Apportionment Paradoxes

Until 1911, the U.S. House of Representatives grew along with the country. Accordingly, when the 1880 census showed an increase in population, C.W. Seaton, chief clerk of the census office, worked out apportionments for all House sizes between 275 and 350, in order to see which states would get the new seats.

He was in for a surprise. The method was straightforward: Take the total U.S. population and divide it by the proposed number of seats in the House, rounding all fractions down. This would dispose of most of the seats; any leftover seats would be awarded to the states whose fractional remainders had been highest. But Seaton discovered an oddity:

alabama paradox

If the House had 299 seats, Alabama would get 8 representatives (because its remainder, .646, was higher than that of Texas or Illinois). But if the House had 300 seats it would get only 7 (the extra representative would now go to Illinois, whose remainder had surpassed Alabama’s). The problem is that the “fair share” of a large state increases more quickly than that of a small state.

Seaton called this the Alabama paradox. A related problem is the population paradox: If the method above had been used in 1901 to reallocate 386 seats in the House, Virginia would have lost a seat to Maine even though the ratio of their populations had increased from 2.67 to 2.68:

population paradox

Here, even though the size of the House has not changed, a fast-growing state receives fewer representatives than a slow-growing one.

In 1982 mathematicians Michel Balinski and Peyton Young showed that if each party gets one of the two numbers closest to its fair share of seats, then any system of apportionment will run into one of these paradoxes. The solution, it seems clear, is to start cutting legislators into pieces.

(These data are from Hannu Nurmi’s Voting Paradoxes and How to Deal With Them, 1999. Balinski and Young’s book is Fair Representation: Meeting the Ideal of One Man, One Vote.)

Posture Guard

This “scholar’s shoulder brace,” patented by Isidor Keller in 1884, is advertised as “a brace for supporting the shoulders in writing”:

In using my shoulder-brace, I propose to secure the bracket A on a school-desk, as shown in Figs. 1 and 3, then I adjust the standard B to suit the scholar occupying the seat in front of said desk, and finally I pass the loops f f of the shoulder-straps over the shoulders of the scholar, and adjust said loops so as to retain the scholar in a position that will not be injurious to the health or to the eyes.

What if there’s a fire?

Derbes’ Law

Most of our pleasures come from filling or emptying cavities, and vice versa.

Contributed by Dr. Vincent J. Derbes of New Orleans to More of Mould’s Medical Anecdotes, 1989.

Time Series,_1973_(Today_Series,_%22Tuesday%22)_On_Kawara.JPG

Since 1966, Japanese artist On Kawara has been producing paintings that depict only the date of their creation, executed in liquitex on canvas in eight standard sizes. If he can’t complete a painting on the day he starts it, he destroys it. Each entry in the series is painted carefully by hand in the language of the country in which he produces it; to date he’s completed more than 2,000 paintings in 112 countries, and he says he’ll continue until he dies.

Is this valuable? Yes: In 2006 Christie’s sold Nov. 8, 1989 for £310,000.


“There is an astonishing imagination, even in the science of mathematics. An inventor must begin with painting correctly in his mind the figure, the machine invented by him, and its properties or effects. We repeat there was far more imagination in the head of Archimedes than in that of Homer.” — Voltaire

Brood War

A newlywed couple are planning their family. They’d like to have four children, a mix of girls and boys. Which is more likely: (1) two girls and two boys or (2) three children of one sex and one of the other? (Assume that each birth has an equal chance of being a boy or a girl.)

Click for Answer

The Christmas Truce

As Christmas approached in 1914, a number of impromptu cease-fires broke out on the Western Front in which German and British troops exchanged greetings, song, and even food. Rifleman Oswald Tilley of the London Rifle Brigade wrote to his parents on Dec. 27 regarding an incident near Ploegsteert, just north of the Franco-Belgian border:

On Christmas morning as we had practically ceased to fire at them, one of them started beckoning to us so one of our Tommies went out in front of our trenches and met him halfway amidst cheering. After a bit a few of our chaps went out to meet theirs until literally hundreds of each side were out in No Man’s Land shaking hands and exchanging cigarettes, chocolate and tobacco etc. … Just you think that while you were eating your turkey etc. I was out talking and shaking hands with the very men I had been trying to kill a few hours before. It was astonishing!

