In December 1914 a remarkable thing happened on the Western Front: British and German soldiers stopped fighting and left their trenches to greet one another, exchange souvenirs, bury their dead, and sing carols in the spirit of the holiday season. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Christmas truce, which one participant called “one of the highlights of my life.”

We’ll also remember James Thurber’s Aunt Sarah and puzzle over an anachronistic twin.

The score for British composer Cornelius Cardew’s Treatise is 193 pages of abstract and geometric shapes. There’s no indication as to how to interpret these, but Cardew suggested that the players work out a plan in advance.

Sylvano Bussotti’s Five Pieces for David Tudor drives conventional notation in the direction of graphics and visual art. “For Bussotti, musical results, whatever they may be, flow directly from the visual,” writes Simon Shaw-Miller in Visible Deeds of Music (2002). “The ear plays no part until the work is performed.”

Stripsody, by Bussotti’s friend Cathy Berberian, is composed as a cartoon strip, complete with characters (including Tarzan and Superman) and sound effects at approximate pitch (including oink, zzzzzz, pwuitt, bang, uhu, and kerplunk). The instructions explain, “The score should be performed as if [by] a radio sound man, without any props, who must provide all the sound effects with his voice.” Here’s an example:

In 1834, engineer John Scott Russell was experimenting with boats in Scotland’s Union Canal when he made a strange discovery:

I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped — not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.

They’re known today as solitons. He found that such waves can travel over very large distances, at a speed that depends on their size and width and the depth of the water. Remarkably, as shown above, they emerge from a collision unchanged, simply “passing through” one another.

Inspired by Isaac Newton’s theory that the seven notes of the diatonic scale were related to the colors of the spectrum, French mathematician Louis Bertrand Castel in 1725 invented an “ocular harpsichord” outfitted with lanterns so that “the pressing of the keys would bring out the colours with their combinations and their chords; in one word, with all their harmony, which would correspond exactly to that of any kind of music.” Voltaire devoted Chapter 14 of his Eléments de la philosophie de Newton to the the theory and to Castel’s instrument, and Telemann composed several pieces for it.

The Great Stalacpipe Organ in Luray Caverns, Virginia, produces its tones by striking stalactites with rubber mallets. Leland W. Sprinkle spent three years in the 1950s identifying promising stalactites, shaving them to pitch, and wiring solenoids to trigger the mallets. The tones can be heard throughout the cavern even without amplification, but a loudspeaker system is normallly used.

I think I’ve written elsewhere about the Katzenklavier, a thankfully imaginary instrument first described by Athanasius Kircher in 1650. In the words of one writer, “if a key was pressed on the keyboard, the corresponding tail would be pulled hard, and it would produce each time a lamentable meow.”

Allegedly Louis XI of France challenged Abbé de Baigne to do the same thing with pigs to produce a “piganino”:

That brutal monarch, Louis XI of France, is said to have constructed, with the assistance of the Abbé de Baigne, an instrument designated a ‘pig organ,’ for the production of natural sounds. The master of the royal music, having made a very large and varied assortment of swine, embracing specimens of all breeds and ages, these were carefully voiced, and placed in order, according to their several tones and semitones, and so arranged that a key-board communicated with them, severally and individually, by means of rods ending in sharp spikes. In this way a player, by touching any note, could instantly sound a corresponding note in nature, and was enabled to produce at will either natural melody or harmony!

“The result is said to have been striking, but not very grateful to human ears.”

After our civilization has destroyed itself, the Adriatic will still be playing harmonies on the “sea organ” in Zadar, Croatia. Wind and waves interact with a system of polyethylene tubes to produce sound in a resonating cavity. In 2006 architect Nikola Bašic received the European Prize for Urban Public Space for the project, voted the best among 207 candidate projects from across Europe.

12/17/2016 UPDATE: I completely forgot the mouse organ! (Thanks, Gavin.)

