. The device could be linked to the waiting for more Information on one problem, it spends ifs ..

. player by radio. It could instruct him automatically· upon · precious time solving another. interrogation.

~amford tried to persuade two casinos to play his Science Y ers~ Chance macbin~ but, in a comedy of double-talk and evasion re- nie IJlathcmaticaJ theory of probability originated in ported by the Pasadena Sw-News. the casinos frustrated the sixteenth and seventeenth centuries with the considera-his_ test. . tion by. Cardano, Pascal, apd others of various gambli118

But a ptatch between another computer ~d a casino games, and With their investigations as to whether or not did talc« place. At the time of -the 1963 Western Joint there were systems ·for beating them. Most notably, these Compater· Conference, a. trio of Los Angeles computermen games were the forerunners and relatives of craps. From ~ on the Tropicana Hotel. The three operated the eight that. time until the recent past, a. series of persons whose· pound LGP-21 for an hour. The machine's method of play names are illustrious in ma~tics and physics have was inspired by .the ideas of the first edition of B.eat the thought seriously about gambling games (and have often Dealer. . made important related contributions to the mathematical

In a one-hour match, witnessed by Paul O'Neil of theory of probability!).ln addition to Cltfdano and Pascal, . Life, myself, and many others, the machine won $36o [44]. . lOme· earlier examples arc Fermat, James and Daniel Bern-Bets varied from $2 to $42. And the JD8Chine won in spite· oaUJi, Laplace, and Poisson. · of. the fact that the harried computennen · made several At the tum of this century, the great mathematician costly errors. and physicist Henri Poincare considered the possibility of ·

do BBAT 'T~B bRAtlft

' predicelng the ~tcome· of a trial 'of roulette by pllysical rather than mathematical methods. He concluded .that ·this was impossible via an argument based on thC mathematical 4oncept of a continuous function. Ho~r. the concepts in­wived illustrated certain philosophical concepts in science (see [52), pages 69-70 and pages 76-77). Also early in this Century the great English statistician Karl ·Pearson spent many years analyzing the records of certain roulette wheels. However, for more than forty years there seem to have been 110 successful scientific attempts to devise winning,gamblinB .......

The modern high-speed computer, essential to a care­ful analysis of blackjack, has been widely available for only 1he last ten or fifteen years; without such a computer the IDalysis on which this book is based would have been im· ~'ble. • With the continuing rapid growth in the number of scielltlsts and enginee~ and the rise of fantastic new

··ldeatiftc tools, the interest in the possibility of winning pmbling systems is increasing.

• . .In the first edition we predicted tha~ scientifically based WIJ1D11lg systems for other games would appear. Within a few months of publication, a team of trained players and I went to Nevada with a winning systeD1 for the baccarat &ide bets (S9o70],

We averaged $Ioo an hour for seven nights in casino 'nuniber one. It cried uncle and barred us. Later it removed the side bet. At casino number two we upped the bets. We· averaged $r,ooo an hour for two homs and they barred us.r

·The side bets then disappeared in Nevada. • • · · · . Allan Wilson gives an interesting and entertainine ac-

count of attempts to identify and beat defective ("biased"' •ne mM 704 hiaiHpeecl computer which we used spent about

Claree houri caJculatiog. It cak:ulatca maD)' millions ot times 11 fast 11 a bwDaD lllCl II aearly error-free. It would have taken roughly tea thoulaacl ~yean to do the eamo cal~latioal with the aid of a desk calcalator. Still bigger and faster mac:blnes are now available.

t Scame'a jambled account of out ploy, wbich he c:laima to hau ~ from his casiao pals, Is pathetically fk:titious. . -

•• As this is written, the Carousel in Las Veps is trvtn,. a moclifiecl ad "'afcr'" venioll of thole lidO betl. -r "" · ·

':?',.

~~« V,~~ I$1

,roul~ wheels in [So]. There ar&also.~ peqple ·(in­~<:luding myself) who~ a method.for beating roulette

_ wheels whether or 119t they are defectivcl ,_ I. played roQ}ette on a regulation wheel in ~ base­

ment.lab of a world-famous scientist. We used the method .and. steadily averaged 44 pc?r cent profit. In an hour't run, betting no more than $25 per number, we won a fiction&l $8,0001 There.are certain electronic:: problems which have SQ f~ ~pt the method from being used on a large $C8le in :he casmos. (Tho few times I have used it to turn two or three dim~ suddenly into a pile of silver dollars baa caused enormous excitement.)

The method works, and the story behind its discovery and development is a long and fascinating one. It will be even more fascinating when, sometime in the next few ye&rs, some of the few who possess the idea cash in on it ia the casinos.

The game of poker has perhaps received more intense mathematical study than any other game. With the. exten­sive theoretical research that has already been done en the game, it should be possible to construct a practical playing strategy superior to that currently used by any ~pert.

The Stock Market The greatest gambling game on earth is the one played

daily through the brokerage hpuses across the country. The customers.-~ $2so,ooo~ooo or so each trading day. A ~·s ac~n-·exceeds $6o,ooo,ooo,ooo. The advantage of this ~bling game are two. First. it presumably serves a. social purpose by hs:lping to finance companies (when stock issues are first sold on the market). Secqnd, the average "value" of stocks has tended strongty upward over the last century so that the game has. an "advantage," on "'average," for the player.• ·

• ~ Quotes are used . euensivciy heN to .indicate that I am usins famtliar words with weB-defined meaainp to roughly india1te other ideas. Pn=clse oxplaDatiolls of tbeso other ideas. to which I have givca much thoupt. arc too complex and lengthy for me to discuss them bere

··tb .. BBA'I' 'I'HB DlfA:ili-t·

_ · ·ne iimDarity bet\veelt1be casinos and the brokerage ~louses is·strifdng. The customers' mea ~1he croupfen. ·-~-~ 'correspond·to the house percentage. the board ·1'00DlJ are the casinos themselves. The stock ex­chiDges and the ticker tape are the gambJiDg devices. The superstitions, unfounded slogans, and sayings of wan Street correspond to those of the gamblers': "The· dice are hot."

To a good first approximation, stocks show the same mathematical characteristics of randomness that are shown by the chance devices in the gaming houses [7]. But a aumber of patterns are now being discovered. To convince yourself tliat there are patterns in stock prices. pick up tomorrow morning's newspaper. (I assure you I haven't ~een it yet.) Notice that the stock prices are giwD as a whote number followed by a fraction, for example 23Ms. Now run down the page and tally the number of each end­·big· tluU ~ You will find whole ~bers ~be most conunon, ;;hen halves, then qu_arters. The lowly eighths occur least often. (I ignore six~ths.) In addition to this pttem of endings exhibited by the market as a whole, each stock has its own characteristic pattern. for these fractional enc!ings. . -

The mathematical analysis of the. stock market is being undertaken by many groups. With the advance in computer technology and Qlathematical theory, We .can expect dra·. matic progress.bl predicting·~ prices. (U~ortunately this

In Beat The Market: A Scientific ~tock Market System. S. Kassouf and I present a stock market system which ap­pears to us to combine high safety with high return. It has averaged more than as% a year for us under widely wcying market conditions.

TheFutur~

In the last part of the twentieth century there wiD be many new applications of scientific and particularly math-

fme called .. chance.~ .We ltave tried to· indicate. a few of the developments that are similar in spirit to.thpse desen"bed in this book. But most of the posstDilitie5 are beyond 1:he reach of our present imagination and dreams. It will be ex· citing to see them unfold.

ematical methods 10 1bc p!Odlction of pbenomeDa hemic>- .[

----------------------------------------~-------------=~----------------~~--~//

Addendum

--

Blackjack in England

·_. Tilt winDing_ systems descn"bed In this boo~ appear to · apply with full 'power to blackjack as played m England. The followplg detail~ ctiscussion suggests th~t the methods -

,; · that/IUCceeaed In Las Vegas and Puerto Rico should do .., well here. The diScussion will be of interest to ~-

- . liSh '1'Nefed Ud to diose who travel in England. It. will " _ aJso be of general interest to readers ~ho ~ .tc:) ~k

~heir. underltalldiDg ·of Table 9.2 ·and 1ts use m analyz1ng catiDos' ·ru~es ~ations. The computations made from Table 9.2-~'differslightly frOm the results in this adden­dum, becalisC I in~uccd extra refinements whenever I was _ awaie of them.

The following is a description of the rules and customs • C)f tbreO big London gambling clubs, ~t in by an English reader, followed by my analysis an~ ~ments. ·The ru~ and customs am those given as "'typical m Chapter 2. witb

· the exceptions noted below._ _ . _ • _ • Ia each case tqere are from one to SIX players besides

·•84·

the dealer, and four decks me used, dealt from a shoo. · (Four decks are disaclvantaP.~ ~ the,RJaycr.)

Casino I Rui.E& (a) No card is bumecL

(b) ·The minimum bet is £1 and the maximum Is £so.

