DAVID I CLARK

CROSSNUMBER PUZZLES F O R S E C O N DA R Y M AT H E M AT I C S S T U D E N T S

Published by

Australian Mathematics TrustUniversity of Canberra Locked Bag 1

Canberra GPO ACT 2601AUSTRALIA

Copyright ©2012 AMT Publishing

Telephone: +61 2 6201 5137www.amt.edu.au

AMTT Limited ACN 083 950 341

Crossnumber Puzzles for Secondary Mathematics Students ISBN 978-1-876420-33-8

Contents

Introduction iiiCrossnumber Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAuthor’s Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Puzzles 1Worked example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Puzzles with simple arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Puzzles with a difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Puzzles from Middle Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32More challenging puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Puzzles with secondary mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Canberra Maths Day puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Guides 67

Solutions 81

Appendices 93Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94A little Maths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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Introduction

Crossnumber Puzzles

Crossnumber puzzles are similar to their more familiar cousins, crossword puzzles, in that they con-sist of interlocked grids of across and down answers, each of which is the answer to a specific clue.The answers are, of course, numbers rather than words. Unlike crossword puzzles the clues typicallyinvolve more than one answer, and answers often appear in more than one clue. It is rare that theanswer to a clue can be determined in isolation.

Another difference from crossword puzzles is that crossnumber puzzles are solved a digit at a timerather than a whole answer at a time. A digit is teased out of one clue, and this in turn helps in findinga digit in another answer.

Crossnumber puzzles are something akin to detective stories. Clues are given, but the implications ofthe clue needs to be worked out before it is applied in furthering the solution of the puzzle. Crossnum-ber puzzles are brain-teasers, and you can anticipate many hours of pleasant occupation in solvingthe puzzles in this book. The more challenging puzzles may take an hour or more, but the puzzlesthemselves and the solvers’ experience are so varied that it is not easy to suggest a par time. The onlypuzzles constructed with a particular solution time in mind are the Canberra Maths Day puzzles (seebelow).

A unique feature of this book is that each puzzle comes with a solution guide which gives one possibleorder for solving the puzzle. If you are stuck, the solution guide will tell you which cell or cells totackle next, without telling you how to do it. You can then still have the fun and satisfaction of solvingthe puzzle. If you are really stuck, you can look up the solution. But again, the solution guide willtell you which cells to look up so that you can continue solving the rest of the puzzle.

Many of the clues in the puzzles in this book have operations that go beyond simple arithmeticoperations of addition, subtraction, multiplication, division and averaging, but they are still accessibleto solvers who are unfamiliar with them. There is a glossary of mathematical terms used in the book.

There is also a fully worked solution to a simple puzzle and some useful hints about solving cross-number puzzles, particularly starting them.

The puzzles are organized into six sections as described below.

Section 1. These puzzles mainly use standard arithmetic operations — addition, subtraction, multi-plication, division and averages.

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Section 2. These puzzles are a bit different. The contexts are different, and therefore solution tech-niques have to be adapted. There tends to be a bit more logic in them.

Section 3. Each of these puzzles has a small story associated with it. You are asked to solve a problemakin to a logic problem, but using a crossnumber puzzle.

Section 4. These problems are rather harder than the others. You will generally need to use morethan one clue to make progress.

Section 5. These problems have more mathematics in them. They incorporate most of the conceptsin secondary maths, including algebra, trigonometry, differentiation and integration, summation andpowers and logarithms. Students will need a thorough understanding of these topics to solve thepuzzles.

Section 6. These problems were devised for the Canberra Maths Day, where schools send a teamof five final-year college students to compete against other teams from other schools in a fun andchallenging day. The crossnumber contest is one of four events on the day. An interesting featureof the event is that each team is split into two halves, one half receiving the across clues, the otherreceiving the down clues. When one half of the team deduces the digit in a particular cell they tellthe supervisor who tells them whether they were correct and lets the other half of the team know.Thus each half of the team relies on the other half to make progress. Typically just a few of the teamsfinish in the allotted 45 minutes, with most of the others fairly close.

Acknowledgments

This book grew out of the puzzles I constructed for the Canberra Maths Day, and includes some ofthem. The maths day has run annually since the middle 1980s when it started life as the University ofCanberra Maths Day. It was given generous administrative and financial support from the AustralianMathematics Trust.

Thanks are due to John Matthews, Ian Lisle, Tracy Huang, Vance Brown and Malcolm Brooks fortheir checking of puzzles in the book, and to Malcolm Brooks for helping to work out how thecrossnumber contest could be made to work as part of the maths day.

This book was typeset in TEX, the mathematical typesetting system designed by Donald Knuth. Manycontributors have given freely of their time and expertise to extend the capabilities of TEX. The TikZ/ PGF package written by Till Tantau was used in the diagrams in this book. Finally, special thanksare due to Ian Lisle for expert advice on all things TEX.

