Second Strike

The Paradox of the Court is a logic problem from ancient Greece. Protogoras took on a pupil, Euathlus, on the understanding that Euathlus would pay him after he won his first court case. After Protogoras taught him the law, Euathlus decided not to practice, and Protogoras sued him for the amount owed.

Protagoras argued that if he won this lawsuit, he’d be paid the money he was owed, and if Euathlus won the suit, then he’d have won his first case and would owe Protagoras the money anyway under the terms of their contract. So he ought to be paid either way.

Euathlus argued that if he won the suit then by the court’s decision he owed nothing, and if he lost the suit then he still would not have won his first case, and thus owed Protagoras nothing under the contract.

One lawyer suggested that the court should decide in favor of the student and declare that he doesn’t have to pay for his education. Then Protagoras should sue him a second time — since then, incontrovertibly, the student will have won his first case!

| Science & Math

Lodging a Complaint

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

Among the most compelling anecdotes suggesting that dolphins have concepts of ‘wrong’ behavior is Thomas White’s description of how a human snorkeler observing Atlantic spotted dolphins off the Bahamas went outside the bounds of the norms of behavior expected by the dolphins of human observers at that site. The swimmer approached a calf engaged in learning to fish with its mother, a no-no in the rules of engagement between swimmers and these dolphins built up over years. When this happened, the mother then swam not to the hapless trespasser but to the leader of the group of swimmers, whom she could identify, and tail-slapped, her displeasure apparently directed at the leader who had not controlled the behavior of those being led.

— Hal Whitehead and Luke Rendell, The Cultural Lives of Whales and Dolphins, 2015

| Science & Math · Society

A One-Sided Score

https://ztfnews.wordpress.com/2013/08/10/moebius-strip-tease/

Conductor and musical lexicographer Nicolas Slonimsky composed a “Möbius Strip Tease” in 1965, while he was teaching at UCLA. The text reads:

Ach! Professor Möbius, glörious Möbius
Ach, we love your topological,
And, ach, so logical strip!
One-sided inside and two-sided outside!
Ach! euphörius, glörius Möbius Strip-Tease!

Slonimsky described the piece as “a unilateral perpetual rondo in a linearly dodecaphonic vertically consonant counterpoint.” The instructions on the score read: “Copy the music for each performer on a strip of 110-b card stock, 68″ by 6″. Give the strip a half twist to turn it into a Möbius strip.” In performance the endless score rotates perpetually around each musician’s head. (That’s Slonimsky above, trying it out with John Cage.)

The score is here if you’d like to try it yourself. Be careful.

| Art · Science & Math

Just Checking

https://www.flickr.com/photos/nicubunuphotos/5262645427/in/photolist-923r8Z-dPcqJ1-4ez6tk-2kqF3P7-ao1rbe-yGrd-khVgg-FV3As-LJeX5q-bDLLB9-dy8fWu-dNuFWh-25FHiqb-fDvS6E-6YwWga-5cXuUv-fHhNjS-ad3hRR-bUDThS-bDBFcu-38uU9o-6nFxS9-2kqzhS8-2kquYCX-58PD49-2mfHYoz-b9fsev-7dCDZp-aATSWL-HHG2pH-7WRGL3-4jbFC8-8hLWUQ-KFWSLy-7WJn9g-4jfJxf-2fDZt2e-ifV2aH-2mmekFq-87NNJY-2m4RoVr-8iwAjQ-24ZaVhr-ej26bZ-GauLXW-FFDY8-fDegSr-FFEex-6hTQfw-DTTfYK
Image: Flickr

South African statistician J.E. Kerrich’s 1946 textbook An Experimental Introduction to the Theory of Probability has an odd origin: Kerrich happened to be visiting Denmark during the Nazi invasion of 1940, and the Danes agreed to intern him, along with other British citizens, to prevent their being taken to Germany. While in confinement he tossed a coin 10,000 times and recorded the results, and he wrote up his analysis afterward in the book.