(c) ~ cards are dealt face up, except for oae Of the dealer's cards. (This is a considerable

-aid in card counting.) . {d) A player who splits a pair and receives a

· third card of the same value is permitted to split agaiD. (This increases the player's- ad­vantage.)

(') The izJsuranc8 bet may be up to the whtM amount of the player's origiDal bet. not merely half the value. (This is very favorable to the system player. When- insurance is adVDDo tageous, the carck:o1111tio'g. player sbouJd ... - <

ways insure for the whole IIII10UDl. '· Customs. ~ dealer always deals dowa to ·~·

ctactmg the Iast c.ans. <This .is extremely fa~ tor the card count~· ... .t\Jso. end play can be used m ~­advantage .h~-No casino. can long contilp.Je tO. play..._.. on -gainst aif expert player with tbis · nm, in e.tfeet.) :

... Conclusions. The b~tegy player has a ~­~tage of o.27%l An average or better~ plaJef wUI win at more ~/the ,typical rate (assun»os 1le ~ 1101110 advantage ot eild play). . . ·.

CasillO 2

Rt1tis. (a) See f(a). _ - (/1)' The D1inbn1111l ts Ss. · an4 the muimUIIl is

£so. - · · -~· '(C) See, I(d)_. _ -- ,-· . "

(d) A plaJcrwllinplifS a p&Jr is DOt peijmued- to '>.. -·- • ~ ~ - •

~r-_,

j

1 -:··!

1.6' B BAT TJIB DB.At.;Btt·

-dou'ble dowa: on thC ·JleW llaDds. (This de­creases the player's advantage.)

(t') Doubling down is only permitted on totals of bard 9. to, and 11, and soft 19, 20, and u. (This decreases the player's advantage.)

(f) A player who doubles down on 9 and gets a 2 may draw another w-d. (This increases the player's advantage. A correct strategy calls

· for sUghtly more frequent doubling down on 9, but this refinement can be ignored.)

Customs. See Ca5ino 1. Conclusion.t. ·The basiHtrategy player bas a disad·

vantage of about o.6o%. An average or better system .player will win at about the typical rate (assuming he takes some advantage of end· play._)

Casino 3 ~ (a) :ne minimum bet is lOS. and the maximum

is £.so. (b) .The cards are all dealt face up~ The dealer

does not get his second card until the playm have taken such additional cards as they re­quire. A player who increases his· stake by splitting and/or doubling down loses his in· creased stake if the dealer gets a natural. (This is somewhat unfavorable to the player. When the dealer shows an Ace or Ten, be much more conservative about doubling down or splitting pairs.)

(c) If the dCaler bas soft 17, be may draw or not. as he likes. after looking at the players' Ca.rds. ('Ibis is quite unfavorable to the player, for . \· the dealer can see what they have ·and so . make decisions quite favorable to himself. His preeise gain is not known.) l

(d) A player who splits Aces and obtains a tbird I 1

Ace. may. split again, but he. may not split further. (This is favorable to the player.)

(e) If a player splits a pair of Io-value cards and draws an Ace to one of the spUt hands, it counts as a natural. But a 1o-value card drawn to a split Ace counts merely as' 21. (This variation is. favorable to the player. rens should tend to be split slightly more than normally.)

(f) see 2(e). (g) See I (e). .

Customs. The dealer shuftles about twenty cards be­fore the end of the shoe. (Not serious, in my experience.) At one time this casino allowed doubling down. with the

. above totals, on any number of. cards. A player whose total after doubling down was bard IO or 1 I or soft 20 or 21 was allowed to redouble and. receive a fourth card. Also. the dealer dealt down· to the last card. ·

Conclusions. The basic-strategy player would appear to be at a disadvantage of roughly o.6I% or moi.'C. A sys­tem player will win, but at perhaps half or less of the typical rate.

In this way readers may analyze any casino with the aid of Table 9.2.

l r t j l f

APPENDIX

Basic· Probabilities for

the Complete Deck

An understanding of this appendix is not essential to. the reSt of the book. It is included for the interest of math­ematically inclined readers. . . '

The tables in this appendix are an extract of the com­puter's results for the case in which cards are dealt from one complete deck. Results. like these. including one full set for each of the decks described in Table 4- I, were used to construct the blackjack theory givt.n in this book. Because of· the extreme length of the data-there are enough final ·results alone to fill several books the size of this one with numbe~ litnit ourselves to presenting complete-deck figures, and only ari extract of these. Since in our discussion and application of these figures we generally do not need more than three decimal places, the tables given here are usually to three decimal places.

All figures are to be understood as having a decimal point on the left, though the decimal point is omitted. For example, -039 is to be read as -0.039. . ·

We emphasize again that figures in this appendix were.. • rsa•

~Basic Probablli~.Jor iM Comp~ Deck

computed assuming one complete deck and the rules of Chapter 3, including the rule of a dealer's soft at8ncting number of 17. Since the figures may vary c:oDSldcrably if these conditious .-re al,tered, any·deductious one makes on the basis of this appendix are precisely applicable only to the situation just described. Such deductions do, however, give rough insight into situatioDS in which there are several decks in play or in which the roles are different. ·

In detetmruing the player's strategy, the errors in our figures may cause enors, but only when the decision is very close. And in that case the errors caused by the erroneous strategy will be very small. This, plus the low frequency of close decisions, meaus the effect on the player's advantage of .any strategic errors of this type are 8J.so very small. "nlis IS a virtue of the running count. Roughly speakiilg. certain dOse decisions, such as whether or not to draw on hard 16

· wtlen the dealer holds a Ten, will depend on what cards the player bas drawn to make up his total of 16. For ex­amplC. ttis known that holding (10.4,2) the player Should stand, while holding (Io.6) he should draw. If several siDati cards have been drawn to make the total of I6, the decision may be fairly clear-cuL For example, if the cards . drawn were (4'.4.4.4), Julian Braun has shown that the player's disadvantage in drawing against a Ten is preclselJ (!) 6.382 per cent, as compared with the average. play~ advantage of 2.9 per cent (3.2 per cent if (8.8) is split)~ in drawing to two-card hard I 6 against a Ten.

One could attempt to improve the basic strategy by. calculating the advantage or disadvantage of standing ~01' drawing for each combination. of cards the player can draw to make a total of hard 16. Then the ptayer could consult a -list of card· cOmbinations to see Whether to draw or staml This refinement,. in aU its precise detiill, is impractical be> ·

. ca1lSe its hulk (many hundreds of eiltrles) would prevent

. the player from memorizing amf using it in play. Further­more, the net gain is quite small.

~---------------..... - .......... --"""'r._'!!!l';_~~~-'!~~1!!'!' __ (:_-!""_ ---..-.---"7·~~-=:::--: ~, .... ' 190 BBAT TBB DBALBlL

-, However, the running count in J;Ollj~ _With tho -Tens strategy does-take into account the cants the player. dra'\\1& It is not as ~ as the detailed strategy outlined above because it only classifies cards into two crude cate­gories. Tens ud others. But it does gain much-even most-of the difference..

'I'AJII.B 1. Dear.t'. Pro1Mbllltl& · J)ea1cr Dealer's total ~ 17 11 J9 20 21 natural busts

z ·1390 1318 1318 1239 1~05 3530 s 1303 1309 1238 1233 1160 3756

• 1310 1142. 1207 1163 1151 4023 5 il97 1235 1169 1047 106) 4289

' 1670 1065 1o72. 1007 om 4208

' 3723 1386 0773 0789 0730 2599 I 1309 3630 1294 0683 0698 2386 , 1219 1039 3574 1223 0611 2334

11 1144 1129 1147 3289 0365 0784 2143

• 1261 1310 1295· 1316 OS16 3137 U6S ov.aJl

probability 1458 1381 1348 1758 0736 0483 2836

-~~#iobatiiiJies tOr the Compltte Declc 191 . ·. l:AIII.B 2a. Pl41er'1 GW. ~ DttZwtng twll' Slandlllf

- with HMil 2'ot111l ' ·

·b!e!er . . PlayeR hard total •. lbowa D 1S 14 U 16 17 18 U

- 3 038 ...016 ...(T17 -141 -171 -383 -753 -1.135 -1.474 3 013 -045 -117 -179 -212 -417 -775 -J.JIH -lAC • -o17 -o86 -158 -222 -258 -467 -761 -1.116 -1·.491 -. -046 -:117 -191 -260 -297 -448 -793 -1.157 -1.S1t ' -(l2S -()94 -167 -233 -220 -470 -853 -Ll90 -l.SC ' 20J 166 114 u' no -331 -'J57 -uoa -uoa I 189 148 145 108 '102 -{119 -657 -1.274 -1.626 ' _141 ~145 103 062 055 -114 .....tOO -964 -1.586

10 156 119 07J 038 029 -148 -471. -813 -L420 A 246 221 186 159 146 -o89 -554 -1.050 -1.533

- To illustrate the use Or Table 2a1 suppose you have a hard total. of ta and the dealer shows a Two. If you decide to draw rather than stand. your gain is 0.038. TJUs ..-na that on the average over a large number of situatioas Ute­this one.:il you -always draw rather than stand. witia hatd 12