In 1921 aeronautical engineer Giovanni Caproni designed a 100-seat transatlantic airliner with nine wings. With an empty weight of 14,000 kg, the Caproni Ca.60 Transaereo did tolerably well on its first test flight on Lake Maggiore, but it crashed on the second and never flew again. Caproni said, “So the fruit of years of work, an aircraft that was to form the basis of future aviation, all is lost in a moment. But one must not be shocked if one wants to progress. The path of progress is strewn with suffering.”

Nine wings isn’t even the record — that might belong to the “clever but somewhat dogmatic” Victorian engineer Horatio Phillips, who devised aircraft with up to 200 airfoils, basing them on a multi-vaned marine hydrofoil that he had designed. “But air and water do not behave similarly,” notes James Gilbert in The World’s Worst Aircraft (1976). “Air is compressible, while water, as you will know if you have ever belly-flopped into a swimming pool, hardly is. Multiple vanes lift well in water, poorly if at all in air.” Phillips spent £4,000 and gave up.

Suppose you’re working on an algebraic expression that involves variables, addition, multiplication, and parentheses. You try repeatedly to expand it using the distributive law. How do you know that the expression won’t continue to expand forever?

For example, expanding

(x + y)(s(u + v) + t)

gives

x(s(u + v) + t) + y(s(u + v) + t),

which has more parentheses than the original expression.

Set all the variables equal to 2! “The point of the distributive law is that its application doesn’t change the value of the expression,” writes Dartmouth mathematician Peter Winkler. “The value of the initial expression limits the size of anything you can get from it by expansion.”

To kill some time before a meeting of chess grandmasters, Burt Hochberg offered this anonymous puzzle from the 15th century. White must place four white rooks on the board, one at a time, giving check with each one. After each placement the black king can respond with any normal legal move. How can White plan his moves so that the fourth rook reliably gives checkmate?

There’s no trick, and in fact there are several solutions, but Hochberg says the grandmasters studied the position for several minutes before Paul Keres came up with an answer. What was it?

The first rook goes on d2. Now if the king flees to the left, check him from the a-file, and if he flees to the right, check him from the g-file. So, for example, if he goes to e4, then play 2. g4+:

That’s the key — this plan traps the king in a narrow pair of files, and now it’s an easy matter to attack those with the remaining rooks (above, for example: 2. … Kf3 3. Rf4+ 4. Ke3 Re2#).

Working alone in his fields on June 8, 1948, Saskatchewan farmer Cecil George Harris accidentally put his tractor into reverse. It rolled backward, pinning his left leg under the rear wheel. His wife didn’t find him until 10:30 that night, and he died at the hospital.

Days later, surveying the scene of the accident, neighbors noticed that Harris had scratched an inscription into the tractor’s fender using his pocketknife:

In case I die in this mess, I leave all to the wife. Cecil Geo Harris.

This is said to have been the most popular problem presented in the American Mathematical Monthly. It was proposed by P.L. Chessin of Westinghouse in the April 1954 issue. Each of the digits in this long division problem has been replaced with an x — except for a single 8 in the quotient. Can you reconstruct the problem?

In long division, whenever two digits are brought down instead of one — as happens twice in this problem — a zero must appear in the quotient. That tells us that the quotient is x080x.

The visible 8 multiplied by the divisor produces a three-digit number. But when the last number in the quotient is multiplied by the divisor, it produces a four-digit number. So the last digit in the quotient must be 9.

The divisor has three digits and produces a three-digit number when multiplied by the visible 8. So the divisor must be less than 125, because 8 × 125 = 1000.

That tells us something further. When the divisor is multiplied by the first digit of the quotient, it produces a three-digit number, and when this is subtracted from the first four digits of the dividend it yields a two-digit difference. That means that the first digit of the quotient must be greater than 7, because 7 × 124 = 868, and that would yield a three-digit difference when subtracted from even the smallest possible four-digit number, 1000 (1000 – 868 = 132). An 8 or a 9 could produce a two-digit difference, but a 9 would produce a four-digit number when multiplied by the divisor. So the first digit of the quotient is 8, and the full quotient is 80809.

Now, 80809 × 123 = 9939507, which has only seven digits, and the dividend has eight. So the divisor is less than 125 and greater than 123, which means it’s 124. That gives us everything we need: