The Pythagoras Paradox

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Draw a right triangle whose legs a and b each measure 1. Draw d and e to complete a unit square. Clearly d + e = 2.

Now if we cut a “step” into the square as shown, then f + h = 1 and g + i = 1, so the total length of the “staircase” is still 2. Cut still finer steps and j + k + l + m + n + o + p + q is likewise 2.

And so on: The more finely we cut the steps, the more closely their shape approximates that of the original triangle’s diagonal. Yet the total length of the stairstep shape remains 2, the sum of its horizontal and vertical elements. At the limit, then, it would seem that c must measure 2 … but we know that the length of a unit square’s diagonal is the square root of 2. Where is the error?

(Thanks, Alex.)

One Two Three

Each point on a straight line is either red or blue. Show that it’s always possible to find three points of the same color in which one is the midpoint of the other two.

Click for Answer

Gender in Swearing

In An Encyclopedia of Swearing (2006), University of the Witwatersrand linguist Geoffrey Hughes notes that terms of vehement personal abuse seem to attach disproportionately to the male sex:

gender in swearing table

In his analysis, even terms derived from female anatomy are applied to men rather than women (at least in British usage). Terms such as bugger, motherfucker, and sod[omite] understandably derive from sexual role, but why are devil, fucker, moron, and cretin applied generally to men and not women?

“All the indeterminate terms, such as bastard, idiot, and shit, which should logically be ‘bisexual’ in application, are invariably applied only to males,” Hughes writes. (Also, strangely, there seems to be no vehement term of abuse that’s used freely of both sexes.) “However, the historical perspective shows one significant trend, namely that several of the terms, like bitch and sow, were first used of males (or of both sexes) and only later applied exclusively to women.”

Mixed Greens

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Professor Starr Jordan, President of Leland Stanford University, told of a case where nature had juggled with real estate during the San Francisco earthquake. An earthquake crack had passed directly in front of three cottages, and moved the rose-garden from the middle cottage to the furthest one, and the raspberry patch from the near cottage exactly opposite the middle one. History does not relate how the law decided who owned the roses and the raspberries after their rearrangement.

— M.E. David, Professor David: The Life of Sir Edgeworth David, 1937

The Winner’s Curse

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In 1983, Max Bazerman and William Samuelson asked M.B.A. students in 12 Boston University microeconomics classes to estimate the value of each of four commodities: jars containing 800 pennies, 160 nickels, 200 large paper clips each worth 4 cents, and 400 small paper clips each worth 2 cents. Thus each jar had a value of $8.00, though the students didn’t know this. They asked the students to bid on the value of each commodity. The student whose bid came closest to the true value in each auction would win a $2 prize.

The average estimated value of all the commodities was $5.13, $2.87 less than the true value. But the average winning bid was $10.01, resulting in an average loss to the winner of $2.01. The average winning bid produced a loss in more than half of all the auctions.

This is the “winner’s curse”: The winner of an auction tends to be one of those who form the highest estimate of an item’s value — and hence one of those most at risk of overpaying.

“If an individual assumes that his or her bid will win the auction, this piece of data should indicate that the bidder has probably overestimated the value of the commodity in comparison to other competitors,” write Bazerman and Samuelson. “When the correct inference is drawn, the bidder should revise the estimate of the true value of the item downward and lower the bid accordingly. By failing to take this inference into account, the winning bidder risks paying too much for the ‘prize.'”

(Max Bazerman and William Samuelson, “I Won the Auction But Don’t Want the Prize,” Journal of Conflict Resolution 27:4 [December 1983], 618-634.)

Round Numbers

A curiosity attributed to a Professor E. Ducci in the 1930s:

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Arrange four nonnegative integers in a circle, as above. Now construct further “cyclic quadruples” of integers by subtracting consecutive pairs, always subtracting the smaller number from the larger. So the quadruple above would produce 22, 8, 38, 8, then 14, 30, 30, 14, and so on.

Ducci found that eventually four equal numbers will occur.

A proof appears in Ross Honsberger’s Ingenuity in Mathematics (1970).

There and Back Again

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John and Mary drive from Westville to Eastville. John drives the first 40 miles, and Mary drives the rest of the way. That afternoon they return by the same route, with John driving the first leg and Mary driving the last 50 miles. Who drives the farthest, and by what distance?

Click for Answer

Turn, Turn, Turn

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Image: Flickr

The Hoover Dam contains a star map depicting the sky of the Northern Hemisphere as it appeared at the moment that Franklin Roosevelt dedicated the dam. Artist Oskar Hansen imagined that the massive structure might outlive our civilization, and that the map could help future astronomers to calculate the date of its creation. The center star on the map, Alcyone, is the brightest star in the Pleiades, and our sun occupies a position at the center of a flagpole. The whole map traces a complete sidereal revolution of the equinox, a period of 25,694 of our years, and marks the point of the dam’s dedication in that period.

“Man has always sought to express and preserve the magnitude of his exploits in symbols,” Hansen said in 1935. “The written words are symbols arranged so as to preserve in objectified form the thought of man and to record his variant states, both mental and physical. All other arts are similar as to their symbolic significance. They take their place among the category of human endeavor simply as the interpreter of life to itself. They serve as an outer object typifying the inner process. They form the connecting link between the spiritual and the material world. They are the shadows cast by the realities of the soul.”

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Image: Wikimedia Commons

Naming Change

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When Canada introduced its 1-dollar coin in 1987, it became known as the “loonie” for the loon on its back.

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When the Royal Canadian Mint introduced the 2-dollar coin in 1996, Canadians tried hard to find a comparable nickname. Though “toonie” or “twoonie” eventually won out, the list of failed suggestions included “doubloonie,” “doozie,” and the charming “moonie.”

Why moonie? Because the coin depicts the queen “with a bear behind.”

(Thanks, Ethan.)