,against a 1\Vo, you win be better o1f by appmximately aa additional 3.8 per cent of yOU!' Initial bet. If an entry is

- Table 1- gives the probabUity 'that the dealer wm positive in the table, the player should draw rather thaa achii:Ve a speci1ied total for eacb possible value of his up Jtalld. Con~ly, if an entry is =gative the player shot,tld card. Tho rows of the table do not generally quite add up s.tand• not draw. The inspection of this table immediately to-one because of small round-off and approximation errors. J!$elds the hard standing numbers. This is, in fact. how they The defect is 110 ~ than Io-• and so for practical pur- were first obtained. poses is negUgJ.'ble. The column totals show slight discrepan-. : - .• Similar remarks apply to Table 2b except that the _cies with the overall probability figures because the original entrieS yield_ the soft standing .numbers. table laad five- figures and was rounded off after the columns _ Thero ~ lwo extremely close decisions, one each iD were summed. Tables 2a and 2b~ In Table_ 2a, the player who standS rather

- ThiS table is of course valid only if we assume that the ;than draws on two-card hard 16 against a Ten loses, in dealer plays all his hands ~ut to a conclusion even though ~ situations, ~ average amount of about 2.9 per cent his opponents all bust. ln an ordinary game the dealer tloes :pf his W8F'· .CW1th (Io,6) the loss is 3.8 per cc;nt. with not do this. . (9,7) .it.is o.8 per cent. and :Mth ~8,8) it iJ o.g .per cent. · From this point on. all tables are computed on 1M .~~numb«~ with worgbts 3z:8~ frQJ:D prob-

IUSIImpticm-thizt .the dealer does not have a natural. . . llbitilf theory gtY~ 2.9 per cent. Jf (8,8) is Split SO tbat.Jt .. -iS DOt inclUded. the figw"e changes to 3·2 p~r cent.) -

,. il 1:

~~ •'

! ~ " ll lj l

l.

,.ABI.B ·2b. PIQen ·o• t~t DlwnP6It-.. ~· · SDft TDitlll. ·

Dealer Pia,.... lOft total tl llhowa- 17 18 u

I 141 -on -28$ -470 s 131 -G74 -251 -45J

• 118 ·-cMS -233 -430 I 141 -o46 -235 -419

' 131 -fJ67 ~ -418 f 151 -230 -388 -528 I 319 -G71 -442 -f08 , 270 on -280 ~

H l33 04S -157 --541 •• 291 -oot .-303 ~14

In Table· 2b. ~ player wbo draws rather than stands on soft 18 against an Ace loses about 0.1 per cent in such

_ situations. Some:playersl blow of attempted to solve bJack· jack empirically; .that is, they dealt out many hund.reds or

. evea thousands of handt and recorded results. in an effort to decide whiCh standing numbers were correct for varioUs

.· up cards of. the dealer. As might _be expected, these players were sharply divided on these two cJoso declsfoas.

Table 3 was computed dJrectly from Table I as f.ol-. lows. Suppose tho player holds a given total. say 19- when tile dealer's up card is a 6. The player's. ad~tage is then tho sum of the probabilities that the dealer will receive a poorer total (18,17. or a bust), ·0.1065 + 0.1670 + o..po8 = o.6943, minus the probability that the dealer wiD have a better total (20 or 21), o.1007 + 0-0979 = 0.1986. Tho diffetence 0.6943 - 0.1986 equals 04957,

_and IOlilading ol to tbreo significant figures we get 0496. the entry iD Table 3· .

We are assnming in Table 3, u we said euJjer that tho dealer does ll()t have a naturaL In · thia situation tho player holding a naturallllways wins I.S times hia ~ bot; that is, hia advanta&Ot in tho aeuse that ..,. are U11J11

i93 the t~,is JSO ~.cent.1hwi thereJs no need to Ust that alterriative in the table.

TABLB 3. Plap¥'1 Advantage 8tantllng tm Vmlou.r 2'olil!f.

Dealer Player's total shows 16 . 17 18 u tl u

2 -294 -155 116 '" Ci35 m 3 -249 -119 143 ,,., 644 884 4 -194 -663 l8l 417 ,, 88S 5 -142 -o23 l21 461 683 8H

' -159 009 281 496 704 9G 7 -480 -108 403 Cil9 115 927 i -$23 -391 lOl 594 19l 930

' -533 ' -411 -185 216 156 '" 10 -535 -411 -16-4 083 564 -A -660 -477 -102 . 218 Ci58 tlS

Table 4 gives the player's advantage for aU possib1o pairs of hOle cards against a given ttp card-of ·the dealer • assuming first that the player simply atands or draws viUlg the proper standing numbers (deduced from Tables · 2a and 2b). Then the player's advantage if he doubles clown is given. Finally~ the plaF"s advantage is given. for the cases in which his hole cards are numerically equal a he split$ the pair and then does the most advantageous Of die two alternatives of doubling down or drawing and ltiDding. The table is divided into ten m~ sections, ODO for each value of the dealer's up cant

The basi<: «ategy for each value of the dealer's ap ·card can be deduced ftom the table as.follows. Fust, sup­pose the hole cards form a pair. Compare tho player's ad­vantage from splitting the pair with his ac:lvaDtago from doubling down and from drawing. If it is greater than these.

· he should split. Otherwise he should do the JDORt advlmta­. geous of doubling .doWn and drawing or standing. For •· ample, on holding (4.4) against a Ten. splitting gives an . advantage of -o.ss~; doubling down gives an advantage

-- ---~- , •. --.=..,....------ ---~-------~- -T=- ---- ---------~-~ ~-

194 -B B A "r TKB J) I. A 1.11 ~ _ 11IIPc ProiNIIJlliJiel_ /or 1M Complel6 Deck

Of -o.739; and drawing and standing. using a hard staJld. lng num'ber of 16 and a soft standing DUmber of 19. pves an advantage of -0.241. Because this last figure is tho best of the tbreet drawing and standing is the best allemative. Thus the player should not double dOwn or apiit in this instance. '

If the player's hole cards are (.4,2 ), this is the same as_(2,..t); so only one of these two spots is filled in tho table. Thus tho subsections of Table 4 get their triangUlar ahape. .

Table 4 sheds further Ught on points about the basic arrategy. In tho discussion of pair splitting for tho basic strategy, we say ·that if a pair Qf Mel is not split, the hand i is "only fair for doubling down or drawing or standing," ~ whereas splitting gives us a good chance for a winning . .:: ~ Table 4 gives us tho preclse advantage for the most -l poiitablo of the two al~ of. doubling down and ' II. 4rawing or $UUlding. We see that the numbers waver , ~ around o, with some positive and some negative. However ,: the corresponding advantage for .rpUttlng the Aces is gen- ·

8

eralJy Considerably positive, as shown by the table. Simi- · . Jarly, tho rough oversimplification that the -splitting of a . ,. pair of Bights against 7 through ..t breaks up a bad hand and .replaces it by two average l1ands. is (only roughly) "' borne out by the table.

For the complete deck, Table 4 was used to compute the player's average advantage against varlous up .cards of tho dealer's and then the players overall advantage was founcL Similar results. were obtained for the other decks from the appropriate data. The results are liste4 in Tables 4-1 ud 9.2. They give some evidence of how much tho ~figures we give are a1fecte4 when the deck <:Omposi­. don or tU.rules vary.

•

"' "" f'\N f'\0

i-

-~-----------__. ________ ~-~~---~--------

195 l

=,.,..,-_::;..";,?;1

TA•t.a 4b. btt~fn'l Up Ctutll1 Nln&.

Playcr'a bolO o8rds ....

f A

f: 1 :

{j A 2

J : 1110 $

:§ ' ! :

A

-oo%1 -o141 -o600 -1134• -1668 -1347 -:0870

2880 76$6

1.$000

-4206

2

-2225 -2625 -3046 -2840• -2084 -om

120S 1425

-3444

-3727 -1.o46J

3

-3100 -3039• -2171• -o5u

U76 1392

-3921 -3588

-3935 -1.0659 -1.0653 -4118 -1.0453 -9462 -4520 -!1263 -7029 -3451 -6916 -2897 -2545 -2741 1537 ~ 1744 2152

nos 2138 -8404

..

-2044• -osu•

1174 1495

-3949 -3652 -m5

-7011 -2790

1646 2247

-8331 -1581

I

1204• 1517

-3896 -4134 -41St -4753

1746 2399

-8260 or-8476 -8598

,, ., MH=17 MS=It

•MH=I&

-3862 -418$ -4746

8

-437:1 -4432 -4811

'

-4191 -4820 -4116 -1964 -4793 -4161 -1961 2643

-8174 -8694 -9782 -8879 -8936 -9742 -9668 -!1641 -1.0638 -t.189S

••

7440

' lt 1165 ..;.7455 -7521 -8394 I -9519 -9586 -1.0529 -1.1864 -1.3956 -1.6841

Pair Splitting 2898 . '7373 -395 -459 -560 -541 -535 -383 -()93

TABLE 4c. Dealer's. Up Card Is Eight,

Player's hole cards-

! A 2 A

l ! Cl

~ 4 "':t 5 ll 6 110 7

·m 8 i'! 9 Q 10

A 2

i !

