jeff adams - cardano and the case of the cubic

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Cardano and the Case of the CubicAuthor(s): Jeff AdamsSource: Math Horizons, Vol. 12, No. 4 (April 2005), pp. 28-30Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/25678551 .Accessed: 28/03/2013 22:38

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I"It's impossible. It an't be done. It'snever been done before "

Cardano and the

Case of the CubicJeff damsSouthernOregon University

It was a dark and stormy night in San

Francisco, but fortunately, worked inMilan where things were bright and

beautiful. Bright and beautiful that isuntil she walked into

myoffice. Her face

said that she had a problem, then hermouth .said it too.

'Tve got a problem."All the while her legswere tellingme

that I should be the one to solve it, butthe rest of her body was saying that she

was trouble. Except for her left elbow,which was as quiet as the aluminum potof cold lamb stew in my refrigerator.Yeah, that lamb stew wasn't saying anything to anybody and neither was her leftelbow. That's what had me worried.Girls with quiet elbows can't be trusted.I deduce these things. I'm amathematician. My name's Cardano.

"So what," I said. "We all got problems. I had three already this morningand I'm expecting six more by parcelpost from my mother this afternoon."

"Mr. Cardano, I need your help," shecried." I don't know who else to turn to."She fell, weeping, into the chair.

"Sure" I said. "Take a seat and cry.That's what it's there for."

"I'm sorry. I just..."

"I know the story, ister. I already gotthe punchline figured. I'm supposed to

play the schmoe. Supposed to fall for

you with your pretty knees and daintysilent elbows sitting there between yourthighs and calves and upper arms and

forearms, respectively. Well, I don't gofor that I'm not some sentimental Sam

What I do I do well and I get paid wellfor doing what I do well; although some

times, I don't do what I do well as goodas I normally do what I do well and thenI

get paida little less than well.

But it'salways in cash I don't take checks."

"Yeah, okay," she said, "Whatever."

"Good, as long as we got that

straight. hoot me the dope. Give me the

scoop.What's the case?"She reached into her handbag and her

fingers withdrew a folded sheet of paper.She offered it to me, and I took it.

Unfolding it, I almost fell out of mychair for what it had written on it.

"No..." I said standing. I pushedaway from my desk and I went to thewindow. Milan was a beautiful town,radiant in the sunlight. T should be out

there,' I thought, 'walking across theverdant grass with a lime snow cone inone hand and bunch of green helium balloons in the other.' Then I thought'when did green become my favorite

color?' My reverie was broken by hervoice.

"Mr. Cardano?"

"Leave." I told her. "Leave now, anddon't come back "

"Why?" She cried."It's impossible. It can't be done. It's

never been done before ""But your ad, your ad said you were

the best ""Darn." She had me there.She crossed the room, shoving the

paper intomy face so I could take a real

good look at it.

"I'm far-sighted." I said, so she

pulled it back a bit.I sneered: "a bit more..."

There itwas: x3+6x=20.

I snatched the paper from her fingers.Itwas a cubic. No one had ever solved acubic. But for fifty liras a day plusexpenses, I knew I could do it.

Next time we met was at her place. Iwas giving her the advantage by meet

ing her there but she had insisted. Bynow, I calculated, we needed each other.I had her solution and she needed that.also had her bill and I definitely neededher to pay it.

Her house was a niceplace

ifyou'rethe kind who goes in for gold, silver, and

ivory, or if you happen to like originalworks of fine art, oriental silk carpets,and crystal fountains bubbling over with

champagne. So what if her sidewalkswere flowing with milk and honey? Allitmeant to me was that you'd better

watch out for loose bees and cows or

god forbid some monstrous genetichybrid of the two. Even in the sixteenth

century, if you had money like she did,

you could accomplish anything youwanted.

Her servant ledme in and we passedthrough three atriums and an anteroombefore we came to the front hall.

"Madam is in the study," he said tome,

"Wait here." As I stood there look

ing around I realized she had everythingyou could ever want plus a littlemore. In

hiring a guy like me she'd gotten off

cheap. She could've hired any of the

28 APRIL2005

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MATH HORIZONS

Cartoon by Greg Nemec"Mr.Cardano, Ihave a problem."

greats: ScipiFerro,

TonyFior,

Nickythe

Stutterer or Isaac Newton and he hadn'teven been born yet. With the change shehad lying around in her Grecian urns,she could've even had Archimedesraised from his grave. At fifty iras a day,I'd been bamboozled and worst of all,I'd done it tomyself. She'd told me toname my price.

"Who are you?" From behind me thatsweet jackhammer of a voice I remembered ratcheted off the gleaming marble

faces of the corridor."C'mon doll face," I glowered, saun

tering up to her. "Don't get cute withme. It doesn't match your personality.It's me Cardano and I got your answer."

"Cardano," she coughed. "It's been

thirteen years ""Yeah, that's right. Thirteen years at

fifty liras a day plus expenses whichmeans if you want this answer," myhand slipped into the inner pocket of myovercoat and

reappearedwith a small

pieceof

paper carefullyfolded,

"youowe me about two hundred and thirtyseven thousand liras."

"Heavens Gerolamo, haven't you

bathed at all in that time?"I laughed, maybe a little too hard,

maybe a little too long. "You would liketo think, that wouldn't you?"

"Actually, wishing.. .praying..."

