# 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?

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.

# Gray Area

A legal conundrum from Jonathan Swift and Alexander Pope’s Memoirs of Martinus Scriblerus (1741): Sir John Swale bequeaths to Matthew Stradling “all my black and white Horses.” Sir John has six black, six white, and six pied horses. Should Stradling get the pied ones?

On the one hand, “Whatever is Black and White, is Pyed, and whatever is Pyed is Black and White; ergo, Black and White is Pyed, and, vice versa, Pyed is Black and White.”

On the other, “A pyed Horse is not a white Horse, neither is a pyed a black Horse; how then can pyed Horses come under the Words of black and white Horses?”

Perhaps this will help — a proof that all horses are the same color, condensed from Joel E. Cohen, “On the Nature of Mathematical Proofs,” Opus, May 1961, from A Random Walk in Science:

It is obvious that one horse is the same colour. Let us assume the proposition P(k) that k horses are the same colour and use this to imply that k+1 horses are the same colour. Given the set of k+1 horses, we remove one horse; then the remaining k horses are the same colour, by hypothesis. We remove another horse and replace the first; the k horses, by hypothesis, are again the same colour. We repeat this until by exhaustion the k+1 sets of k horses have each been shown to be the same color. It follows then that since every horse is the same colour as every other horse, P(k) entails P(k+1). But since we have shown P(1) to be true, P is true for all succeeding values of k, that is, all horses are the same colour.

# Turn, Turn, Turn

The hedge maze at Hampton Court has been entertaining visitors since 1695, occasionally belying its reputation for ease. In Jerome K. Jerome’s Three Men in a Boat (1889), Harris says, “We’ll just go in here, so that you can say you’ve been, but it’s very simple. It’s absurd to call it a maze.” Then, after two miles of wandering:

‘The map may be all right enough,’ said one of the party, ‘if you know whereabouts in it we are now.’

Harris didn’t know, and suggested that the best thing to do would be to go back to the entrance, and begin again. For the beginning again part of it there was not much enthusiasm; but with regard to the advisability of going back to the entrance there was complete unanimity, and so they turned, and trailed after Harris again, in the opposite direction. About ten minutes more passed, and then they found themselves in the centre.

Mazes have exercised a peculiar fascination for the mathematically minded. The young Lewis Carroll composed this one for a family magazine — the object is to make your way from the outside to the central space; it’s acceptable to pass over or under another path, but a single line means your way is blocked.

Cambridge University mathematician W.W. Rouse Ball constructed this maze in his garden. He notes that unless a loop surrounds the goal, the wanderer can defeat any maze by trailing one hand along a wall, and “no labyrinth is worthy of the name of a puzzle which can be threaded in this way.”

Hampton Court is modest in comparison to the modern hedge maze at Longleat, a stately home in Somerset. Its 16,000 English yews enclose 1.75 miles of paths that require an hour and a half to traverse; the course includes six wooden bridges from which to plot a path to the goal, an observation tower.

In solving any of these, as Harris discovered, the chief danger is overconfidence:

Said a boastful young student from Hayes,
As he entered the Hampton Court maze:
“There’s nothing in it.
I won’t be a minute.”
He’s been missing for forty-one days.

— Frank Richards

# Math Notes

From Pedro A. Pisa in Scripta Mathematica, September 1954 — this identity:

1234 + 2484 + 3674 = 1254 + 2444 + 3694

… remains valid when the digits in each term are permuted in the same way:

1234 + 2484 + 3674 = 1254 + 2444 + 3694
1243 + 2448 + 3647 = 1245 + 2444 + 3649
1324 + 2844 + 3764 = 1524 + 2444 + 3964
1342 + 2844 + 3746 = 1542 + 2444 + 3946
1423 + 2448 + 3467 = 1425 + 2444 + 3469
1432 + 2484 + 3476 = 1452 + 2444 + 3496
2134 + 4284 + 6374 = 2154 + 4244 + 6394
2143 + 4248 + 6347 = 2145 + 4244 + 6349
2314 + 4824 + 6734 = 2514 + 4424 + 6934
2341 + 4842 + 6743 = 2541 + 4442 + 6943
2413 + 4428 + 6437 = 2415 + 4424 + 6439
2431 + 4482 + 6473 = 2451 + 4442 + 6493
3124 + 8244 + 7364 = 5124 + 4244 + 9364
3142 + 8244 + 7346 = 5142 + 4244 + 9346
3214 + 8424 + 7634 = 5214 + 4424 + 9634
3241 + 8442 + 7643 = 5241 + 4442 + 9643
3412 + 8424 + 7436 = 5412 + 4424 + 9436
3421 + 8442 + 7463 = 5421 + 4442 + 9463
4123 + 4248 + 4367 = 4125 + 4244 + 4369
4132 + 4284 + 4376 = 4152 + 4244 + 4396
4213 + 4428 + 4637 = 4215 + 4424 + 4639
4231 + 4482 + 4673 = 4251 + 4442 + 4693
4312 + 4824 + 4736 = 4512 + 4424 + 4936
4321 + 4842 + 4763 = 4521 + 4442 + 4963

And everything above holds true if each term is squared.

# Enforced Rest

At a certain moment yesterday evening I coughed and at a certain moment yesterday I went to bed. It was therefore true on Saturday that on Sunday I would cough at the one moment and go to bed at the other. … But if it was true beforehand … that I was to cough and go to bed at those two moments on Sunday, 25 January 1953, then it was impossible for me not to do so.

— Gilbert Ryle, Dilemmas, 1954