I : 7 8

' .lt

0930 0391 -1410 0350 -1808

-Q355 -2342 -Q843 -2178 -o649 -o5s9

1209 1076 6078 2073 7848 2153

1.5000 -2745

-2956 -3124 -1.0298 -2542 -1.0254' -3141 -1.0231 -3326 -853S -2297 -4371 -Q153 0074

1902 2945 2298 3277 3317' ' -6259

3

-2311 -2284 -Q570

1175 2171 2203

-3161 -3282

-!1997 -8486 -4282

0142 3269 3300

-7037 -7056

4

-Q548 1081 2075 2217

-3192 -3389 -3574

-4471 0007 3172 3402

-7193 -7241 -7393

5

2075 2297

-3210 -3868 -3701 -4180

3229 3657

-7112 -8291 -7619 ·-s4~

6 7 8 9

MH=17 MS=t8

-3217 -3944 -3691 -4207 -4248

-7114 -8429 -7746 -8556 -8496

-4019 -3796 -4263 -4278 -4149 -3942 0955

-8519 -7774 -8526

0645 5768

-8556 -!1506 -1.1325 ~97 -:1.1303 -1.3947

hit SpUttblg ~~~~'__:-!9~ ___=244 -291. -391. ·-374 -378 -o59 207

171

10

7832

-U8S4 345

... '&

w tit.· >-... .... · = tit

"· til ~ ... > ~

= .i;;. h tl:i

~· "tt

_g. ~ ;::: ~: Cot

l :i­~ ~ "" ~ ~ ~

.... -~

(

:' \ t

•• ~ r~

... -~

TAIILB 4cl. D«dtl• Up Cartll1 In&

Plajer"s bole c:anls-

f A 2 A

Ill) 2

1 : "' 5 fil ' f: a ' 10

1584 1073 0604. 0337

-o238 0596 4120 614$ :7732

1.5000

A -1370

-o918 -1192 -1639 -o674

0918 1836 2676 288!)

-2120

2 -1572 -9421

s

. -1645 -o706 -0926

1977 2772 2918

-2471 -2704

4

1106 2013 2856 2938

-2456 -2741 -3422

~ 3 -1745 -9378 -8712. -8 4 -1409 -8703 -5426 -1084 Ill) 5 -1891 -5672 -1323 1905 ;§ 6 0142 -1481 1757 4754 il 7 2402 1600 4447 4874 iS 8 3253 4166 4726 -5636

9 3513 4554 -5188 -6001

I

278G 2974

-258% -3275 -3480 -3645

<1663 5005'

-5852 -7065 -7276

' ' 8 9

MH=17 MS=18

-'J.011 -3307 -3m -348S -3241 -3736 -3632 -3750 -122!) «<ll -3761 -1213 3887 6101

-5985 -7141 -8230 -7346 -6660 -7471 -7446 -7499 -8976 -1.1321

fD

7647

10 4676 -4958 -5830 -7068 -7363 -1524 -9039 -1.1345 -1.3971 -1.6860

P~SpUtting 5401 -oo6 -o68 -160 -228 -228 -os6 259 364 ·

TAIILI! 4e. Dtaltt'l Up Cartl I• SIZ.

Player's b'ole cards-

f A 2 3 4

A 1996

f 2 1685 0320 3 1472 0192 0139 4 1203 0141 0592 1753

'!2 5 liS!) 0696 1637 2633 Iii ' 1332 1321 2313 3316 .r 7 2622 2097 3180 3657 'II! 8 4824 3075 3561 -1473 a 9 6941 3458 -1538 -1514

18 1.5000 -1604 -1578 -1554 A 2479 2 2302 -2490

~ 3 2218 -2438 -2145 o 4 2001 -2150 -os69 1932

I

3618 3807

-1435 -1460 -1501 -1542

' 7 8

NH=11 MS=IB

-1651 -1697 -1142 -1723 -1770 -1782 -176S -1796 -o114 -1790 · -om 2681

-: 5 2i67 -o610 1899 4433 7236 ;§ ' 2665 1242 3792 6633 7614 -3871 i 7 3849 3574 6361 7315 -3577 -5613 -733~ a a 4826 6151 1122 -3682 -$351 -6465 -7241 -7934

9

2651 4841

' 5598 6916 -3876 -4563 -6115 -8018 -7944 -9616 -1.1734

478

10

6974

10 68~ -3189 -4578 -6161 -7746 -7926 -9667 -1.1721 '-1.4140 -1.6900

PaftSpHtUns 758J 240 220 183 131 151 220 356 437 543-

... ' ~ GO

• til' >. ~

~.

= til

tJ til > l"' til ...

:~ f)

~ ~ at i' l i­&' t it

r

I ..,_,__,, --

,_...,_".'·-~5'-"?."'::":·;~~~-":""?+·--~-~iii!~~-~!I'!~!HJ'"".~""o":4£'!¥l. 2EUL_!¥4¥. £&_~ ___ ¥¥QC1§ ___ L- .B .IJ .. ~Ubf!§ .· .. W.4L4-%=-41f-41£!4,W¥W ifi $i]iiiiij!!i!i'_~lllf'l'·-- 2~

t'.\11.1 .. ,. Dltlfd•, Coni ,.·Fir .. PlaJ'l'a hole cards..,.

__ ! A t . S 4 I ' 1 I 9 ·1e A 1820

f 2 1587 0359 3 1366 0215 0083 4 1078 0089 0490 1539

MH=12 MS=18

1 5 0821 0568 14i1 2464 3473 ' 1400 1306 2350 338& 3936 -1022

J 7 2123 1PS.S 3075 '3617 -1234 -1288 111 8 4608 2960 341»5 -1308 -1283 -1338 a ' m1 33PS -1!68 -1339 -1314 -1369

te 1.5000 -1440 -1412 -13Bl .:.1357 -1411

-1555 -1605 -1654 -1636 -0444 -()432 2023

A 2157 2 2123 -2140 = s 2036 -2085 -1840

-8 4 1750 -tass -os14 1m liD 5 1482 -()70'J 1550 4150 6947 ;§ ' 2800 130& 3916 6776 7873 -3247 ~ 7 3491 3321 6150 7234 -3555 -5381 -7407

-7462 -9058

201' 4478

-a • 4s3o 5920 6991 -3732 -s4oo -6344 ' 5362 6796 -3932 -4641 -6277 -8041 -9041 -9852 -1.1805

18 6630 -3171 -470& -6293 -7992 -8884 -9849 -uau -1.4175

PairSpllttln& 7322 239

Player'& bolo cards-

! .A 2 A

:f:a I : 1 : ., 7

! : 10 A 2

J : ·:~~ 5

:a ' ! : ' 10.

Pair SpUtting

1421 1102 0908 0614 0380 0773 2040 4155 6539

1.5000 1366 1151 1091 0849 0626 1S4S 3127 3731 4592 5?51

66116

-G348 -o410 -GSS2 -G103

0811 1676 2525

'3000 -1940•

-3840 -3517 -3237 -1946

0133 2718 '5049 6001

-:-3879

112

228 ,197 133 172 200 ' 312 415

T.UU! 4g. Dtaler'l Up Card 11 FoUl'.

3

-G474 -G166

0866 1860 2852 3126

-1913 -1971

-3022

4

0978 1896' 2928 3425

-1829 -1885 -1934

-1777 0441 0411 2970 2896 sass 5104 6851 6252 ' -4281

-4510 -5086 -5159 -ci729

102 076

5

2949 3520

-1545 -1813 -1859 -1907

5898 7039

-3836 -5836 -6710 -8360

014

' 7

MH=12 MS=18 •MH=13

-1519 -1584 . -1640

8

-1841 -1896 -2153 -1886 -1942 -G844 -1935 ..:.o644 1641

-3802 -5541 -7376 -6751 -7601 -9113

'

1670 4041

-8392 -9059 -1.0823 -1.1940 -9091 -1.0702 -1.1946 -1.4234

047 103 215 320

67'J7

-1.6912

521

10

6448

-1.6925 444

8:

w w >­... ..

-= ., 0 ld >' ,, t"' llf ... ,

r· :l ~ :::: ·f l i' ~ .e i" t:::l a.

