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.)

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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.

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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.

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Benham’s Top

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Cut out this disc, pierce it with a pencil, and spin it like a top. The colors that appear are not entirely understood; it’s thought that they arise due to the different rates of stimulation of color receptors in the retina. The effect was discovered by the French monk Benedict Prévost in 1826, and then rediscovered 12 times, most famously by the toy maker Charles E. Benham, who marketed an “artificial spectrum top” in 1894. Nature remarked on it that November: “If the direction of rotation is reversed, the order of these tints is also reversed. The cause of these appearances does not appear to have been exactly worked out.”

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Upstanding

You can distinguish a raw egg from a hard-boiled one by spinning it.

The reason for this was puzzled out only in 2002 by mathematicians Keith Moffat of Cambridge University and Yatuka Shimomura of Keio University. Friction between the egg and the table produces a gyroscopic effect, and the egg trades some kinetic energy for potential energy, raising its center of gravity. The raw egg can’t do this because its runny interior lags behind the shell. Moffat wrote:

Place a hard-boiled egg on a table,
And spin it as fast as you’re able;
It will stand on one end
With vectorial blend
Of precession and spin that’s quite stable.

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Right and Wrong

Can objects have preferences? The rattleback is a top that seems to prefer spinning in a certain direction — when spun clockwise, this one arrests its motion, shakes itself peevishly, and then sweeps grandly counterclockwise as if forgiving an insult.

There’s no trick here — the reversal arises due to a coupling of instabilities in the top’s other axes of rotation — but prehistoric peoples have attributed it to magic.

See Right Side Up.

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