“That’s me,” I said.

She smiled and held out a hand. “I’mThelma O’Keefe. We were in the samealgebra 101 class.”

We shook hands.

“You won’t remember me,” she said. “Iwas fat in those days, and shy, and notvery pretty.”

“That’s hard to believe,” I said. “Youlook gorgeous now.”

“Well, thank you, kind sir,” she said,

SEPTEMBER 2010 • MATH HORIZONS • WWW.MAA.ORG/MATHHORIZONS 7

Charles Trigg published 54 differentproofs in the Journal of RecreationalMathematics Vol. 4, April 1971, pages90–99. A proof using paper cutting, byAli R. Amir-Moéz, appeared in the samejournal, Vol. 5, Winter 1973, pages 8–9.For other proofs, see Roger North’scontribution to The MathematicalGazette, December 1973, pages 334–36, and its continuation in thesame journal, October 1974, pages212–15. For a generalization of theproblem to a row of n squares, seeTrigg’s “Geometrical Proof of a Resultof Lehmer’s,” in The FibonacciQuarterly, Vol. 11, December 1973,pages 539-40.

Red, White, and Blue WeightsOne way to solve the problem of sixweights—two red, two white, and twoblue—is first to balance a red and awhite weight against a blue and a whiteweight.

If the scales balance, you know thereare a heavy and a light weight on eachpan. Remove both colored weights,leaving the white weights, one on eachside. This establishes which white

weight is the heavier. At thesame time it tells you whichof the other two weightsused before (one red, oneblue) is heavy and which islight. This in turn tells youwhich is heavy and which islight in the red-blue pair notyet used.

If the scales do not balanceon the first weighing, youknow that the white weighton the side that went downmust be the heavier of thetwo whites, but you are stillin the dark about the redand the blue. Weigh theoriginal red against the mate of the original blue (orthe original blue against the mate of theoriginal red). As C. B. Chandler (whosent this simple solution) put it, theresult of the second weighing, plus thememory of which side was heavier inthe first weighing, is now sufficient toidentify the six weights.

For readers who liked working on thisproblem, Ben Braude, a New York City

dentist and amateur magician, devisedthe following variation. The six weightsare alike in all respects (including color)except that three are heavy and threelight. The heavy weights weigh thesame and the light weights weigh thesame. Identify each in three separateweighings on a balance scale.

DOI: 10.4169/194762110X524684

Figure 2. Construction for proof of the three-squaretheorem.

Several years ago I was agraduate student at theUniversity of Chicago. I was

working on my doctorate in physics,about possible ways to test superstringtheory, when my brother in Tulsa diedsuddenly from a heart attack. Bothparents had earlier passed away. Afterthe funeral I drove past my past,marveling at the enormous changesthat had taken place since I grew upthere. I drove by the red brick building,now an enormous warehouse, that had

once been Tulsa Central High. Mygrades in history, Latin, and English litwere low, but I was good in math andhad a great physics teacher. He wasmainly responsible for my majoring inphysics after a scholarship took me tothe University of Chicago.

While I was having dinner at a popularrestaurant on the corner of Main andSixth streets, the waitress stared at mewith a look of surprise. “Are youMichael Brown?”

Short Fiction

Superstrings and ThelmaMartin Gardner

smiling. “You were a whiz at algebra.Do you remember when you caught Mr.Miller in a mistake he made on theblackboard, and how embarrassed hewas?”

“I remember. He was a miserableteacher. I think he hated math.”

“I know I hated it,” Thelma said.

“I’m sorry to hear that. Math can beexciting and beautiful when it’s taughtproperly.”

After Thelma brought my receipt andcredit card, I said, “Any chance I couldsee you after work? Maybe you couldsteer me to a late night bar where wecould chat about old times?”

“I’m free at eleven,” she said.

I followed Thelma’s car to a pleasantlittle bar on the outskirts of town, nearwhere she lived. She was divorced, shetold me, and had a boy of ten who wasprobably asleep in her apartment. Thebar served only beer. She said shedidn’t drink anything with more alcoholthan beer. Her ex, she added, had analcohol problem. I didn’t press her fordetails. Instead, I fear I talked too muchabout myself, andmainly aboutsuperstrings.

I did my best toexplain that stringswere inconceivably tiny loops, likerubber bands, that vibrated at differentrates. Their frequencies generated allthe properties of the fundamentalparticles, such as electrons and quarks.The simplest string vibration producesgravitons, conjectured particles thatcarry gravity waves.

“You have nice dark eyes,” sheinterrupted.

“Thanks,” I said. “Your eyes aren’t sobad either.”

I tried hard to explain how a famousphysicist named Ed Witten hadgeneralized strings to what he calledM-theory. The M stands for

membranes, or branes forshort. A superstring is abrane of one dimension.Other branes have high-er dimensions. Our uni-verse, I said, has ten oreleven dimensions, ofwhich six or seven aresqueezed intocompact tinyspheres that areattached to everypoint in our space-time.