·-~ .... ' /

~~·*-"-c-~~~...,_-¥¥£¥.::.fY.,..:.J. fo#S: ·': --:..·¥!J¥8¥L.~J£-i.)A_ &a&.- .. A4 ---~:-. ____ ;; .;__ J __ .a ---- --· .; ,.,-..._.._""!",~--,-·~·:--...,....""'!'r._<:

'

··~'

TAII.I 4fL· ~141ffl Vp Ctml.ls TltrH. -- ---~- ---"'------

Player's bolo cards-+

! A 2

A

f: 1 :

f: 10 A 2

J : i: ;g 7 .3 8

' 10

1203 0105 0442° 0234•

-oD19• 0369• 1668 4191 6441

1.5000

0549 0184 0108 0026

-o192 0739 188!) 3460 3912 $363

-oBIS -o983 -1062• -o61.S•

0238• 1308 2411• 1690°

-2193

-5015 -49.70 -4455 -3107 -0783

1946 4825 5380

-4386

3

-1181 -oats•

0194• 1389• 2461• 2952•

-2551 -1657

-4721 -3237 -o832

1891 .C921 5904

-5114 -5664

Pair Splitting 6128 024 . -oJJ

4

0288• 1411° 2504 3056•

-2202• -2460 -2509

-o825 1956 5009 6111

-4595 -5607 -7114

-o48

5

2548° 3147°

-2124° -2167 -2436 -2474

5096 6294

-4439 -6112 -7111 -8595

-104

' 7

MH=13 MS=18 ·•MH=ll

-2111° -1148 -2194

8

-1201 -2239 -2284

9

-1460 -2497 -1207 1225 -2499 -1190 1444 3835

-4443 -6081 -7765 -6939 -7626 -899P -8646 -9180 -1.0710 ...,1.2766 -9228 -1.0728 -1.2673 -1.4251

-o69 -o20 132 . 24.

TABLE 4l. Dealer'1 Up Card 11 T,.·o. Player's bolo cards-

! A A ·0948 l 2 0392

1:1 3 0169 · .a 4 -om '1:1 5 -o317 fl ' 0071 ., 7 1358 IJ 11 4016 s 9 6559'

10 1.5000

A -D193

2 3

-1132 -1314 -1530 -1507 -1230 -o994 -ot65 -o130 0921

0835 . 2144 2130 2682 2636

1 -2663

-2434• '-3042•

2 -o~20 -5816

4

-ot26 0933 2175 2723

-2598 -2858 -3102

1 3 -0468 -5866 -5615 .a 4 -o700 -5601 -42!)1 -1846

5

2238 2840

-2526 -2650 -2694 -2948

' 7 8

MH=1l 0 MH=14

MS=18

-2527 -2651 -2684 -2679 -2721 -2749 -2734 -2766 -1366 -2977 -1582 1188

. 5 -o819 -4018 -1678 1135 4464 I , o1n -1628 ms 4322 56n -so54 1 7 1276 1174 4260 5417 -5053 -6639 -8128 ~ 8 2372 4245 ~36 -5197 -6626 1 -7252 -7808 -9082

9

1370 3848

10

6361

-1.6926

383

10

a272

t . -3798 . $248 -5325 -5860 -7179 -8605 -9120 -1.0603 -1.2550 1. ~870 -4868 -6084 .. 7376 -8724 -9308' -1.0769 -1.2651 -1.4993 -1.6t33 !_alrsFut~-~CI.S,---(147-. --11& -156 ..:.193 -t6S~IOS~ 064 188 331

~-.,

w &If >

"" "ll

" ttl cf &If > "· &If' ..

-,; '·,\

·r.

I " l It n

i i' a.

! .._ __________________ ............. ~,'>J.?."f'l\._,•_,,,_,;.c_J'..<!i:F~Iolo.L .. .i......-,-lu:&.'·'" "F-~"1'.111

,...---"·~-"-"

' aG4-

• "'

• 1: .. ! t II ,, ll 4 w

I! 3 ,, ,, I, ~ !! •

~"" .. J_

t .. • ,; .. ..

~ f· ~; t ..

1

I -5 ~ oA .. .. .t-

IG "" 1:::

i-q, i"

!',

BBAT TBB DB.ALBB

... : ""' ~ ... s 1": T

£!s l;:t: :::: , ... ~~ i Ti -~~ ~~~ ... G\tt\CO CD 00 1D ID --- TTY 'C!C'! 'I _ .... --~i I I

!;t:;~~ ~00\0 ~. co-- :::: 1111 s:r~= ID .... ..: 1..: I - I I :1!: ref~~~= ! ........ "' 2 r--. .... ., ..... ""' 7f17Y M"''I::"O- 'f T"a't=:!=:!

