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Single Cases


If we roll a fair die an infinite number of times, the outcome 4 occurs in 1/6 of the cases. In this light we can say that the probability of rolling a 4 with this die is 1/6. But suppose that, instead of repeating the experiment forever, we roll the die only once. Now it still seems natural to say that there’s a 1/6 chance of rolling a 4, but in fact either we’ll roll a 4 … or we won’t. Can it make sense to assign a probability to a single outcome? Charles Sanders Peirce writes:

If a man had to choose between drawing a card from a pack containing twenty-five red cards and a black one, or from a pack containing twenty-five black cards and a red one, and if the drawing of a red card were destined to transport him to eternal felicity, and that of a black one to consign him to everlasting woe, it would be folly to deny that he ought to prefer the pack containing the larger proportion of red cards, although, from the nature of the risk, it could not be repeated. It is not easy to reconcile this with our analysis of the conception of chance. But suppose he should choose the red pack, and should draw the wrong card, what consolation would he have? He might say that he had acted in accordance with reason, but that would only show that his reason was absolutely worthless. And if he should choose the right card, how could he regard it as anything but a happy accident? He could not say that if he had drawn from the other pack, he might have drawn the wrong one, because an hypothetical proposition such as, ‘if A, then B,’ means nothing with reference to a single case.

Peirce’s solution to this problem is curiously humanistic. Our inferences must extend to include the interests of all races in all epochs. A soldier storms a fort knowing that he may die but that his zeal, if carried through the regiment, will win the day. The man trying to draw a red card “cannot be logical so long as he is concerned only with his own fate” but “should care equally for what was to happen in all possible cases … and would draw from the pack with the most red cards.”

“He who would not sacrifice his own soul to save the whole world, is, as it seems to me, illogical in all his inferences, collectively.”

A Living Casualty

On Feb. 1, 1918, a French soldier appeared in the railway station in Lyon. He had lost his memory: He muttered that his name was Anthelme Mangin, but he didn’t know who he was or where he belonged. His uniform lacked unit tags, and his pockets held only a cigarette lighter. The authorities placed him in an asylum and published his photograph in newspapers, hoping that his family would recognize him.

This gave desperate hope to scores of families whose loved ones had disappeared. World War I had claimed the lives of 1.4 million Frenchmen, and 300,000 of their bodies were unidentified or never found. Three hundred families claimed Mangin as their own, and dozens of these were given personal interviews with him. But he responded to none of them.

In 1930 he was identified tentatively as Octave Monjoin, a French waiter in the London embassy of the Ottoman State who had returned to his homeland to fight and been taken prisoner on the western front in August 1914. Judicial officers dropped him off near Monjoin’s hometown and observed him from a distance. He went from the railway station to the village, sat in a café that Monjoin had once enjoyed, walked to the house of Monjoin’s father, whom he did not recognize, and said, “The church has changed.”

But others, who had different hopes for Mangin’s identity, refused to accept the validity of the test, and Mangin remained in official limbo until his death in a French mental institution in 1942 — in the midst of another wrenching war.


“I maintain that there is no common language or medium of understanding between people of education and without it — between those who judge of things from books or from their senses. Ignorance has so far the advantage over learning; for it can make an appeal to you from what you know; but you cannot re-act upon it through that which it is a perfect stranger to. Ignorance is, therefore, power.” — William Hazlitt

Pursuit of Truth


Can animals reason without using language? Sextus Empiricus writes:

[Chrysippus] declares that the dog makes use of the fifth complex indemonstrable syllogism when, on arriving at a spot where three ways meet …, after smelling at the two roads by which the quarry did not pass, he rushes off at once by the third without stopping to smell. For, says the old writer, the dog implicitly reasons thus: ‘The animal went either by this road, or by that, or by the other: but it did not go by this or that, therefore he went the other way.’

So, perhaps. There’s a limit, though.


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?

Click for Answer

Turn, Turn, Turn

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.

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The Ghost House


Architect Robert Venturi got a tricky assignment in 1976: The National Park Service wanted to commemorate Ben Franklin’s residence in Philadelphia, but Franklin’s house had been demolished in 1812 and no record of its appearance had survived.

“There were fire insurance descriptions and archaeological remains of the house, so you could tell the exact configuration of the walls, and there were letters about the house exchanged between Franklin and his wife while he was in London and the house was in construction,” Venturi said. “But, with so little visual information, the Park Service was pretty easily persuaded not to try to reproduce the house.”

He solved the problem with a “ghost house” outlining the dimensions of the original house with a steel armature. “The aim was to create a delightful open place in the center of the dense texture of the city,” he said. “So the courtyard became a pleasant neighborhood amenity for people who live there, as well as for tourists.”

Object Lessons

oldenburg good humor monument

In 1965, sculptor Claes Oldenburg proposed building a colossal Good Humor bar on Park Avenue in Manhattan. All the traffic would have to be routed through a bite in the bar’s corner.

He also proposed replacing the Statue of Liberty with a giant electric fan and the Washington Monument with a pair of scissors.

The fan, he said, would guarantee workers in lower Manhattan a “steady breeze.” “You can also think of the Fan as a sort of substitute image of America. The suggestion is probably there but I haven’t drawn a conclusion.”

The Bargain Bin

Unusual book titles collected by Russell Ash and Brian Lake for Bizarre Books (1998):

Jokes Cracked by Lord Aberdeen by Lord Aberdeen, 1929
The Romance of Proctology by Charles Elton Blanchard, 1938
Atomic Bombing: How to Protect Yourself by Watson Davis, 1950
God Drives a Flying Saucer by R.L. Dione, 1973
The Fangs of Suet Pudding by Adams Farr, 1944
The Benefit of Farting Explain’d by “Don Fart-inhando Puffindorst” (Jonathan Swift), 1727
Handbook for the Limbless by Geoffrey Howson, 1922
A Toddler’s Guide to the Rubber Industry by D. Lowe, 1947
Be Married and Like It by Bernarr Macfadden, 1937
Hand Grenade Throwing As a College Sport by Lewis Omer, 1918

In 1963 the Athens publisher Harmi Press published an edition of Oliver Twist by “Mark Twain.” They managed to credit it to Charles Dickens on the title page.

Black and White

wiehe chess problem

By Christian Wiehe. White to mate in two moves.

Click for Answer
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