In subsequent years the authorities tried to discourage such truces. Apart from reproving the breakdown in discipline, they had trouble getting the war started again. In late 1915 Ethel Cooper, an Australian woman living in Germany, met a soldier home on leave from the XIX Saxon Corps who told her that his unit had fraternized extensively with a British battalion for two days beginning that Christmas Eve. She wrote, “The trouble began on the 26th, when the order to fire was given, for the men struck. Herr Lange says that in the accumulated years he had never heard such language as the officers indulged in, while they stormed up and down, and got, as the only result, the answer: ‘We can’t — they are good fellows, and we can’t.’ Finally, the officers turned on the men, ‘Fire, or we do — and not at the enemy.’ Not a shot had come from the other side, but at last they fired, and an answering fire came back, but not a man fell. ‘We spent that day and the next day,’ said Herr Lange, ‘wasting ammunition in trying to shoot the stars down from the sky.'”

(From Marc Ferro et al., Meetings in No Man’s Land, 2007)


In 1981, when science journalist Marcus Chown was an undergraduate physics student, his mother watched a profile of Richard Feynman on the BBC series Horizon. She had never shown an interest in science before, and he wanted to encourage her, so when he advanced to Caltech to study astrophysics, he told Feynman of his mother’s interest and asked him to send her a birthday note. She received this:

Happy Birthday Mrs. Chown!

Tell your son to stop trying to fill your head with science — for to fill your heart with love is enough!

Richard P. Feynman (the man you watched on BBC “Horizons”)

The Centipede Game

Before you are two piles of coins. One contains 4 coins and the other contains 1. If you like, you can keep the larger pile, give me the smaller, and end the game. Or you can pass both piles to me. In that case the size of each pile doubles and I’m given the same option — I can keep the larger pile and give you the smaller one, or I can pass both piles back to you, in which case they’ll double again.

We both know that the game will end after six rounds. At that point I’ll have the coins and will win 128 coins to your 32. You’d be better off stopping the game in round 5, when you’ll have 64 coins and I have 16. But, by similar reasoning, I’d prefer round 4 to round 5, and you’d prefer round 3 to round 4 … if we rely on each other to be purely rational, it seems your best opening move is to end the game at once and keep 4 coins. This is less than you’d make in round 6, but it appears that purely rational play will never reach that round.

In practice, interestingly, human beings don’t do this — almost no one stops at the first opportunity, even after several repetitions of the game. Why they do so is not clear — possibly they’re hoping that their opponent has not reasoned through the whole game, or perhaps they’re agreeing tacitly to cultivate the pot in hopes of being the first one to cash out abruptly; perhaps the satisfaction of anticipating such a victory makes the risk worthwhile.

In lab tests in 2009, economists Ignacio Palacios-Huerta and Oscar Volij found that only 3 percent of games between students ended in the first round, but 69 percent of games between chess players did so. This rose to 100 percent when the first player was a grandmaster. They conclude that the most important factor is common knowledge of the players’ rationality, rather than altruism or social preferences.

(Ignacio Palacios-Huerta and Oscar Volij, “Field Centipedes,” American Economic Review 99 (4): 1619–1635.)

Party Planning

An Englishman buys a horse and hires porters to take the horse up to his apartment on the fourth floor. The porters exert themselves and sweat. Finally they succeed in getting the horse to his apartment.

He asks them to put the horse in the bathtub.

After they finish the job, one of the porters asks him, “Why do you need a horse in the bathtub?”

The Englishman says, “Well, tomorrow evening I’m having a party at home. One of the guests will go into the bathroom, see the horse, come to me and say, ‘You know you have a horse in your bathtub.’ And I’ll tell him, ‘So what?'”

— Sion Rubi, Intelligent Jokes, 2004