I I

§ G\ 00 0 f'4·""' ;;!~s~~ - s;;¥; ... It\ C'f\M .-.4

'f Q Ttt=:! •

~ ... ~::~~::::~ u~=fj!i2~ ~ 0 o~a ... oo 1Y 0 ~T7T ~"':f!. TT; i' ii

~~;:~~!~~ u~~::::g~ru~ "' .... .... "co..,..~Q. ~ ... o.yf! -~T \0

771Y 0 aT "S7 I T I 1

~~~=·:s:z;~ -· .... a- ... -~s ... ~0\~fO\~oo~ ... :; ~~~ ~,soo- ... a7•17Y 1 "l"l --:'; ... , T

• TTii

~'§si~§§i~ =::t~~:;~:!;:t,~ 0\ ...

0\0\G\-V\C"ftt\G\ .... ~ 'ri'i''f'fi'TT ... I I I I I ...

I -<jf.,..,,.anw"'•~:a ~ ...... IIQ\0 .... ~=

:IIJ!PUlql pU8 :IIIJdl(l 1L1AOP iunCluoa ~ ·a ""

I a)

' 12)

I 31

[ 4) I sl

(6] [ 7)

[ 8)

References

AsHsAUOH, DoN, "'Game for Gaming," l.tu Vegu Review lottriUll (Sunday Feature Section), December 25, 196o, pp. _ 20~ 22.

BALDWIN, ROGER; CucnY, WILBERT; MAISEL, HERBERT; ancl McDERMOTT, lAMES, "1bc Optimum Strategy in Blackjack. • JouriUll of the .American Slllli6ticol .Auociation. VoL 51, 429-439 (1956). -. Playing Blackjack to Win; .A New Slratetf1/or thl Gt~~M of :11 (M. Barrows 8t Co., Inc.., New York, 1957) • &Mton Globe, January 24. 1961, pp. 1, 11~ Cuu>.ulo, GEROLAMO. Book em Gsmu of Cb.tnrc. ,(writtea about 1520 and first published in 1663), Trailslated b)' SIDNEY H. Gotn.o (Holt, RiDeban and Winston. Inc.. New York and San Fl'IUKlisc:o, 1961). Collllflb• Dl6patrh, January 301 1961, p. I•B. Coo1"ND. PAUL H., ed., The Random Character of Stoci Prices (M. I. T. Press, Cambridge. Massachusetts) • CRAWFORD, 10HN R.. How to Be tz Consistent Wbuter In tlw Most Popular Card GtUnes (Doubledq and Co.. IDe.. New York, 1953). ·

I 91. CULBERTSON. ELY; MOIU!IIEAD, ALBERT; MOTT.sMml, GBOPo FREY, Culbertson':~ Card Gamel Complete, with OlficltJl RW.. -(The Greystone Press, New York. -•9si).

•1o.s•

2o6 BEAT THE DEAL ••

(10] DAIWAS, NICHOI.AS, Wall Slreet, TM Second Lu Vega~ (Stuart. Lyle, New York. rg62)". .

[II) PELLER. WIWAM. An lntroductloll to ProbllbUlty ThetWI lllld.ltl Appllctllltml, VoL J. Second Edition (John Wlley &: Sons. Inc.. New York. 1957). •

[12] Fox. PHILLIP G. (as told to SrANLBY Fox), .. A Pnmer for Chumps." Stllurtbzy Evening Post. November :n, 1959, PP. 31ff. . b'"

[13] Fli.EY, RICRA1tD L., At:t:ordlng to Hoyle (Fawcett Pu .tca-tions. Inc.. Greenwich. Conn., 1956).

[14] Fuan DocToR BatJNO. TM Practktzl Way to a Better­Mtmo", (Fawcett Publications. Inc., Greenwich, Coma.. 1957>· . •

[15] GARCIA. FaANK. Mar~ Cartfl and Loaded Dice (PrentiCe-Hall, fnc.. New York, I!)6~).

[16} GooDMAN, MDCE. How to Will at Cards, Dice, Races flll4 Roulette (HoUoway House Publishing Co., Los Angeles. • 196)).

[17] GIU!ENSPtJN, HANK. "'Where I Stand." Lu Y ttgu Sun, Jan-uary 26. 196~ p. lo .

(18) HUFF, DAUBLL, Thtt Mtllhttnuztb D/ Sex. Gambling flll4 IIUrll'tlMe (Harper & Brothen, New York, 1959).

(19} JoNEs. JACJC. GoiJM N~ Gtunbtg Guille (Silver State Publishing Co.. Las Vegas. 1949).

(20) JONES, Sri.AT (AP), '"Thorp's Book Brings About Vegas Sbateup. .. lAI Cruea Sun-Nftll. AprU 3, 1964. p. I.

(21) KATCHEB. LEo. TM Big Btlllkroll: thtt Llftt and. Times of Artaold Roth8ulll (Harper lc BrotJiers. New York, 1959).

(22) K. C. Card Co.. F~ Annlven/l1'] Blue Book, 196o, - Chicago. I959-

ra31 KI!LLY, J. L;."•A New Interpretation of Information Rate," IRE TrtJ~~S«tlOIII on 111/ormtllloia Theory, Vol. IT-~ No. 3o September, 1956. BeU Systma Tech. J., Vol. 35, 917-926 •• (1956).

1241 l.tu Cruce• Sllrt-New. "'Mobster Swears Gang Boss Has In­terests in Las Vegas," October I, 1963: p. I. "

(25] lAI Vegu Rm.w-IOUTIIIIl. "'Silver Slipper Raided." April 4e 1964, p. i.

(26] l.tu Yep Rnkw-1011111tll. "'State Casinos Change Rules oa •a1' Games," AprD ~ 1964, p. 1.

[27] lm Y1gar Sun. January 25 and 27, 1961. (28] Lta VegtU Sun, "U.S. to Smash Mob-Ruled LV easrnos.•

. January 29. 19~ P. lo

207 {i9J LEWIS, OscAR, Sagehnah C~ Thtt StDry oflAIJfll G­

bling in Nevada (Doubleday & Co .. Inc., New Y0rt.1953). (3o) . Life, "Senators Survey Low-Belly ~~ippers," Septelnber. le

I96I,p.39· . • (31) Los Angeles Herald Examlller, -can YOU Beat BlackJa47"

June to, 1962, p. HI. · . · (32] Los Angeles Herald Examiner, "Crooked Dice Charge: Vegaa

l331

l341

l35l

(36]

l37l

(38)

l391 (401

(41) (42)

(431 [441

[451

"[461

Casino Oosed. First Case of Cheating in Nevada (nc)." April 4. 1964. p •. 1. · . · Los Angeles Times, "Federal Extortion Case May Link Lat Vegas Gambling to Underworld." February 6, 1964. p. 1. Los Angeles Times, "Vegas Casinos Cry Uncle, CbanJe Rules," April 2, 1964-MAcDou<iALL. MICHAEL, "'Even 'HonesJ' Vegas Hause Cheats." Sunday Star-Ledger, Newark. New Jersey, J>c:cem. ber 2, 1962, p. 35· . -, MacDougall on Dice tuUI Cards (Coward-Mceana. Inc., New York. 1944). --. "Nevada Trumps a Blackjack Dealer," Sunday Sttlllo Ledger, Newark, New Jersey, Apn1 19, 1964. section~. p. 2.. MCKINSEY, JoHN C.. lniroduction to tM Theoq. of Gama (McGraw-Hill Book Co •• Inc., New .York, 195~). Miami News. JanQary 25, 1961, p. 6A. · · MoNROE, KEITH, "William Harrah: The New. Gamb1bJa King. and the Social Scientists," Harpers, January, 1!)62. The Nation, February 4. 1!)61. . Newsweek, "Gambling: Hello Suckers." September 4. 1961. pp. 22tf. . ' New York Herald Tlihu'nl, January 29, 1!)61, pp. I, 24o New York Journql American, "Computer Beats House II '21' in Las Vegas.• Nqvember 15, 1963, p. 1. . New York Joul7ial American. "How W"IZIU"d of Odds Beat Las Vegas Cards, .. Apri1·3, 1¢4, p. 1.

New Yor/c Times. Western Edition, "Las Vegas~ GambiJna Take Creates New Force in U. S.; MiJiions in Untaxed 'Black Money' Give Obscure Figures Power that &tend~ from Underworld to Government," November 18, 1963. New Yor! Times Book Review~ Best.Seller IJst. Ap.rillt and May J.

[481 OLSEN, Eo, Lett~ to the Editor, Llje. AJJrill7, 1964, p. 27. [49~ O'NEIL, PAUL, "'The Professor Who Breaks the Bank." Ll/tt,

• March 27~ 1964. pp. 8()-!)1. · · (Sol ORE, OYsTEIN, Catdano, The Gambling Scholarlwith a trans-

r -- ~--~- --

~-r r BBAT TJIB J)IJALJltt · 'B.e/ereilca 20, I

f ~ol ,d I " Jadon (from t1nt l.atlll ~ Cardano's book. Gamu of Cham:e) '(68) -.'"A Favorable Strategy for Twenty-One.• .l'rocud· ' . f

i by SIDNEY .HENRY. Gout.i>) (PriDc:etoll University Press, lngs of the Nt:lllonDl Jfctldenry of Sciences, Vol. 47, No. r, Princetoa, N. J,. 1953). • • pp. no-u2 · (l!lfjl). t

(SI] PIP'ade Sunday Magar)M, Intelligence Report: "Cri~ [~) -.'"A Prof Beats the Gamblers, • Th~ Atkmdc Montlalf, i August 2S. 1963- · June. 1962.

(52] Po~ H&NJUo Scknce tmd Mahotl. Translated by Fma- [7o] -and WAI.DI!N, w .. A Favorable- S"J.de Bet In Nevada ,H cis Maidand (Dover Publications. Inc.. New York. 1958). Baccarat. Jounllll of the American Sttlllstlclll Anocilltlon, VoL ji fs3] RADNea. SIDNEY H.. How to Spot Card Sharps tiiUl Their Ch1 JIJ-J18 (f966), . Methbtls (Key Publishing Co.. New York. 1957). [71) TUM. Moderll Uving: "Eight Days to Wm." .January 13.

~ THa RAND CoRPOJtA110N, A Mmlon Random Digits wltla 1961, p. 82ff. ;

~1 ' 100.000 Normlll Devlota (Free Press of Glencoe. -Diinois, . [721 ~. "'Games: 'Beating the Dealer,'" .January 25. 1963. (~

1955). . P· 70. . 'l bsl ReiD, ED and DeMARIS, OVID, Th~ Grun Felt Jungle (Trl- (73] Tim~. Non-Fictioa Best Seller List. May 29. 1964. p. 4o ..

dent. New York. 1963). Reprinted and enlarged (Pocket 174~ TuaNea. WALLACl!o "Nevada Gambling Faces New Test. • '~ Books. Inc.. New York. 1964). All references are to the New York Tlmn, April 12, 1964. p. 53· . ;i

enlargecl Pocket Boot verslon. . [75, ---:--.New York Tlmu. November 18. 15163 to Novem~ ~.~ ,.

fs6) RIDDLE, MAJOR A.. as told to Hyams. Joe. The Weekt:ntl 22, 1963, p. •• Gtunblet'• iltlllllbook (Random House, New York. 1963). · [76) W~EN. W .. Ph.D. 1besis. .New Mexico State University

fs7l Rosa, INI!Zo •Bets Are Off,• New York JVDrld-Telegrtun on4 (unpublished). Sun, February 7o 1961. [77) W ANNISKJ, .Juoe. "'Gamblers ShufDe Blackjack Rut~:~ Bact

