A logic exercise by Lewis Carroll: What conclusion can be drawn from these premises?
- No shark ever doubts that it is well fitted out.
- A fish that cannot dance a minuet is contemptible.
- No fish is quite certain that it is well fitted out unless it has three rows of teeth.
- All fishes except sharks are kind to children.
- No heavy fish can dance a minuet.
- A fish with three rows of teeth is not to be despised.
Longfellow thought that Dante Gabriel Rossetti, the Victorian poet and painter, was two different people. On leaving Rossetti’s house he said, “I have been very glad to meet you, Mr. Rossetti, and should like to have met your brother also. Pray tell him how much I admire his beautiful poem, ‘The Blessed Damozel.’”
In Philosophical Troubles, Saul A. Kripke offers a related puzzle. Peter believes that politicians never have musical talent. He knows of Paderewski, the great pianist and composer, and he has heard of Paderewski the Polish statesman, but he does not know that they are the same person. Does Peter believe that Paderewski had musical talent?
A riddle by Horatio Walpole:
Before my birth I had a name,
But soon as born I chang’d the same;
And when I’m laid within the tomb,
I shall my father’s name assume.
I change my name three days together
Yet live but one in any weather.
Another puzzle by Boris Kordemsky: Jack London tells of racing from Skagway, Alaska, to a camp where a friend lay dying. London drove a sled pulled by five huskies, which pulled the sled at full speed for 24 hours. But then two dogs ran off with a pack of wolves. Left with three dogs and slowed down proportionally, London reached the camp 48 hours later than he had planned. If the two lost huskies had remained in harness for 50 more miles, he would have been only 24 hours late. How far is the camp from Skagway?
In the February 1926 issue of the National Puzzlers’ League publication Enigma, “Remardo” offered this mock-Latin verse:
Justa sibi dama ne
Luci dat eas qua re
Ibi dama id per se
Veret odo thesa me
What does it mean?
A Russian problem from the 1999 Mathematical Olympiad:
Each cell in an 8×8 grid contains an arrow that points up, down, left, or right. There’s an exit at the top edge of the top right square. You begin in the bottom left square. On each turn, you move one square in the direction of the arrow, and then the square you have departed turns 90° clockwise. If you’re not able to move because the edge of the board blocks your path, then you remain on the square and it turns 90° clockwise. Prove that eventually you’ll leave the maze.
From the 2001 Moscow Mathematical Olympiad:
Before you are three piles of stones. One contains 51 stones, one 49 stones, and one 5 stones. On each move you can either combine two piles into one or divide any pile with an even number of stones into two equal piles. Is it possible to end up with 105 piles, each containing a single stone?
On Nov. 25, 1862, Abraham Lincoln sent this dispatch to Gen. Ambrose Burnside at Aquia Creek, Va.:
Can Inn Ale me withe 2 oar our Ann pas Ann me flesh ends NV Corn Inn out with U cud Inn heaven day nest Wed roe Moore Tom darkey hat Greek Why Hawk of abbott Inn B chewed I if.
What did it mean?
In 1887, Irish journalist Richard Pigott sold a series of letters to the Times of London. Purportedly written by Irish Parliamentary Party leader Charles Parnell, they seemed to show that Parnell had approved of a savage political assassination five years earlier. Parnell denounced the letters as a “villainous and bare-faced forgery.”
In the ensuing investigation, Parnell’s attorney asked Pigott to write a series of words and submit them to the court:
What was the point of this?
From Russian puzzle maven Boris Kordemsky:
Two diesel ships leave a pier at the same time. One travels upstream, the other downstream, each with the same motive power. As they depart, one drops a lifebuoy into the water. An hour later, both ships reverse course. Which will reach the buoy first?
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?