Science & Math

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 colour. 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

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.

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.

And Tennessee’s Odyssey mirror maze covers 5,000 square feet with a field of eight-foot mirrors set at 60-degree angles to create the illusion of an enormous vaulted crypt:

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

Nobrow Cartoons by Mark Heath

nobrow klein bottles

After Hours

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.

In a Word

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.)

Command Performance

J.B.S. Haldane’s father was a physiologist who would sometimes take his son along while investigating mines in order to teach him the rudiments of science. At one point they were lowered by a bucket into a pit in North Staffordshire, where a tunnel’s low roof forced their party to crawl:

“After a while, we got to a place where the roof was about eight feet high and a man could stand up. One of the party lifted his safety lamp. It filled with blue flame and went out with a pop. If it had been a candle this would have started an explosion, and we should probably have been killed. But of course the flame of the explosion inside the safety lamp was kept in by the wire gauze. The air near the roof was full of methane, or firedamp, which is a gas lighter than air, so the air on the floor was not dangerous.

“To demonstrate the effects of breathing firedamp, my father told me to stand up and recite Mark Anthony’s speech from Shakespeare’s Julius Caesar, beginning ‘Friends, Romans, countrymen.’ I soon began to pant, and somewhere about ‘the noble Brutus’ my legs gave way and I collapsed on to the floor, where, of course, the air was all right. In this way I learnt that firedamp is lighter than air and not dangerous to breathe.”

So It Goes

The Man who thought about Proteids sat by the roadside, writing with an indelible pencil in a little notebook. And Spring, all in pink and white, came tripping by, and cried to him: ‘I will dance for you! Watch me dance!’ She danced very prettily, but the Man went on writing, and never looked at her once. So Spring, being young, burst into tears, and told her sister, Summer.

Summer said to herself: ‘Spring is very foolish to cry. Probably he does not like dancing. I will sing to him.’ She sang a beautiful sleepy song to him, but he never listened, being busy writing in his little notebook. Summer was indignant, and told her sister, Autumn.

Autumn said: ‘There are many good men who do not like dancing. I will give him some of my wine.’ So she went to the Man and offered him her purple wine, but he merely said, ‘I do not drink wine,’ and resumed his writing. Then Autumn was very angry indeed, and told her big brother, Winter, all that had passed.

Winter was an enormous fellow, with a dreadful roar and howl, and every time he moved, snowflakes came whirling from his flowing robes. ‘Show me the fellow,’ he bellowed, puffing out his cheeks. Then he saw the Man who thought about Proteids, still sitting by the roadside.

‘Do you know me?’ roared Winter, and the Man looked and his teeth chattered like dead men’s bones.

Then Winter seized him by the neck and whirled him round and round, and finally flung him over his left shoulder into space.

And the Man who thought about Proteids has not been seen since, but, the other day, a boy found the little note-book lying by the roadside.

— J.B. Priestley, Brief Diversions, 1922

That’s Logic

“If it is feared by a schizophrenic that nothing feared by a schizophrenic is the case, then there must be at least one other schizophrenic fear besides this one.”

— P.T. Geach, quoted in A.N. Prior, “On a Family of Paradoxes,” Notre Dame Journal of Formal Logic, 1961