Science & Math

The Perfect Crime

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Suspect A has shot a man through the heart during the last half minute. But Suspect B shot him through the heart during the preceding 1/4 minute, Suspect C shot him through the heart during the 1/8 minute before that, and so on. Assuming that a bullet through the heart kills a man instantly, the victim must already have been dead before any given suspect shot him.

Indeed, notes José Benardete, he cannot be said to have died of a bullet wound.

“The Research Man’s Prayer”

Help me be MANIC so I may be joyous though the results are equivocal.

Help me be DEPRESSIVE for when a prediction is verified, I must know that it will not later be confirmed.

Help me be SADISTIC so I suffer not though the subjects be sorely anguished.

Help me be MASOCHISTIC for even the most obstinate experimental animal should be a pleasure to me.

Help me be PSYCHOPATHIC to quiet the guilt when I tell loved ones that the experiment is going well.

Help me be SCHIZOPHRENIC to sustain myself by finding hopeful trends in random data.

Help me be PARANOID so I can see in the hostile attitudes of others the supremacy of my own work.

Help me to have ANXIETY ATTACKS so that even on holidays I find myself toiling in the laboratory.

And finally,

Help my wife get a job! for when I cross over the shadowy border of normalcy, somebody will have to support the kids. Amen.

— R.A McCleary in the Worm Runner’s Digest, November 1960

Double Trouble

The properties of the simple Möbius strip are well understood: Take a strip of paper, give it a half-twist, and tape the ends together. Now an ant can traverse the full length of the loop, on both sides, and return to its starting point without ever crossing an edge.

But try doing the same thing with two strips of paper. Pair the strips, give them a half-twist, and connect the ends. Now it’s possible to insert a toothpick between the bands and to draw the toothpick along the entire length of the loop, which seems to show that they’re two distinct objects. But if you draw a line along either strip, starting anywhere, you’ll find that you traverse both strips and return to your starting point.

“I have known people to ponder this for hours while listening to Pink Floyd without ever fully appreciating what they have beheld,” writes Clifford Pickover in The Möbius Strip. Are you holding one object or two?

The Paradox of the Second Ace

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You’re watching four statisticians play bridge. After a hand is dealt, you choose a player and ask, “Do you have at least one ace?” If she answers yes, the chance that she’s holding more than one ace is 5359/14498, which is less than 37 percent.

On a later hand, you choose a player and ask, “Do you have the ace of spades?” Strangely, if she says yes now the chance that she has more than one ace is 11686/20825, which is more than 56 percent.

Why does specifying the suit of her ace improve the odds that she’s holding more than one ace? Because, though a smaller number of potential hands contain that particular ace, a greater proportion of those hands contain a second ace. It’s counterintuitive, but it’s true.

A Change of Key

5 × 55 × 555 = 152625

remains true if each digit is increased by 1:

6 × 66 × 666 = 263736

Brunnian Links

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.

Number Forms

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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.”

Misc

  • A TOYOTA’S A TOYOTA is a palindrome.
  • Lee Trevino was struck by lightning in 1975.
  • KILIMANJARO contains IJKLMNO.
  • 39343 = 39 + 343
  • “Money often costs too much.” — Emerson

Soul Support

“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

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.”