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.


Benjamin Dickman of Columbia University Teachers College offered this problem in the April 2011 issue of Math Horizons.

On the first day of math class, 36 desks are arranged in a circle to accommodate 18 girls and 18 boys. The teacher says that students can choose their own seats so long as girls and boys take turns in making their choices. The last two students to choose will be Amanda and Bill, not necessarily in that order.

Bill has a crush on Amanda and wants to sit next to her. She can’t stand him and is determined to prevent this from happening. All the girls want to help Amanda, and all the boys want to help Bill. If Amanda gets to decide which gender will pick first, what should she say?

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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.)

The Black Beetle


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.


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.


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.”)

The Switchback Puzzle


“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?

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Local Materials

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.

All Out

Image: Wikimedia Commons

Played by Japanese priests in the 16th century, taikyoku shogi may be the largest variant of chess ever devised. Each player deploys 402 pieces of 209 types on a board of 1,296 squares to try to capture his opponent’s king(s) and prince(s).

It’s not clear precisely how it was played, but Wikipedia takes more than 10,000 words to describe one likely set of rules.

(Thanks, Alejandro.)

The Case of the Speluncean Explorers

Image: Wikimedia Commons

In the year 4299, five cave explorers are trapped by a landslide. To stay alive they decide to engage in cannibalism, choosing the victim by throwing dice. When the four survivors are rescued, they’re convicted of murder and face a mandatory sentence of death. After a public outcry, the “Supreme Court of Newgarth” takes up the case. Its five judges subscribe to five different legal philosophies, with the result that two vote to affirm the convictions, two vote to overturn them, and one recuses himself. As this is a tie, the original conviction is upheld and the four explorers face death.

Harvard philosopher Lon L. Fuller presented this story in 1949 to contrast various legal philosophies prevailing in the 20th century, primarily natural law and legal positivism.

But in the ensuing years, dozens of further hypothetical judgments have been offered by writers from perspectives ranging from historical contextualism to process theory. Frank Easterbrook wrote in 1999 that Fuller’s essay combines “a timely consideration of contemporaneous debates with a timeless quality that continues to entice students and scholars to think and write about [it] some half a century later — and will doubtless engage our successors well into the next millennium.”

Fuller had written, “The case was constructed for the sole purpose of bringing into a common focus certain divergent philosophies of law and government. These philosophies presented men with live questions of choice in the days of Plato and Aristotle. Perhaps they will continue to do so when our era has had its say about them. If there is any element of prediction in the case, it does not go beyond a suggestion that the questions involved are among the permanent problems of the human race.”

(Lon L. Fuller, “The Case of the Speluncean Explorers,” Harvard Law Review 62:4 [February 1949], 616–645.)