From the 2001 Moscow Mathematical Olympiad:
Two stones, one black and one white, are placed on a chessboard. A move consists of moving one stone up, down, left, or right. The two stones may not occupy the same square. Does a sequence of moves exist that will produce every possible arrangement of the stones, each occurring exactly once?
You’re fitted with a watch that imparts an electric current to your skin in increments too small to distinguish. Initially it’s set to 0 (off), and the settings run up to 1000. At the start of each week you’re allowed a period of experimentation to compare various settings, and then the watch is returned to its last setting. Then you have the option to increase the setting by 1; if you do this, you get $10,000. You may never reduce the setting.
What should you do? On the first day your experimentation shows you that the highest setting is completely intolerable; at that setting you’d pay any amount of money to get rid of the watch. But on this first week your decision is simply whether to advance from 0 to 1, getting $10,000 for accepting an imperceptible amount of pain. That seems attractive.
The trouble seems to be that your evaluations are “transitive” only at a large scale. If you prefer 0 to 500 and 500 to 1000, then it’s valid to conclude that you’ll prefer 0 to 1000. But if you prefer 51 to 4 (because of the financial reward) and 103 to 51, can we conclude that you’ll prefer 103 to 4? Not necessarily.
Unfortunately for all of us, this describes a lot of life. “The self-torturer is not alone in his predicament,” writes philosopher Warren S. Quinn, who proposed this puzzle in 1990. “Most of us are like him in one way or another. We like to eat but also care about our appearance. Just one more bite will give us pleasure and won’t make us look fatter; but very many bites will. And there may be similar connections between puffs of pleasant smoking and lung cancer, or between pleasurable moments of idleness and wasted lives.” What’s the best course?
(Warren S. Quinn, “The Puzzle of the Self-Torturer,” Philosophical Studies, May 1990)
A poser by Cyril Pearson, puzzle editor for the London Evening Standard at the turn of the 20th century:
Upon the shutters of a barber’s shop the legend above was painted in bold letters. One evening about 8:30, when it was blowing great guns, quite a crowd gathered round the window, and seemed to be enjoying some excellent joke. What was amusing them when the shutter blew open?
In 1996, MIT philosopher George Boolos published this puzzle by Raymond Smullyan in The Harvard Review of Philosophy, calling it “the hardest logical puzzle ever”:
Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.
Boolos made three clarifying points:
- It could be that some god gets asked more than one question (and hence that some god is not asked any question at all).
- What the second question is, and to which god it is put, may depend on the answer to the first question. (And of course similarly for the third question.)
- Whether Random speaks truly or not should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely.
What’s the solution?
By Friedrich Freiherr Von Wardener, 1904. White to mate in two moves.
The people of Delos were arguing before the Athenians the claims of their country,– a sacred island, they said, in which no one is ever born and no one is ever buried. ‘Then,’ asked Pausanias, ‘how can that be your country?’
— F.A. Paley, Greek Wit, 1888
What’s unusual about this paragraph, composed by Lawrence Cowan?
Trade was arrested as a base act after federated reserves regressed faster as extracted free trade was saved as extra reverted waste. Deserted as better fates were created, a few brave castes feared effects as excess stargazers were severed. Statecraft fretted, staggered, braced as steadfast braggarts beat state stewards, stewardesses. Tested as a great craze, trade traversed war; zest was dead as a few eager asses abated better treats, detested street fracases. Facts were effaced as we attested a great faded age.
From Henry Dudeney’s 300 Best Word Puzzles:
- What is that from which you may take away the whole and yet have some left?
- What is it which goes with an automobile, and comes with it; is of no use to it, and yet the automobile cannot move without it?
- Take away my first letter and I remain unchanged; take away my second letter and I remain unchanged; take away my third letter and I remain unchanged; take away all my letters and still I remain exactly the same.
In 1951 James Thurber’s friend Mitchell challenged him to think of an English word that contains the four consecutive letters SGRA. Lying in bed that night, Thurber came up with these:
kissgranny. A man who seeks the company of older women, especially older women with money; a designing fellow, a fortune hunter.
blessgravy. A minister or cleric; the head of a family; one who says grace.
hossgrace. Innate or native dignity, similar to that of the thoroughbred hoss.
bussgranite. Literally, a stonekisser; a man who persists in trying to win the favor or attention of cold, indifferent, or capricious women.
tossgravel. A male human being who tosses gravel, usually at night, at the window of a female human being’s bedroom, usually that of a young virgin; hence, a lover, a male sweetheart, and an eloper.
Unfortunately, none of these is in the dictionary. What word was Mitchell thinking of?
A problem from the 1999 Russian mathematical olympiad:
Show that the numbers from 1 to 15 can’t be divided into a group A of 13 numbers and a group B of 2 numbers so that the sum of the numbers in A equals the product of the numbers in B.
Enigma, the official publication of the National Puzzlers’ League, published this item in the “Chat” column of its August 1916 issue:
“The police department of Lima, O., is greatly puzzled over a cryptic message received in connection with the robbery of a Western Ohio ticket agent. Here it is: WAS NVKVAFT BY AAKAT TXPXSCK UPBK TXPHN OHAY YBTX CPT MXHG WAE SXFP ZAV FZ ACK THERE FIRST TXLK WEEK WAYZA WITH THX.”
As far as I can tell, in the ensuing 97 years it has never been solved. Any ideas?
n. a place where four roads meet
Traveling between country towns, you arrive at a lonely crossroads where some mischief-maker has uprooted the signpost and left it lying by the side of the road.
Without help, how can you choose the right road and continue your journey?
A room contains more than one Martian. Each Martian has two hands, with at least one finger on each hand, and all Martians have the same number of fingers. Altogether there are between 200 and 300 Martian fingers in the room; if you knew the exact number, you could deduce the exact number of Martians. How many Martians are there, and how many fingers does each one have?
In Lord Dunsany’s Fourth Book of Jorkens, a member of the Billiards Club observes a book called On the Other Side of the Sun and says, “On the other side of the sun. I wonder what’s there.”
Jorkens, to everyone’s surprise, says, “I have been there.”
Terbut challenges this, but Jorkens insists he was on the other side of the sun six months ago. Terbut knows perfectly well that Jorkens was at the club six months ago, so he wagers £5 that Jorkens is wrong. Jorkens accepts.
“You have witnesses, I suppose,” says Terbut.
“Oh, yes,” says Jorkens.
“My first witness will be the hall-porter,” says Terbut. “And yours?”
“I am only calling one witness,” says Jorkens.
“Went with you to the other side of the sun?” asks Terbut.
“Oh, yes,” says Jorkens. “Six months ago.”
“And who is he?” asks Terbut.
Whom did Jorkens call?
A bottle of fine wine normally improves with age for a while, but then goes bad. Consider, however, a bottle of EverBetter Wine, which continues to get better forever. When should we drink it?
— John L. Pollock, “How Do You Maximize Expectation Value?”, Noûs, September 1983
See The Devil’s Game.
Several spherical planets of equal size are floating in space. The surface of each planet includes a region that is invisible from the other planets. Prove that the sum of these regions is equal to the surface area of one planet.
You and I drive from Los Angeles to Las Vegas in separate cars. We depart simultaneously, and you stay always ahead of me, dutifully driving the speed limit throughout the trip. Nonetheless I get ticketed for speeding. How?