(58] ScARNEo JOHN, Scai-ne'l Complete Guide to Gtunbling (SbnOD to Old Deal.• The NIIIIINull Obuner, June 15, 1964. p; & ancl Shuster. Inc.. New York.. 1961). [78) WMhlngtoll Pc tmd Tlma Hnrl14. January 25, 19fj1, p. 3;

b91 ScHERMAN,· DAVID E.. '"It's ByeJ Bye! Blackjack. • Sportl editorial. "High Stakes. • p. ~16, January 26, 19fj1. IUustrated, January 13, 196.4- .. [79) Wsu.wo. JOHN D .. TAe Compktlt Strategyst CMcGra.,.

[6o] SciENTIFIC AMERICAN, ·"How to Beat the Game. • April. Hut Boot Co.. Inc., New York. 1954). 1961. p. 84- (So] W11.SON. ALLAN, The Casino Gtunblln'• Gidde (Harper ancl

(61] SHeiNWOLO. ALFREDo "It's in the Cards: Blao:kjack-Count- Row, Ne\V York. 1965). ing the Cards," Argory, August, 1961. ;1

(62] SHEllMAN, GENe. • "Off The Top' Plagues Gambling Au-

l, tboritles. Pocketing Moaey Without Being Reported for Tax Purposes Called Impossible to Prave." lA Angela Tlma. October 28, 1963.

!I (631 Showboat Hotel. Las Vegas. Nevada, '"The Univac '21' FOlio mula for Standing or Drawing."

ll i64J SMmf. HAROLD S.. I Want to Quit Winnm (Prentice HaJJ. Englewood Cliffs. New Jersey, 1961). 11

(65) Sports IUustratt:d. "Calculated Risk. • February 6, 1961. pp. ,:j!

4. 5· ~!

(66] STEEN. JOAN. "Exposing Crooked Gambler's Tricks." Popular Science Monthly. January. 1962, pp. 61ft.

(67] THORP. EDwARD o.. "Fortune's Formula: The Game of Blackjack." Notlca of the Anwlaua Mlllhemallctzl Soclety. December, 196o. pp. ~35-936.

List of Figures and Tables

Figure 3.1 The Player's Key Decisions 20

Table 3.1 Drawing or Standing with Hard llaDds 20

Table 3.2 Drawing or Standing with Soft Hands ,. Table 3·3 Doubling Down '5 Table 3-4 Pair Splitting 28

'Table 3·5 A Complete Miniature Version of the Basic Strategy 30

Table 3.6 . Results Using the Basic Strategy 32 Table 3·7 Results of Baldwin, et aL 32 Table 3.8 The Basic Strategy Compared with Other

Blackjack Strategic$ " Table 3·9 The Basic Strategy Compated . with Best Play in Other CasinO Games, 34

Table 4-1 P~yer's Advantage or Disadvantage for eer.. tain Spcclal Decks 48

Tablo 4-2 The Best Strategy When It Is Ouly JCaowu That no Fives CaD Appear Ia tho Next Round of Play . ~ 49

•211• I , '

212

Tabto·.a.s

BBAT THB DBALJfJt When the Deck Probably Win Be Adequate for a FuU ROUild of PJa_y, According to the Count of Used Cards 53

Table 44 Variation iD tho Numbers of Known Favor­ablo Situlatlcms, When Fives ODly Arc Count-ed. as a Function of tho Number of Players s6

'!'able 6.1 Basic Strategy When the Dealer Exposes His Hole Carel 90

f!gwe 7·1 Player"s Advantage as High-Low Index Varies 97 ·

Table 7.1 Using the High-Low Index to Draw or Stand with .Bard Hands g8

· Tablo 7.2. Using the High-Low IncJox to Draw or Stand with Soft Hands gS

Table 7·3 Hard Doubling Down with tho High-Low ~ g8

. Table 7-4 Soft Doubling Down with the High-Low Ind~ 98

Table 7.S Using the High-Low Index to Split Pairs 98 -Tablo 7.6 Advantage and Frequency of Favorable

Situations Tablo 8.1 Approximate Player Advantage iD the Ten­

Count Strategy Table 8.2 A CoDservatlvc Betting Scheme for the Ten­

Count Strategy Tablo 8.3 The Tea-Count Strategy, Based on the Value

of the Ratio When a Rulming Count of the Cards Is .Kept

Tablo 8-4 A First ·Approximation to the Ten-Count Strategy

Table S.s Frequency of Favorable Situations Arising iD the Tea-Count Strategy

Table 9.1 'The Effect of tho (rcmporary) Las Vegas . Rules Cbauges

101

104

Jo6

no

117

'" • . List of Flgl & Tablel 213 _,. -· ~~-- • - 1

! . Table 9.2 Approximate Effect of ComnJon Rules van.. ' ations OD Player's Advantage When the Basic . f Strategy Ia Used 131

Table I Dealer's Probabilities 190 Tablo 2a Player's Gain by Drawing over Standing with

' Hard Totals 191 it 'J

Table 2b Playefs Gaia by Drawing over StaDdlDg with )i( ' Soft Totals 192 'I :;i

Table 3 Playefs Advantage 'Standing OD Varioas ' Totals 193 1

i[ Tele 48 Dealer's Up Cird Is Tea 195 ll Table 4b Dealer's Up card Is N"mo 196 !

Table 4C Dealer's Up Carella Eight 197 ~t Table 4d Dealer's Up Card Is Scvea 191 }'

)

Table 40 Dealer"s Up Card Is Silt 199 Table 4f Dealer's Up Card Is F11o 200 Table 4g Doaler"s Up Card Is Four 201

Table 4h Dea1er's Up Card Ia Tme 203

Table 4i Dealer"s Up Card Is Two 203 Tablo 4) Dealer's Up card Is N» 204

!"' I

---- ...._.--.- ...... _~

Index·

Ace-richness, correction for, 118-121

Aces. including in ~t. 118-121

splitting of, 13-14 Action, 41 Advantage of player, drawing

over standing, 191-193 for various deck compositions,

47 with various bole cards and

· decisions, 195-204 (see olso Fives strategy, basic strategy, point count. Tens strategy)

American Mathematical Society, announcement of Fives strategy to, 59

Automatic blackjack machines, 11, 13, 14, 18, 136-140

Baldwin. Roger R. .. 12-13 fn. " basic strategy first published

by, 4-S disadvantage of player who

mimics the dealer, cat. culatioo by, 12-12 tu. 16fn

Bamford, Robert, 178-179 Barber, mao who trimmed, 40 Barring of players by casillOI,

. 124fn. 164, 176-177 . Basic strategy, 16-40

advantage with, 4, S, 18 compared with best play Ia

other casino games, 34 compared wi(b other blacJc.

jack strategies, 33 definition, 18 doubling down with, 24-27 drawing and standing with,

19-24 Baccarat, winning system for aperiments to verify, 35-38

side bets in Nevada, 94, Jriioiaturized summary, Table 181 3.S, 30 .

• fn denotes that the particutar item may be found in a footnote CID the Jivcn page.

. ~ I

:n6 ~c strategy (ctmtlnu~d)

resUltS of using. 31-·~33 splitting pairs with. 27-29

approximate rules for, 27 Thorp's first trial of, 16-18 when dealer's bolo card Ia

known, 88-91 Bean, Tom, 178 Beat The Dealer Co., 178 fn Beat The Deater Computer, 178 Beat The Dealer, Pirst Edition

of, on best seller lists, 83 Bernouilli. Daniel, 180 kDoui11i. James, 180 Bets, maximum, 9, 10, 81, 184-.

187 minimum 9, 10, 81, 184-187 pJaced before ~. 9 ieUJomcDt of, 12.-13

in cue of tie, 12-13 iavolviag • Datura~. 1 l

ftriAtioa in lize of. 132 in Plves strategy, 55-S7 in Teos ttrategy, 102

Blackjack. common emn Ia. 33-34

. other aamei for, 3 . where "Played. 3 . (sea olio

casino blackjack) Blattner, John, 40 Bonus,17 Bottom card. advantage from

eeeing. 129 Braun, Julian. 'Viii, 5, 35, 45 m.

8S. 89, 93, 94, 131, 189 Burned card, advantage from

seeing. 119 If dealer Conceals, 52 may or may DOt be shown. 9

Bust, 11 . deliberate by player, 8l

' c&m.ouftage, 131-132 Cantey, Wilbert E., 11 fn (.rH

also Baldwin. Roger R.)

lnJn:i .,.. earct'c:oU!iting. tJasr.torwiimins··. Claalleoge ~J~&tch with x..s vega

strategies, 46-50 ·. casmo. 79 c1etecting cheatins by, n-54 · ~g bJ the player, 12, us-effect of errors in, 85 fn 136 · . .. with aid of chips. 87 Chips, refusal to~-~ c:asino. (.rea olio Fives strategy,. Tens 118 ·

strategy, complete point Complete deck. tables of proba-count) bilities for, 190-204

Cardano, Gerolamo, 180 Complete point count, 93-100 quoted on sixteenth-century advantage, 100-101 . cheating, 150 announcocJ. 93

Carousel Club,. 181 fn bot sizes in, 96 Casino blackjack, where played, 3 c!Qubling down, 98 Casinos. closed for cheating. 146 drawing and standing. 96-.98 Cheating by the dealer, 68, 133, favorable lituations, 99-100

134, 141-172 index, 96 adding cards to the deck, 107, iasuranc:e, 98

108 pair splitting, 98 anchor men, 158-160 playel''s advantage, 97

·avoiding, 41, 140. 162-163· Computer, high-speed (.su IBM bridging, 156 704) . dumps. 157 Cosa Nostra, 142 decks pocketed, 157 Countermeasures by casinOI, defense against by ""paper · 121-140. 17.6-177 .

