You are sharing a train compartment with two strangers, A and B. You and A each think of a positive integer and whisper it to B. B stands up and says, “This is my stop. You have each thought of a different number. Neither of you can deduce which number is bigger.” He leaves.

You and A regard one another across the compartment. You think, “He cannot have chosen 1, because then he would know immediately that my number is bigger. And he can apply the same reasoning to my number. So it’s certain that neither of us has chosen 1. But once we’ve established that, then it’s equally clear that neither of us can have chosen 2, for the same reason.”

It would seem that you can extend this line of reasoning to include any number you like — including 157, the number you picked. How can this be?

See The Necktie Paradox and Tug of War.

Faced with the simultaneous equations

and

a lazy student ignored the second equation and solved the first in this way:

and hence x = 5 and y = 6. As it happens, that’s correct. Maurice Kraitchik writes, “Can you blame him for his answer?”

In Mathematical Circles Revisited, Howard Eves tells of “Hapless Harry,” who tackled

by writing

and

1/998001 = 0.000001002003004005006007008009010011012013014015016017018019020
0210220230240250260270280290300310320330340350360370380390400410420430440450
4604704804905005105205305405505605705805906006106206306406506606706806907007
1072073074075076077078079080081082083084085086087088089090091092093094095096
0970980991001011021031041051061071081091101111121131141151161171181191201211
2212312412512612712812913013113213313413513613713813914014114214314414514614
7148149150151152153154155156157158159160161162163164165166167168169170171172
1731741751761771781791801811821831841851861871881891901911921931941951961971
9819920020120220320420520620720820921021121221321421521621721821922022122222
3224225226227228229230231232233234235236237238239240241242243244245246247248
2492502512522532542552562572582592602612622632642652662672682692702712722732
7427527627727827928028128228328428528628728828929029129229329429529629729829
9300301302303304305306307308309310311312313314315316317318319320321322323324
3253263273283293303313323333343353363373383393403413423433443453463473483493
5035135235335435535635735835936036136236336436536636736836937037137237337437
5376377378379380381382383384385386387388389390391392393394395396397398399400
4014024034044054064074084094104114124134144154164174184194204214224234244254
2642742842943043143243343443543643743843944044144244344444544644744844945045
1452453454455456457458459460461462463464465466467468469470471472473474475476
4774784794804814824834844854864874884894904914924934944954964974984995005015
0250350450550650750850951051151251351451551651751851952052152252352452552652
7528529530531532533534535536537538539540541542543544545546547548549550551552
5535545555565575585595605615625635645655665675685695705715725735745755765775
7857958058158258358458558658758858959059159259359459559659759859960060160260
3604605606607608609610611612613614615616617618619620621622623624625626627628
6296306316326336346356366376386396406416426436446456466476486496506516526536
5465565665765865966066166266366466566666766866967067167267367467567667767867
9680681682683684685686687688689690691692693694695696697698699700701702703704
7057067077087097107117127137147157167177187197207217227237247257267277287297
3073173273373473573673773873974074174274374474574674774874975075175275375475
5756757758759760761762763764765766767768769770771772773774775776777778779780
7817827837847857867877887897907917927937947957967977987998008018028038048058
0680780880981081181281381481581681781881982082182282382482582682782882983083
1832833834835836837838839840841842843844845846847848849850851852853854855856
8578588598608618628638648658668678688698708718728738748758768778788798808818
8288388488588688788888989089189289389489589689789889990090190290390490590690
7908909910911912913914915916917918919920921922923924925926927928929930931932
9339349359369379389399409419429439449459469479489499509519529539549559569579
5895996096196296396496596696796896997097197297397497597697797897998098198298
3984985986987988989990991992993994995996997999 ...

(Thanks, William.)

In logic, the contrapositive of “If A, then B” is “If not B, then not A”: “If Socrates is a man, then Socrates is human” carries the same message as “If Socrates is not human, then Socrates is not a man.”

To make this vivid for his geometry students, W.P. Cooke of West Texas State University enlisted Tex Ritter’s song “Rye Whiskey”:

If the ocean was whiskey and I was a duck,
I’d swim to the bottom and never come up.
But the ocean ain’t whiskey and I ain’t no duck,
So I’ll play Jack-O-Diamonds and trust to my luck.
For it’s whiskey, rye whiskey, rye whiskey, I cry,
If I don’t get rye whiskey, I surely will die.

In contrapositive form, Cooke said, Ritter would sing:

If I never reach bottom or sometimes come up,
Then the ocean ain’t whiskey or I ain’t a duck.
But my luck can’t be trusted or the cards I’ll not buck,
So the ocean is whiskey or I am a duck.
For it’s whiskey, rye whiskey, rye whiskey, I cry,
If my death is uncertain then I get whiskey (rye).

(American Mathematical Monthly, November 1969)

01/31/2012 Related:

Andrew Marvell’s “To His Coy Mistress” is constructed as a three-paragraph logical argument:

“Had we but world enough, and time,
This coyness, lady, were no crime…”
(If A, then B)

“But at my back I always hear
Time’s winged chariot hurrying near…”
(Not A)

“Now therefore, while the youthful hue
Sits on thy skin like morning dew…
Now let us sport us while we may…”
(Therefore Not B)

Unfortunately, he’s trying to argue his way from “If A, then B” to “Not A, therefore not B,” which is invalid. One wonders how his coy mistress responded.

(Thanks, Cleve.)

The loneliest animal in the world is George, a 100-year-old Pinta Island tortoise who is thought to be the last of his subspecies. When goats overran their island in the Galápagos, George’s companions died out, and no one has discovered a Pinta female, despite a $10,000 reward. If none is found in time for George to breed, he’ll be the last of his kind.

The second loneliest animal is Khanzir, “the only pig in Afghanistan,” where pork products are illegal. Khanzir, whose name means “pig,” was one of a pair of white pigs that China donated to the Kabul zoo in 2002, but his mate has died.

See Spectacularly Bad Driving.

03/09/2018 UPDATE: George passed away in 2012, marking the extinction of the Pinta species.

Hold a match horizontally before a mirror. If the match’s head is to the right, then so is that of its reflection — the match is not reversed left to right. But now hold up the matchbox. Its writing is reversed. Why?

“Here is a related puzzle,” writes psychologist Richard Gregory in Mirrors in Mind (1997). “Hold a mug with writing on it to a mirror. What do you see in the mirror? The reflection of the handle is unchanged — but the writing is right-left reversed. Can a mirror read?!”

Set two mirrors together at a 60° angle and rotate the pair around your line of sight. Your image is preserved despite the mirrors’ rotation. This is not the case if the mirrors are set at 45° or 90°. Why?