Double Talk

A logical curiosity by L.J. Cohen: A policeman testifies that nothing a prisoner says is true, and the prisoner testifies that something the policeman says is true. The policeman’s statement can’t be right, as that leads immediately to a contradiction. This means that something the prisoner says is true — either a new statement or his current one. If it’s a new statement, then we establish that the prisoner says something else. If it’s his current statement, then the policeman must say something else (as we know that his current statement is false).

J.L. Mackie writes, “From the mere fact that each of them says these things — not from their being true — it follows logically, as an interpretation of a formally valid proof, that one of them — either of them — must say something else. And hence, by contraposition, if neither said anything else they logically could not both say what they are supposed to say, though each could say what he is supposed to say so long as the other did not.”

| Science & Math

The Devil’s Game

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Ms. C dies and goes to hell, where the devil offers a game of chance. If she plays today, she has a 1/2 chance of winning; if she plays tomorrow, the chance will be 2/3; and so on. If she wins, she can go to heaven, but if she loses she must stay in hell forever. When should she play?

The answer is not clear. If she waits a full year, her probability of winning will have risen to about 0.997268. At that point, waiting an additional day will improve her chances by only about 0.000007. But at stake is infinite joy, and 0.000007 multiplied by infinity is infinite. And the additional day spent waiting will contain (presumably) only a finite amount of torment. So it seems that the expected benefit from a further delay will always outweigh the cost.

“This logic might suggest that Ms. C should wait forever, but clearly such a strategy would be self-defeating,” wrote Edward J. Gracely in proposing this conundrum in Analysis in 1988. “Why should she stay forever in a place in order to increase her chances of leaving it? So the question remains: what should Ms. C do?”

(Edward J. Gracely, “Playing Games With Eternity: The Devil’s Offer,” Analysis 48:3 [1988]: 113-113.)

| Science & Math

The Copernicus Method

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Image: Wikimedia Commons

Princeton astrophysicist J. Richard Gott was visiting the Berlin Wall in 1969 when a curious thought occurred to him. His visit occurred at a random moment in the wall’s existence. So it seemed reasonable to assume that there was a 50 percent chance that he was observing it in the middle two quarters of its lifetime. “If I was at the beginning of this interval, then one-quarter of the wall’s life had passed and three-quarters remained,” he wrote later in New Scientist. “On the other hand, if I was at the end of of this interval, then three-quarters had passed and only one-quarter lay in the future. In this way I reckoned that there was a 50 per cent chance the wall would last from 1/3 to 3 times as long as it had already.”

At the time, the wall was 8 years old, so Gott concluded that there was a 50 percent chance that it would last more than 2-2/3 years but fewer than 24. The 24 years would have elapsed in 1993. The wall came down in 1989.

Encouraged, Gott applied the same principle to estimate the lifetime of the human race. In an article published in Nature in 1993, he argued that there was a 95 percent chance that our species would survive for between 5,100 and 7.8 million years.

When and whether the method is valid is still a matter of debate among physicists and philosophers. But it’s worth noting that on the day Gott’s paper was published, he used it to predict the longevities of 44 plays and musicals on and off Broadway. His accuracy rate was more than 90 percent.

| Science & Math

World Cuisine

This would be incredible if it weren’t so well documented — in the early 19th century a Frenchman known as Tarrare became famous as a “polyphagist,” an eater of everything. From the London Medical and Physical Journal, September 1819:

He would eat dogs and cats. One day, in the presence of the chief physician of the army, Dr. Lorence, he seized by the neck and paws a large living cat, tore open its belly with his teeth, sucked its blood, and devoured it, leaving no part of it but the bare skeleton: half an hour afterwards he threw up the hairs of the cat, just as birds of prey, and other carnivorous animals, do. Tarrare liked the flesh of serpents; he managed them familiarly, and ate alive the largest snakes (couleuvres) without leaving any part of them. He swallowed a large eel alive without chewing it, but we thought we perceived him crush its head between his teeth. He ate, in a few instants, the dinner prepared for fifteen German labourers: this repast was composed of four bowls of curdled milk, and two enormous hard puddings. After this the belly of Tarrare, commonly lank and wrinkled, was distended like a balloon: he went away, and slept until the next day, and was not incommoded by it. M. Comville, the surgeon-major of the hospital where Tarrare then was, made him swallow a wooden case, enclosing a sheet of white paper: he voided it the following day by the anus, and the paper was uninjured. The general-in-chief had him brought before him; and, after having devoured in his presence nearly thirty pounds of raw liver and lights, Tarrare again swallowed the wooden case, in which was placed a letter to a French officer, who was a prisoner to the enemy. Tarrare set out, was taken, flogged, imprisoned; voided the wooden case, which he had retained thirty hours, and had the address to swallow it again, to conceal the knowledge of its contents from the enemy. They tried to cure him of this insatiable hunger, by the use of acids, preparations of opium, and pills of tobacco; but nothing diminished his appetite and his gluttony. He went about the slaughter-houses and bye-places, to dispute with dogs and wolves the most disgusting aliments. The servants of the hospital surprised him drinking the blood of patients who had been bled, and in the dead-room devouring the bodies. A child fourteen months old disappeared suddenly; fearful suspicions fell on Tarrare; they drove him from the hospital. M. Percy lost sight of him for four years: at the end of this time he saw Tarrare at the civil hospital at Versailles, where he was perishing in a tabid state.

