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!”

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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

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Gray Area

http://commons.wikimedia.org/wiki/File:Karel_Dujardin_-_Italian_Landscape_with_Herdsman_and_a_Piebald_Horse_WGA.jpg

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.

| Language · Science & Math

Turn, Turn, Turn

hampton court maze

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.

http://books.google.com/books?id=t80kAQAAIAAJ

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.

http://books.google.com/books?id=hvDuAAAAMAAJ

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.”

http://commons.wikimedia.org/wiki/File:Longleat_maze.jpg

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

| Entertainment · Science & Math

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.

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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

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After Hours

http://commons.wikimedia.org/wiki/File:PSM_V48_D855_Crookes_tubes.jpg

When the Cavendish Laboratory at Cambridge University instituted an annual dinner in 1897, it began a tradition of “postprandial proceedings” — typically songs sung around a piano. This air, “Ions Mine,” was sung to the tune of “Clementine”:

In the dusty lab’ratory,
‘Mid the coils and wax and twine,
There the atoms in their glory
Ionize and recombine.

(chorus) Oh my darlings! Oh my darlings!
Oh my darling ions mine!
You are lost and gone forever
When just once you recombine!

In a tube quite electrodeless,
They discharge around a line,
And the glow they leave behind them
Is quite corking for a time.

(repeat chorus)

And with quite a small expansion,
1.8 or 1.9,
You can get a cloud delightful,
Which explains both snow and rain.

(repeat chorus)

In the weird magnetic circuit
See how lovingly they twine,
As each ion describes a spiral
Round its own magnetic line.

(repeat chorus)

Ultra-violet radiation
From the arc of glowing lime,
Soon discharges a conductor
If it’s charged with minus sign.

(repeat chorus)

Alpha rays from radium bromide
Cause a zinc-blende screen to shine,
Set it glowing, clearly showing
Scintillations all the time.

(repeat chorus)

Radium bromide emanation,
Rutherford did first divine,
Turns to helium, then Sir William
Got the spectrum, every line.

(repeat chorus)

The fourth verse was contributed by J.J. Thomson himself.

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In a Word

satanophany
n. a visible manifestation of Satan

Potassium chlorate brings out the worst in gummy bears.

In their 1996 manual Chemical Curiosities, H.W. Roesky and K. Möckel introduce this demonstration with an invocation from the Talmud: “He who ponders long over four things were better never to have been born: that which is above, that which is below, that which came before, and that which comes hereafter.”

(Please don’t try this yourself.)

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