Author’s Comments

This book has been a labour of love. I have enjoyed constructing the puzzles, and coming backand solving them some months or even years later. I hope that it will be a fun and challenging forthose who enjoy mathematics. I also hope that using mathematics in a different context will enhanceunderstanding and occasionally lead to further explorations – I am a firm believer in indirect learning.

David ClarkUniversity of CanberraFebruary, 2012.

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Puzzle 1.41 2 3 4 5

6 7

8 9

10 11 12

13 14 15

16 17

Across Down1 One fifth of 13d3 see 16a, 2d6 One less than a square7 A multiple of 4d8 see 3d, 14d

10 see 14d13 A multiple of 715 12d− 17a

see also 17a, 11d16 Three times 3a

see also 2d17 15a+ 5

see also 15a

1 5d+ 13d2 3a+ 16a3 A permutation of 8a4 see 7a5 see 1d9 The square of 13d

11 15a+ 12d12 see 15a, 11d13 see 1a, 1d, 9d14 A divisor (10a− 8a)

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Puzzle 2.9In this puzzle each of the letters stands for a different digit.

1 2 3 4 5

6 7 8

9 10

11 12

13 14 15

16 17

A B

C

D

E F G H

I J

CluesAcross Down

1 12d− 14a3 see 4d6 Half of 11a8 A multiple of 13a9 11a− 7d

11 A squaresee 6a, 9a

13 see 8a14 see 1a, 10d16 see 10d17 A multiple of 13d

1 A power of 22 13d = 11d mod 2d4 3a+ 5d5 see 4d7 see 9a

10 16a− 14a11 see 2d12 see 1a13 see 17a, 2d, 15d15 13d× 5

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Puzzle 3.1The Middle Earth Treasure Hunt

The Middle Earth treasure hunt was contested by a team of hobbits and a team of elves. The hobbitswere Bilbo, Frodo, Merry, Pippin and Sam, while the elves team consisted of Celeborn, Galadriel,Arwen, Legolas and Gimli, the elf friend. A maximum of 100 points was possible, with a prize of amithril ring for a score of 90 or more; a gold ring for a score of 80 or more; a silver ring for a scoreof 70 or more; a bronze ring for a score of 60 or more and an iron ring for a score of 50 or more.Every member of each team won a ring, although only one member of each team won a mithril ringand no one won a gold ring. At least one member of each team only won an iron ring. The mostsuccessful elf was Arwen while Sam was the best of the hobbits. Bilbo scored lowest for the hobbitsand Galadriel lowest for the elves. As to be expected, the scores of Gimli and Legolas were veryclose.

Solve the puzzle below to determine how many points each contestant got.

1 2 3 4

5 6

7

8 9

10 11 12 13

14 15

Across Down1 The average elves’ score3 Galadriel’s score5 Pippin’s score6 Sam’s score7 Frodo’s score8 Merry’s score

10 Total points scored by hobbits andelves

12 Average hobbit’s score14 Celeborn’s score15 The range of hobbits’ scores

1 Gimli’s score2 Total points scored by hobbits3 From the point of view of winning

rings, scores over 90, 80, 70, 60, 50were wasted.3d is the total wasted score.

4 Bilbo’s score9 Total points scored by the elves

10 Legolas’ score11 Those competitors who got iron

rings were a little disappointed.11d is the number of extra pointsthey would have needed to scorebetween them so that they all wonbronze rings.

13 Arwen’s score

3

Puzzle 5.101 2 3 4

5 6

7 8 9

10 11

12 13 14 15

16 17

18 19

CluesAcross Down

1 7a, 2d, 1a and 1d form an increasingAP.

3 log3 3a = 1 + log3 4d5 75a = 71d × 718a

7 4d = 9× 7asee also 1a

9 9a mod 5 = 29a mod 3 = 0see also 15d

10 10a = −1 + 2− 3 + · · ·+ 3dsee also 19a, 12d, 15d

11 see 16d, 17d12 see 9d14 214a = 6d× 213d

16 12d = 16a+ 13d18 see 5a19 8d = 10a+ 19a

1 see 1a, 5a2 see 1a3 3d is a square.

see also 10a4 see 3a, 7a6 see 14a, 16d, 17d8 see 19a9 9d is a multiple of 12a

12 tan(10a+ 12d) = 1see also 16a

13 see 14a, 16a15 15d = 9a+ 10a16 11a = log2 6d+ 16d

17∫ 17d

0

1 dx =

∫ 6d

11a

1 dx

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Appendices

Conventions

The puzzles are presented in a fashion similar to crossword puzzles. And as in crosswordpuzzles, a means “across”, d means “down”.

Because answers can appear in more than one clue, cross references may be given to otherclues. These are in the form of “see ...”.