route" technique. 140 . barring. 124 fD, 164, 175-176 detection by counting, 51 de'alei'S who count. 126 false sbufDe, 154 multiple decks, 127-128 bigh-low pickup, tS2-tSS (see tllso barring of players by marked cards. 147-148 casinoa, cheating by tho mechanics, 160-161 dealer, rules of blackjack, peeldng. 143-144, 148-149, c:hangea bv Las Vegas

160 casinos, ruies variations. recommendations for prevent. shuflle up) ·

ing. 166-167 Counting cards (.see card count-removal of cards from c1eck. ing)

· 52 :' Customs, new decks, 15 Robin-Hood function, 145 Puerto Rico, 79-81 second dealing. 149-151 shill rules, 1S shiner, 149 · sbufDjog. 15 stacking deck. 152-158 . spread decks before use, 118-stepup. seven card, 155-158 119 turnover, S3-S4 (ue tllso sliume up)

strong form of, 53 . ? ,

weak form of, 53 . : Dcalersholecardtnown,playa's (ue also Cardano, GeroJaalo. ,. . advantage when, 88-91

.and card counting) · t player'utrate&Y when, 88-91

217

Dealer's ~hod, U · . (fee u.mimrokfnc the~)

DeaJen totals. table Of prObao bi1itiea of, 190

Dealing bOx (su shoe) Docis1ona of plafel'. order of con-

sideration, 19-20 .. Deck, new, 15 · Demaris, Ovid. 142 Depeadenc:eoftrials,lmportance

of in blackjack, 43-44 Disguises. 132-136 Doubling dowD, 13-14

basic strategy for, 24-27 complete point count strategy

for, 98 distinction between soft and

hard.24-25 Fives strategy for, 49 ·. Tens strategy for, Table 8.3,

109 with bard twenty, 120 with soft Twenty One!, 81

Doubling-up system, 41 Drawing or standing. 11

·basic lfrategy for, 19-24 complete point count strategy

for, 98 Fives strategy for, 49 · player's advantage when stancJ.

ing, table of, 193 .. player's gain 'by dra'WiDJ.

tables of, 191-192. . Tens strategy for, Table 8.3,

109 Dubner, Harvey, 93, 94

End play, 83-84; 86-91, 121-123 .

England, blackjack in, 3, 184-187

Expec:JatioD_ (.fU advaotago) Experimeots, to show ..

doubling down on k'd Ten ·.~ versus aat'Ac:e, Joss Jn, 37 ,

de_ct of h1p-Iow pickuP, ISS

r---

:"SIS '-- .. , Ex~ents '(t:tJmtnud)

· df~ of secoad dealiDg. 149-150 .

hard sixteen against an Ace. loss in standing with, 35-36

epUUing a pair of Sixes against a Five. gain from. 37-38

P,Mr.,176 mistress of. 176

Fall Joint Computer Conference, 1963, 513-94, 164-169, 179

FaYOI'8ble situations, basis for wiDDing strategies, 44-47

example where player must win,4S .

frequency of. 56, 100, 117 Fermat, Pierre, 180 Fivo-count strategy (see Fives

strategy) Fives strategy, 48-58 a~tage o( player~& 48 approximate, SO bet nriatioos in, effect of, 55-

57 . capital required, 51-58 card counting for, 51-54 compared with basic strategy,

48-50 frequency of favorable situa-

tions in, 55 profit from, rate of, 56-51 refined, 55 remaining cards, when ade-

quate for, 53 risk m using, ss simplified Fives strategy, SO table for, 49

Fortune's Formula, 59 Fraikin, Gle,no L, 124 fA

Gambling systems, failme of popular Ones, 42-43

Gaadas'Conlrot Bt!afd.&ite of. Nevada, VIi, 146, 148, 165'

casinos closed by foF cheating. 146

Goren point count, S Greasy John, 174 · Green Felt Jungle, 142, 165

Hard hand, 10 Harold's Club, 65, 19 Harrah's Club, Lake Tahoe, 69 Harvey's Wagon Wheel, 69 Head on, 45 High-Low strategy (see complete

point count) Hit,ll

bard 17, 84 Hole cards, 9

dealt up, 9 iDferences about other play­

ers', 114

IBM 704 computer used, 18, 45 Independent trials processes, 42- .

43 Insurance, 14

house advantage from, with full deck, 106

misconceptions about, 107-108

aot offered by some casinos, 107

I've Got A Secret, appearance on, 78

Junior, 20

Kalaba. Robert E.. 94 Kelly gambling system, 514

La Concha.casino. 79, 8S Laplace, Pierre, 180 Las Vegas Re$01t Hotel Assocfa,.

tion, 128 LGP-21 computer, 179

UfB magazine, ·..m; ·82. t6C-.. 166 Little dark-haired p.y, GS, 121,

123. 176

M,Mr.,78-91 Macao, blackjack fit, !30 McDermott. James P. (see Bald-

wiD, Roger R.) MacDouaaU. Michael (Mickey),

vii, 1S7 I 158 Mafia, 142 Maisel, Herbert (see Baldwin,

Roger R.) Manila, bJactjaak in, 130 Marked cards (se~ cheating) Martingale. Small. 42 Mimiekiog the dealer, . player

strategy of, disadvaota&e using, 12-13 fD, 38

MnemotecbDy, 73 Moore, Gary, 78 Morgan. Henry, plays basic

.atrategy b18ckjack in ~ Puerto Rico. 78-79

N, Mr., 78-91 Natural, 1o-11

Olsen, Edward A., 165' O'Neil, Paul, viii, 164-166, 179 Open End. appearance on, 65 fn Operation Money-Pump, 169-

172 Others, aynonomous with non­

Teus, 102

Packs, number of osed, t Pair splitting, 13-14

.Ac:e:s, 13-14 face cards, 13 (see tdso basic strategy, Fives

strategy, Tens atrategy, point count) -

Pairs, 13 Paper (see marked cards)

119 Paper JVUte techrdque · for·\tla.

Ding. 92, 14(). 16io-1U. Pascal, Blaise, 180 Patbaad. 90

. Pearson, Karl; 180 Per:feaser stoppers. 9 fn Player who aever busts, advaa­

. tage of. 39-40 Players. number of, 8, 83

affects rre,uency of favorable situations, 56

Poincar6, Henri. 180 Point count (see aimpto point

count, complete poin& count)

Point count. systems, 75-76 Poisson, SimCoo., 180 Poker, 182 . PrimadQnna cashlo,-?4 Puerto Rico, blackjack ID.; 3.

79-!n rules changed in, 91

Push. 11

RatiO of others/fens, effect of on player's advantaa,e. 103

Reid, Ed, 142 Roulette, 180, 181 Rules of blackjack, 8-15' .

changed back by Las Vegas casinos, 130

changes by Las Vegas c:astnos. . 6, 13, 14, 128-l]O

changes foreseen in first edi-tion, 6, 129

Rules variations, 131 RuDDing count, 95, 190 RusseJI, Bertrand, ~ of.

40fn

Salmon, The, 78, 82-92 Scarne. 'John. llfn Schweizer, Berthold, vU Scientific ReaCarch Recor~Ine..

178

[. t ' I ' I

~I .,;:.

22.0

SbaiiDml. OaucJe B., 'VII Shill. 15, 12S

ruJea follon4 by, lS Sbiaer,149 Shoe,~ from, 85-86

used ia Puerto JUco. 80 aume ~ 1s. 11-12

wiDDIIiS despite. 72, 124-126 Silver Slipper CaaiDo closed, 166 Simple pOfat count. 15-91

for liwral dec:ka, 77-78 etrategy,76

Sklar, Abe, Yll Smith.~-) P. (8ee System Smltb, H::fc£ Jr .. 65 Smith. Halol4. Sr .. 65 Soft haocl. 10 Sonny (-.Junior) Sorgeufmy, Robert. 16 Sprague. lUc:barc1 E., 94 Stand.11 SlaadiDC..,__ minimum, dOe

.20 Stelf-. LiDcoln, ,170 Stewart, Diet. 59 Stiff. 90 . Stock market, 1D-183 Strategy tables, learning. 114-

117 ~s importanco of parts,

Susstdod. Dmd, 65' fn Sptem Smitty, 174-17S

Tax ewsion by ca.sinof. 167-172

Tea-count strateg (.we Teas strategy)

TCDI strategy, 6, 101-123 · . Aces. iDcludlas in count, 1J9..

120

lntle% ac1vutages over Fives strategy.

102 bet si%es Ia. 119 card counting for, 69-11 casino teat of. 59-74 ~ of favorable .uu..

tioDS ln. 117 insuranc:e iD. 106-111 learning cablea, 109-110 Ne98d& test of, 6, 59-14 rateofp!Ofitwitb, 116-118 ratio iD. 102 tables for, Tables 8.3, 8.-\

109-110 will rate, 71

estimatea pen are coa­servative,-71 fA

. Thorp. J~ vii Thorp poiDt count.. 5

(.ra poiot COUGt) ~ VMan, Vii Tropiiaaa Hotel, 179 Turner, WaDac:e, 142 Turnover (.ra c:beating by the

dealer)

Vltimate strategy, 16 fa

Vopui, OabrieJ. 128

Waldeu. Wdliam E., Yiii, 94 Walkei, Elbert, Vii WamUskl. Jude. 130 Wdsoa, Al1aD, 94, 181

X, Mr .. 60-74, 143-144, 176

Y, Mr., 60-74, 144

z. Mr., 175-176

A NOTB ABOUT THB AUTHOB.

Eow.um 0. "l'Ho1u> is currently teaching at the Irvine campus· of the UDiversity of Califomia. where be Ia an ·Assoc:fate Professor of Mathematics. Prior to this he tau~ at New.Mexico State Uuiversity. Piofessor Thorp was also a C.LB. Moore fDitructor at the Massachusetta lastitute of Tec:hnology, and be taUght at the Unfwnity of CaJifomla at Los Angeles. from which be nceived a Ph.D. In 1958. Professor Thorp li a member of the AJQedc:aQ Methematbl SocietY and of Pbi Beta Kappa.· Professor Thorp.. major fields of interest include probabi1lty 8lld game theory. He Ia the author of a text in probabnity theory and· of numerous articles. botb in Jeamed joumals and In popular magazines. •

~--~---~-~,----"""""" ____ _

~ I