“I didn’t under-stand a word youspoke,” Thelmasaid. “It soundsnutty to me. Doyou believe allthat?”

“Mostly. I think strings are for real,but I’m not so sure about Witten’smembranes.”

“Is everything made of strings?”Thelma asked.

“Everything.”

“And what are the strings made of?”

“Nothing. They’re just puremathematical structures.”

“If the universe isn’t made of anything,”she said, “how come it exists?”

“Good question. Nobody knows.”

“Well, maybe God knows,” she said.

Outside the bar, standing by our cars,Thelma invited me to her apartment forsome coffee.

“No,” I said. “I really can’t stay anotherminute. I have to be up early to catch aplane to Chicago. It was great gettingto know you.”

“Will I see you again?”

“It would be a pleasure,” I said.

Like a fool I failed to ask for heraddress and phone number. We shookhands. She said goodbye, then startledme with a quick kiss on the mouth.

Almost a year drifted by. My thesis was published as abook by theUniversity of Chica-go Press. Mysuggestions for test-

ing string theory were taken seriouslyby most stringers. There was hope thatsome of the tests might actually bemade by a new atom cruncher underconstruction in Switzerland. There werevague rumors about a Nobel prize.

I was unable to get Thelma out of myhead. I kept thinking of her wonderfulsmile, and how good she smelled. Itwasn’t perfume. Was it her hair? Ithought about her more than I thoughtabout superstrings!

The University of Oklahoma, atNorman, hired me as an assistantprofessor. A suburb of Oklahoma City,Norman is only a few hours drive fromTulsa.

8 SEPTEMBER 2010 • MATH HORIZONS • WWW.MAA.ORG/MATHHORIZONS

Photograph courtesy of Adeline C., http://www.flickr.com/photos/adollinn/

I was unable to get Thelma out of my head. I kept thinking of her wonderful smile, and how good she smelled.

None of the waitresses at therestaurant where Thelma had workedknew what had happened to her. Sheleft her job six months before, and theyhadn’t heard from her since.

No Thelma O’Keefe was in the Tulsaphone book. I drove back to Normanfeeling sad and frustrated. Should I hirea detective? The Norman yellow bookhad a long list of “investigators” andtwo detective agencies.

I was planning to call one of theagencies when my telephone rang. Itwas Thelma!

“I heard you were asking about me,”she said.

“Yes. How did you get my phonenumber?”

“It’s on the Internet. How are strings?”

“Not so good. It didn’t predict darkmatter. It didn’t predict dark energy. Iteven failed to pass one of my tests.Lots of stringers are starting to havedoubts, including me.”

“If we meet again,” said Thelma, “don’ttell me about it.”

Further ReadingTwo recent books attacking string/Mtheory as pseudoscience are Not EvenWrong, by mathematician Peter Woit(Basic Books, 2006), and The Troublewith Physics, by Lee Smolin (MarinerBooks, 2007). See Chapter 18, “IsString Theory in Trouble?” in my book,The Jinn from Hyperspace (PrometheusBooks, 2008).

Editor's note: A provocative andwidely discussed article in which string theory is used to suggest thatgravity is not a fundamental force, butis rather a consequence of entropy, isErik Verlinde's “On the Origin of Gravity and the Laws of Newton”(2010), which can be found athttp://arxiv.org/abs/1001.0785.

DOI: 10.4169/194762110X525557

SEPTEMBER 2010 • MATH HORIZONS • WWW.MAA.ORG/MATHHORIZONS 9

Bruce Torrence

Gathering for Gardner

Photographs courtesy of Bruce Torrence

It’s a chaotic scene in the lobby of theRitz-Carlton. There’s a dental con-vention getting under way, and as

travel-weary orthodontic professionalstrickle into the packed room, theyencounter a bewildering display. Everytabletop holds a collection of fascinat-ing objects—puzzles of every imagina-ble shape and design—around whichmagicians, mathematicians, and puzzlemasters gather to discuss their latestinventions. The conversations areanimated, and there is a tangiblefeeling that something important isunfolding. And indeed there is:unbeknownst to the dentists, the ninthGathering for Gardner has just begun.

Atlanta has been host to this unlikelyconvention of tinkerers and freethinkers for almost two decades. Theyassemble to pay homage to MartinGardner, the prolific and magnetic

author whose interests spanned theseemingly disparate disciplines ofmathematics, puzzles, magic, and thespirited debunking of pseudoscience.This invitation-only affair attracts lumi-naries in all of these fields, and despitetheir obvious differences, there is a fer-tile and dynamic common ground anda deep mutual respect among theparticipants.

What Has Tuesday Got to Do with It?The organization of the conference issimple: there is one grand conferenceroom with a single stage. Speakers givebrief presentations in turn and areawarded a dollar coin for each minutethat they finish ahead of their allottedtime—an innovative and surprisinglyinexpensive management tactic. Onlyone speaker really cashed in: GaryFoshee, a mechanical puzzle collector