No answer has first digit zero (this can be useful in solving the puzzles).

Angles are in degrees rather than radians.

Hints

1. A 3-digit number which is the sum of two 2-digit numbers must start with 1.

2. If a 2-digit number is the average of a 2-digit and a 3-digit number, the 3-digit number muststart with 1.

3. If a 3-digit number is n times a 2-digit number, with n < 2, the 3-digit number must start with1 and the 2-digit number must be > 50.

4. A multiple of 5 must end in 0 or 5. (This can be used in conjunction with numbers not startingwith 0.)

5. If a 2-digit number is 5 (or more) times another 2-digit number, the latter must start with 1.

6. Squares must end in 0, 1, 4, 5, 6 or 9.

7. Pythagorean triples are often of the form (a + 1, a, b) with b an odd number. In which case,b2 = 2 × a + 1. For example (5,4,3), (13,12,5), (25,24,7). They may also be multiples of(a+ 1, a, b). For example (50, 40, 30), (39,36,15).

(The first 5 hints are often used in starting a puzzle.)

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Glossary

Arithmetic progression (AP)Three or more numbers a0, a1, a2, . . . form an AP if the difference between successive numbersis constant.That is, ai+1 − ai = ai − ai−1.For example 3, 7, 11, 15 form an AP, with a common difference of 2.

Binary numbersBinary numbers are numbers with base 2.Normal (decimal) numbers have base 10,so that 100010 = 1× 103 + 0× 102 + 0× 101 + 1,whilst 10012 = 1× 23 + 0× 22 + 0× 21 + 1 = 910.

CompositeA composite number is not prime.For example, 6 (= 2× 3) is composite.

CongruentTwo numbers are congruent modulo n if their remainders after being divided by n are equal.Or, equivalently, if n is a divisor of their difference.For example, 64 and 28 are congruent modulo 12. This can be expressed as 64 ≡ 28 (mod 12).

CoprimeTwo numbers are coprime if their only common factor is 1.For example 16 and 21 are coprime, and 16, 21 and 25 are mutually coprime.

divInteger divide. Any remainder is discarded.For example, 14 div 5 is 2.(cf mod)

DivisorA divisor of a number is any number which divides it without leaving a remainder.A (positive) divisor of n that is different from n is a proper divisor of n.For example, the positive divisors of 10 are 2, 5 and 10. The proper divisors of 10 are 2 and 5.

FactorA synonym of divisor.

Fibonacci numbersThe Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, ... Each Fibonacci number after the second isthe sum of the preceding two Fibonacci numbers.

Geometric meanThe geometric mean of two numbers a and b is

√a× b.

For example, the geometric mean of 8 and 18 is 12.

Geometric progression (GP)Three or more numbers a0, a1, a2, . . . form a GP if if the ratio between successive numbers isconstant.That is, ai+1/ai = ai/ai−1.For example 2, 6, 18, 54 form a GP with common ratio 3, and 20, 30, 45 form a GP withcommon ratio 1.5.

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Greatest common divisor (GCD)The greatest common divisor of two numbers is the largest number which is a factor of bothnumbers.For example the greatest common divisor of 12 and 20 is 4.

Lowest common multiple (LCM)The lowest common multiple of two numbers is the smallest number which is a multiple ofboth numbers.For example the lowest common multiple of 12 and 20 is 60.

Magic squareIn a magic square, the sums of the rows, columns and diagonals are equal. Normally, duplicatesare not allowed, but in some of the puzzles in this book there may be duplicates. Examples:

No duplicates Duplicates4 9 23 5 78 1 6

5 1 65 4 32 7 3

mod (modulo)a mod n is the remainder after a is divided by n.For example, 12 mod 5 is 2.(cf div)

Number of ways of choosing(na

)(na

)is the number of ways of choosing a elements from n elements without replacement.(

n2

)= n× (n− 1)/2 and

(n3

)= n× (n− 1)× (n− 2)/6.

For example, the number of ways of choosing 2 elements from 10 elements is 45.

Octal numbersOctal numbers are numbers with base 8.Normal (decimal) numbers have base 10,so that 43210 = 4× 102 + 3× 101 + 2,whilst 4328 = 4× 82 + 3× 81 + 2 = 28210.

PalindromeA number (or word) which is the same when reversed.For example, 1221 is a palindrome.

PrimeThe only factors of a prime number are itself and 1.For example, 11 is prime.

Prime factorA factor which is a prime.For example, 2 is a prime factor of 20, but 4 is not.

Pythagorean tripleThree numbers a, b and c form a Pythagorean triple if a2 = b2 + c2.Examples of Pythagorean triples are (3, 4, 5) and (30, 40, 50).

Triangular numbersThe triangular numbers are 1, 3, 6, 10, 15, 21, ...The nth triangular number is n× (n+ 1)/2.

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