For the record, it landed heads 5,067 times.

| Science & Math

The Blythe Intaglios

https://commons.wikimedia.org/wiki/File:Blythe_Intaglios_-_Human_Figure_1.jpg
Image: Wikimedia Commons

Flying between Las Vegas and Blythe, Calif., in 1932, pilot George Palmer looked down and got a surprise — a group of enormous figures had been carved into the surface of the Colorado Desert. They had lain there for a thousand years, but they’re so large that no one had noticed them before. (The largest human figure is more than 50 meters long.)

No one knows for certain who created them; altogether there are several dozen figures, most probably representing mythic characters from Yuman cosmology. What else have we been overlooking?

| Science & Math

The Trust Game

https://commons.wikimedia.org/wiki/File:US10dollarbill-Series_2004A.jpg

University of Iowa economist Joyce Berg devised this test of social expectation. Two players are each given $10. The two are anonymous to one another and may not communicate. The first player, known as the trustor, is given the option to transfer any part of her $10 to the second player, who is known as the trustee. Whatever she sends will be tripled; if she sends $5 the trustee will receive $15. The trustee then has the option to return any portion of what she’s received. The game is played only once, so the two players have no opportunity to communicate through repeated play.

What should they do? If the two trust one another perfectly, then both stand to double their money — the trustor will give all $10 to the trustee, who now has $40. If she returns half of that, then each player has $20.

The trouble is that rational players, who seek to maximize their personal gains, won’t behave this way. If the trustor gives the trustee $10, she can just keep all of it, walking away with $40 and leaving the trustor with nothing. Realizing this, the trustor should send nothing at all, keeping at least the $10 she was given. This is the rational expectation.

But in actual experiment, Berg found that fully 30 of 32 trustors sent money, and they sent an average of $5.16. This is surprising. “From a rational choice perspective,” she wrote, “subjects who sent money must have believed their expected return would be positive; but given the noncooperative prediction, why would they believe this?”

(Joyce Berg, John Dickhaut, and Kevin McCabe, “Trust, Reciprocity, and Social History,” Games and Economic Behavior 10 [1995], 122-142.)

| Science & Math · Society

Extra Credit

https://commons.wikimedia.org/wiki/File:Bogdanov-Belsky_Ustny_Schet_(Tretyakov).jpg

The boys in Nikolay Bogdanov-Belsky’s 1895 painting Mental Arithmetic are having a difficult time solving the problem on the board:

\displaystyle \frac{10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}}{365}

As it happens, there’s a simple solution: Both (102 + 112 + 122) and (132 + 142) are equal to 365, so the answer is simply (365 + 365) / 365, or 2. They’ll figure it out.

| Art · Science & Math

A Box Code

https://www.gutenberg.org/files/13180/13180-h/13180-h.htm

In Robert Chambers’ 1906 novel The Tracer of Lost Persons, Mr. Keen copies the figure above from a mysterious photograph. He is trying to help Captain Harren find a young woman with whom he has become obsessed.

“It’s the strangest cipher I ever encountered,” he says at length. “The strangest I ever heard of. I have seen hundreds of ciphers — hundreds — secret codes of the State Department, secret military codes, elaborate Oriental ciphers, symbols used in commercial transactions, symbols used by criminals and every species of malefactor. And every one of them can be solved with time and patience and a little knowledge of the subject. But this … this is too simple.”

The message reveals the name of the young woman whom Captain Harren has been seeking. What is it?

Click for Answer
| Literature · Science & Math

Hall’s Marriage Theorem

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Suppose we have a group of n men and n women. Each of the women can find some subset of the men whom she would be happy to marry. And each of the men would be happy with any woman who will have him. Is it always possible to pair everyone off into happy marriages?

Clearly this won’t work if, for example, two of the women have their hearts set on the same man and won’t be happy with anyone else. In general, for any subset of the women, we need to be sure that they can reconcile their preferences so that each of them finds a mate.

Surprisingly, though, that’s all that’s required. So long as every subset of women can collectively express interest in a group of men at least as numerous as their own, it will always be possible to marry off the whole group into happy couples.

The theorem was proved by English mathematician Philip Hall in 1935. Another application of the same principle: Shuffle an ordinary deck of 52 playing cards and deal it into 13 piles of 4 cards each. Now it’s always possible to assemble a run of 13 cards, ace through king, by drawing one card from each pile.

| Science & Math