“He shortly died, and his body almost immediately became a mass of putridity,” wrote Perceval Barton Lord in his 1839 Popular Physiology. “On being opened, his stomach was found to be of an immense size, and, as well as all the intestines, in a state of suppuration.”

| Oddities

Out of Rank

beasley chess puzzle

A puzzle from John Beasley’s The Mathematics of Games (2006): Black has just moved. What is the smallest number of moves that can have been played in this game?

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| Puzzles

The Sofa Problem

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Image: Wikimedia Commons

In 1966, Austrian mathematician Leo Moser asked a pleasingly practical question: If a corridor is 1 meter wide, what’s the largest sofa one could squeeze around a corner?

That was 46 years ago, and it’s still an open question. In 1968 Britain’s John Michael Hammersley showed that a sofa shaped somewhat like a telephone receiver could make the turn even if its area were more than 2 square meters (above). In 1992 Joseph Gerver improved this a bit further, but the world’s tenants await a definitive solution.

Similar problems concern moving ladders and pianos. Perhaps what we need are wider corridors.

| Science & Math

Drinking Problem

I have a 16-ounce bottle of wine and want to make it last as long as possible, so I establish the following plan: On the first day I’ll drink 1 ounce of wine and refill the bottle with water. On the second day I’ll drink 2 ounces of the mixture and refill the bottle with water. On the third day I’ll drink three ounces of the mixture and again refill the bottle with water. If I continue until the bottle is empty, how many ounces of water will I have drunk?

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| Puzzles

A Muddy Grave

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Memories of the first day of the Battle of the Somme, July 1, 1916:

“Before the bombardment started and while everything was peaceful, I could see through my periscope a young Englishman playing his trumpet every evening. We used to wait for this hour but suddenly there was nothing to be heard and we all hoped that nothing had happened to him.” — Feldwebel Karl Stumpf, 169th Regiment

“As the gun-fire died away I saw an infantryman climb onto the parapet into No Man’s Land, beckoning others to follow. As he did so he kicked off a football; a good kick, the ball rose and travelled well towards the German line. That seemed to be the signal to advance.” — Private L.S. Price, 8th Royal Sussex

“For some reason nothing seemed to happen to us at first; we strolled along as though walking in a park. Then, suddenly, we were in the midst of a storm of machine-gun bullets and I saw men beginning to twirl round and fall in all kinds of curious ways as they were hit — quite unlike the way actors do it in films.” — Private W. Slater, 2nd Bradford Pals

“When the English started advancing we were very worried; they looked as though they must overrun our trenches. We were very surprised to see them walking, we had never seen that before. I could see them everywhere; there were hundreds. The officers were in front. I noticed one of them walking calmly, carrying a walking stick. When we started firing, we just had to load and reload. They went down in their hundreds. You didn’t have to aim, we just fired into them. If only they had run, they would have overwhelmed us.” — Musketier Karl Blenk, 169th Regiment

“Imagine stumbling over a ploughed field in a thunderstorm, the incessant roar of the guns and flashes as the shells exploded. Multiply all this and you have some idea of the Hell into which we were heading. To me it seemed a hundred times worse than any storm.” — Private E. Houston, Public Schools Battalion

“The sound was different, not only in magnitude but in quality, from anything known to me. It was not a succession of explosions or a continuous roar; I, at least, never heard either a gun or a bursting shell. It was not a noise, it was a symphony. And it did not move. It hung over us. It seemed as though the air were full of vast and agonised passion, bursting now with groans and sighs, now into shrill screaming and pitiful whimpering, shuddering beneath terrible blows, torn by unearthly whips, vibrating with the solemn pulses of enormous wings. And the supernatural tumult did not pass in this direction or in that. It did not begin, intensify, decline and end. It was poised in the air, a stationary panorama of sound, a condition of the atmosphere, not the creation of man.” — Anonymous NCO, 22nd Manchester Rifles

It would become the bloodiest day in the history of the British Army, with 57,470 casualties. “From that moment all my religion died,” recalled Private C. Bartram of the 94th Trench Mortar Battery. “All my teaching and beliefs in God had left me, never to return.”

| History

Weighty Matters

I’ve accidentally turned the calibration dial on my bathroom scale, so its readings are skewed by a consistent amount. Apart from that it works fine, though. When I stand on the scale it reads 170 pounds, when my wife stands on it it reads 130, and when we stand on it together it reads 292 pounds. How should I adjust the scale?

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| Puzzles