The standard braid has a curious property: If we remove any one of the three strands, the other two are seen to be unconnected. If we remove the black strand above, the blue and red strands simply snake along one above the other. Similarly, removing the red or the blue strand reveals that the remaining strands are not braided together.
See Borromean Rings.
When thinking of numbers, about 5 percent of the population see them arranged on a sort of mental map. The shape varies from person to person, assuming “all sorts of angles, bends, curves, and zigzags,” in the words of Francis Galton, who described them first in The Visions of Sane Persons (1881). Usually the forms are two-dimensional, but occasionally they twist through space or bear color.
People who have forms report that they remain unchanged throughout life, but having one is such a peculiarly personal experience that “it would seem that a person having even a complicated form might live and die without knowing it, or at least without once fixing his attention upon it or speaking of it to his nearest friends,” wrote philosopher G.T.W. Patrick in 1893. One man told mathematician Underwood Dudley that “when he told his wife about his number form, she looked at him oddly, as if he were unusual, when he thought that she was the peculiar one because she did not have one.”
The phenomenon is poorly understood even today; possibly it arises because of a cross-activation between the parts of the brain that recognize spatial relationships and numbers. Two of Dudley’s students were identical twins; both had forms, but the forms were different. “Although our understanding of how the brain works has advanced since 1880, it probably has not advanced enough to deal with number forms,” he writes. “Another hundred years or so may be needed.”
“It seems to me immensely unlikely that mind is a mere by-product of matter. For if my mental processes are determined wholly by the motions of atoms in my brain I have no reason to suppose that my beliefs are true. They may be sound chemically, but that does not make them sound logically. And hence I have no reason for supposing my brain to be composed of atoms.” — J.B.S. Haldane, Possible Worlds, 1927
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.”
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.)
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.
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.
452 = 2025
20 + 25 = 45
453 = 91125
9 + 11 + 25 = 45
454 = 4100625
4 + 10 + 06 + 25 = 45
455 = 184528125
18 + 4 + 5 + 2 + 8 + 1 + 2 + 5 = 45
456 = 8303765625
8 + 3 + 0 + 3 + 7 + 6 + 5 + 6 + 2 + 5 = 45
(Thanks, Dheeraj.)
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.”