"No way, sister, I earned my fee. Ihaven't bathed, shaved or slept. Iadmit Iate but only those things that ame with

inmy grasp. Flies mostly."" For thirteen years?""When I take a job I devote myself toit. You won't find any reason not to pay

me, baby cakes." I could tell my dedication was getting to her. Her eyes were

watering and shewas getting choked up."Okay, I'll pay." She turned but then

froze, without looking back she said,"I'll pay but you have to tellme how youdid it.You have to show me the tech

nique."

"No way, sugarplum."

I said. "We

had a deal, sister, nd our deal didn't saydiddly-squat about me sharing any tech

niques.""But I paid..." She glanced at me

over her shoulder, flashing me with her

pretty puppy dog eyes."Flash your baby blues at me all you

want baby cakes."

"They're green.""Whatever. But I ain't going for it. A

lot f good mathematicians," I said swal

lowing hard,"more brillianter guys than

Ihave fallen for dames with more beautifuller eyes than yours, only to findthemselves face down in a pile ofunworked equations with a knife in theirback and all their good techniquesstashed away in some gal's griddle."

"You mean girdle." She pushed her

swaying bottom towards me.

"That's not gonna be me pumpkin," I

replied icily. "You paid for an answerand that's all

yourtwo hundred and

WWW.MAA.ORG/MATHH0RIZ0NS29







MATH HORIZONS

thirty-seven thousand liras is going to

get you?an answer."

At that point, she seemed to surren

der, her shoulders slumping, her head

hung. I imagined her unseen face staring

disconsolately at the chiffon pink marble of the tiling."I'm sorry, Gerolamo," she said

almost breathlessly and spun around onthe balls of her feet to face me. In the

long bony-knuckled fingers of her righthand, she held a stainless steel .45.

It probably should have dawned onme then that something was up. All theclues lay bare before me like a deck ofTarot cards spread across the tatteredand

thinning purplevelvet of some old

gypsy's table inside a worn and waterstained canvas tent t the last carnival to

pass through town in thewaning days ofOctober. I had only to recognize that theface of the fool was mine.

"You're going to give me that tech

nique, Cardano," she told me. "You haveto."

"Sure, doll face. Right after youshootme," I said confidently. "Don't get

me wrong. It's not that I think youwon't; I think you can't. Dead men don'ttell tales."

"True," she said, tensing her finger onthe trigger, letting the hammer rise."But ones with excruciatingly painfulgunshot wounds often spill their gutslike a penitent to a priest."

Out of the blue, she starts makingsense? I knew the tables had turned, the

coin flipped, the dice thrown, the ballrolled, the car started, and the paradebegun.

For amoment, we stood there onsid

ering each other, considering ourselves.

Each looking backward down the longsad path that had brought us here untilthat uncomfortable silence that alwaysoccurs when a smart gal with a gunholds a dumb guy at bay became theunbearable quiet of a smart gal with a

gun about to plug a dumb guy."Okay then," I said, no longer seeing

the point of holding out. "All you do iscube one third the coefficient of jc, dd itto the square of one half the constant ofthe

equationand take the

squareroot of

thewhole. You will duplicate this and toone of the two you add one half thenumber you have already squared andfrom the other you subtract one half thesame. Then subtracting the cube root ofthe first from the cube root of the sec

ond, the remainder of which is left s thevalue of jc."1

She nodded in a circular sort ofmotion that didn't seem to demonstrateeither agreement or comprehension.

"You want I should write it down?""I think you'd better. And clean it up

a little so the kids can understand it."

"Sure," I said, "but I'm going to needa pencil and some paper."

She turned to call her butler and an

opportunity appeared. The gun was

pointing away from me. I lunged forward and grabbed her right hand with

my left. s the gun discharged and a bullet ricocheted off the hard stone floorsand walls, I punched her in the jaw ashard as I could. Unfortunately, after thirteen years of self-deprivation, itwasn't

very hard and itwasn't nearly enough.Without much effort on her part, shefreed herself and pushed me away, send

ingme tumbling like a Chinese gymnastacross the atrium. I slammed against thewall and lay there, every square inch of

my body slowly bruising like a badlymishandled peach.

"Don't you get it Gerolamo," she

cried, flicking my nose with the barrelof the .45. "I don't want to hurt you. I

justwant the answer."

"Two," Iwhimpered."That's it, x = 2?"

I nodded."Thirteen years for that?"I shrugged."Yeah, okay," she said.

"Whatever."

She whistled through her teeth and in aflash her butler appeared next to her.

"Mr. Cardano needs a pencil andsome paper." He bowed, clicking hisheels together, nd left.

Coda...

When he returned, he brought withhim with a legal pad and a Berol #10Black Beauty. I was a good dog andwrote down the technique as best Icould. I couldn't clean it up much butthat's the way the cubic crumbles |

1This is the actual quote from Cardano's written proof explaining his method.

Answers from page 16

First of all, the ordering of the agecategories makes no sense. The agesstart in the late 30s, then decrease tothose under 25, then jump up to theover-65 category, and finally decreasesfrom there back to the 40s. Youngest tooldest would seem tomake more sensesince that is the natural progression of

ages. Second, open-ended categories

like 'under 25' or 'over 65' should beavoided ifpossible because they re hardto compare to other intervals of a fixedsize. Speaking of fixed size, all themiddle categories are five years in lengthexcept for the interval from 55-64.Since this interval is twice as long as

most of the others it cannot really be

compared to the others. Another prob

lem is that the graph has amissing zero;the lower end of the vertical scale is 6%.

The final problem is that the wrong dis

play type is used. A line graph should beused to show changes over time; the dataare percentages of the whole (they dototal to 100%).

30 April 2005





jeff adams - cardano and the case of the cubic

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