Double Trouble
Image: Wikimedia Commons
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
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
Image: Wikimedia Commons
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
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
Guy Debord’s 1957 autobiography, Mémoires, was bound in a sandpaper cover so that it would destroy any book placed next to it.
(Thanks, Vinny.)
Soul Support
Image: Wikimedia Commons
“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.”
The Devil’s Game
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?”
The Copernicus Method
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.
The Sofa Problem
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.
Regeneration
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.)
Benham’s Top
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.”
Misc
- Dorothy Parker left her entire estate to Martin Luther King Jr.
- SOUTH CAMBRIDGE, NY contains 16 different letters.
- 45927 = ((4 + 5) × 9)2 × 7
- STONE AGE = STAGE ONE
- “You cannot be both fashionable and first-rate.” — Logan Pearsall Smith
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.
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.
Some Coincidences
There are almost exactly 500 million inches in the pole-to-pole diameter of the earth.
The speed of light is within 0.1% of 300,000 kilometers per second.
A cube with a side of 1 mile has nearly the same volume as a sphere with a radius of 1 kilometer.
See Applied Math.
(Thanks, Ben.)
High Hopes
A worm crawls along an elastic band that’s 1 meter long. It starts at one end and covers 1 centimeter per minute. Unfortunately, at the end of each minute the band is instantly and uniformly stretched by an additional meter. Heroically, the worm keeps its grip and continues crawling. Will it ever reach the far end?
Heads and Tails
Let’s play a coin-flipping game. At stake is half the money in my pocket. If the coin comes up heads, you pay me that amount; if it comes up tails, I pay you.
Initially this looks like a bad deal for me. If the coin is fair, then on average we should expect equal numbers of heads and tails, and I’ll lose money steadily. Suppose I start with $100. If we flip heads and then tails, my bankroll will rise to $150 but then drop to $75. If we flip tails and then heads, then it will drop to $50 and then rise to $75. Either way, I’ve lost a quarter of my money after the first two flips.
Strangely, though, the game is fair: In the long run my winnings will exactly offset my losses. How can this be?
Rules of Thumb
“If an elderly but distinguished scientist says that something is possible he is almost certainly right, but if he says that it is impossible he is very probably wrong.” — Arthur C. Clarke
“When, however, the lay public rallies around an idea that is denounced by distinguished but elderly scientists and supports that idea with great fervor and emotion — the distinguished but elderly scientists are then, after all, probably right.” — Isaac Asimov
Midair
A “curious puzzle” from Raymond Smullyan:
Imagine a plane table of infinite extent. Attached perpendicularly to the table is a rod of finite length, and above that, attached by a hinge, is a second vertical rod, this one infinitely long.
Operate the hinge. What happens? The infinite rod descends freely through the first 90 degrees, until it’s parallel to the tabletop. But it can’t go beyond this, because then at some point the solid rod would intersect the solid table.
Thus it’s impossible to “rest” an infinite rod on an infinite plane. “And so, you have the curious phenomenon of the hinged rod being supported at only one end!”
Teamwork
LOGIC, n. The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basic of logic is the syllogism, consisting of a major and a minor premise and a conclusion–thus:
Major Premise: Sixty men can do a piece of work sixty times as quickly as one man.
Minor Premise: One man can dig a posthole in sixty seconds; therefore–
Conclusion: Sixty men can dig a posthole in one second.
This may be called the syllogism arithmetical, in which, by combining logic and mathematics, we obtain a double certainty and are twice blessed.
– Ambrose Bierce, The Devil’s Dictionary, 1911
A Simple Proof
Can an irrational number raised to an irrational power yield a rational result?
Yes. 
is either rational or irrational. If it’s rational then our task is done. If it’s irrational, then 
= 2 proves the statement.
