Langley’s Adventitious Angles

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Image: Wikimedia Commons

Edward Mann Langley, founder of the Mathematical Gazette, posed this problem in its pages in 1922:

ABC is an isosceles triangle. B = C = 80 degrees. CF at 30 degrees to AC cuts AB in F. BE at 20 degrees to AB cuts AC in E. Prove angle BEF = 30 degrees.

(Langley’s description makes no mention of D; perhaps this is at the intersection of BE and CF.)

A number of solutions appeared. One, offered by J.W. Mercer in 1923, proposes drawing BG at 20 degrees to BC, cutting CA in G. Now angle GBF is 60 degrees, and angles BGC and BCG are both 80 degrees, so BC = BG. Also, angles BCF and BFC are both 50 degrees, so BF = BG and triangle BFG is equilateral. But angles GBE and BEG are both 40 degrees, so BG = GE = GF. And angle FGE is 40 degrees, so GEF is 70 degrees and BEF is 30 degrees.

| Science & Math

Miniatures

https://commons.wikimedia.org/wiki/File:Komarikmicro.jpg Image: Wikimedia Commons
Image: Wikimedia Commons

Russian artist Anatoly Konenko works on an absurdly, almost unthinkably tiny scale. Trained as an engineer, he took up microminiature art in 1981, inventing his own instruments and techniques. Soon he was writing on grains of rice, poppy seeds, and even a human hair, and in 1994 he began to publish miniature versions of books by Mikhail Koltsov, Yevgeny Yevtushenko, and Alexander Pushkin. His 1996 edition of Chekhov’s Chameleon assembled 29 pages, three color illustrations, and a portrait of the author into a volume 0.9 millimeters square, at the time the smallest book in the world.

In other media, his feats include a 10-milliliter aquarium, a shod flea, a 1:40,000 scale balalaika, and a 17-millimeter chess set with 2-millimeter pieces. Above: a 3.2-millimeter proboscis midge holds a model of the Eiffel Tower. See his website for more — including a caravan of camels passing through the eye of a needle.

| Oddities

Fleeting

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Pains and other such sensory processes may be long or short, continuous or intermittent; but in spite of Longfellow’s ‘long, long thoughts’, I do not think a thought (say, that the pack of cards is on the table, or that Geach’s arguments are fallacious) can significantly be called long or short; nor are we obliged to say that in that case every thought must be strictly instantaneous.

… [W]hat I am suggesting is that thoughts have not got all the kinds of time-relations that physical events, and I think also sensory processes, have. One may say that during half an hour by the clock such-and-such a series of thoughts occurred to a man; but I think it is impossible to find a stretch of physical events that would be just simultaneous, or even simultaneous to a good approximation, with one of the thoughts in the series. I think Norman Malcolm was right when he said at a meeting in Oxford that a mental image could be before one’s mind’s eye for just as long as a beetle took to crawl across a table; but I think it would be nonsense to say that I ‘was thinking’ a given thought for the period of the beetle’s crawl — the continous past of ‘think’ has no such use.

— Peter Geach, God and the Soul, 1969

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Extra Credit

Critic Harold C. Schonberg called Leopold Godowsky’s Studies on Chopin’s Études “the most impossibly difficult things ever written for the piano”; Godowsky said they were “aimed at the transcendental heights of pianism.” In the “Badinage,” above, the pianist plays Chopin’s “Black Key” étude with the left hand while simultaneously playing the “Butterfly” étude with the right and somehow preserving the melodies of both. One observer calculated that this requires 1,680 independent finger movements in the space of about 80 seconds, an average of 21 notes per second. “The pair go laughing over the keyboard like two friends long ago separated, now happily united,” marveled James Huneker in the New York World. “After them trails a cloud of iridescent glory.”

The studies’ difficulty means that they’re rarely performed even today; Schonberg said they “push piano technique to heights undreamed of even by Liszt.” Only Italian pianist Francesco Libetta, above, has performed the complete set from memory in concert.

| Art

Even Up

rolling die puzzle

Suppose we cover a chessboard with 32 dominoes so that each domino covers two squares. What is the likelihood that there will be an even number of dominoes in each of the two orientations (horizontal and vertical)?

In fact this will always be the case. Consider the 32 squares in the odd-numbered horizontal rows. Each horizontal domino on the board covers either two of these squares or none of them. And each vertical domino covers exactly one of these squares. So the horizontal dominoes cover an even number of these squares (call it n), and the number of squares remaining in this group (32 – n) must also be even. This latter number is also equal to the number of vertical dominoes, so both quantities are even.

(By Vyacheslav Proizvolov.)

| Science & Math

The Black Beetle

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In 1966, with the Interstate Highway System on the rise and increasing competition from airlines, the New York Central Railroad decided to experiment with a high-speed rail service. The result was startling: a jet-powered railcar.

With two secondhand General Electric J47-19 jet engines mounted above a streamlined cowling, this diesel car reached a speed of 183.68 mph that July on the arrow-straight rail segment between Butler, Indiana, and Stryker, Ohio.

Ultimately the project went nowhere — the company was headed for a merger with the rival Pennsylvania Railroad — but that experimental jaunt still holds the rail speed record in the United States.

| Technology

Protocol

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Image: Wikimedia Commons

The malicious spirit Kuchisake-onna of Japanese folklore wears a mask and carries a sharp object. When you meet her, she asks whether you think she is beautiful. If you answer no, she kills you. If you answer yes, she removes her mask to reveal that the corners of her mouth have been slit open to her ears.

Then she repeats her question. If again you answer no, she kills you. If you answer yes, she cuts your mouth to resemble her own.

Happily, there are at least two ways to escape: describe her appearance as average, or throw hard candies to distract her.

| Oddities

Closer

The young specialist in English Lit … lectured me severely on the fact that in every century people have thought they understood the Universe at last, and in every century they were proved to be wrong. It follows that the one thing we can say about our modern ‘knowledge’ is that it is wrong.

… My answer to him was, ‘… when people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.’

— Isaac Asimov, The Relativity of Wrong, 1989

(J.R. Deller Jr. wrote, “Education is the process of telling smaller and smaller lies.”)

| History · Science & Math

The Switchback Puzzle

https://books.google.com/books?id=45MkAQAAIAAJ&pg=PA692

“Some years ago there was a craze for rolling pellet puzzles,” wrote Henry Dudeney in 1909, “though they are really more trials of patience than puzzles.”

One exception was this undulated glass tube, which contained three shots or pellets. The task was to get them into the three depressions at A, B, and C, which are unfortunately positioned at high points in the tube.

This “could be solved by a puzzle trick which I was surprised to notice how few people discovered,” Dudeney wrote. What was it?

Click for Answer
| Puzzles

Local Materials

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Image: Wikimedia Commons

After a national competition in 1938, the hamlet of Monsanto became known as “the most Portuguese village in Portugal.”

That’s an odd epithet, because it’s one of the most distinctive towns in the country — its architecture incorporates enormous boulders from the surrounding landscape.

| Oddities