A Tidy Theorem

If an equilateral triangle is inscribed in a circle, then the distance from any point on the circle to the triangle’s farthest vertex is equal to the sum of its distances to the two nearer vertices (above, q = p + r).

(A corollary of Ptolemy’s theorem.)

Three-Cornered Baseball

http://keymancollectibles.com/publications/images/wpe1.gif

In 1944, eager to help the war effort, a group of New York sportswriters arranged a three-way baseball game among the Brooklyn Dodgers, the New York Yankees, and the New York Giants. The teams took turns batting, fielding, and sitting so that each played a total of six innings, under a scheme concocted by Columbia University math professor Paul A. Smith:

three-cornered baseball

Leo Durocher managed the Dodgers, Joe McCarthy the Yankees, and Mel Ott the Giants. The Dodgers showed a strange talent for this sort of play, winning 5-1-0 and shutting out the Giants entirely. The game “drew a gathering of 50,000, which included 500 wounded service heroes,” reported the New York Times, whose Byzantine box score is reproduced here.

“It helped swell New York’s war bond quota by approximately $56,500,000, and it also demonstrated that no matter how much you may strive to complicate things, the Dodgers, who still insist on doing things in their own fashion, will invariably continue to find their way around.”

Another Car Puzzle

http://commons.wikimedia.org/wiki/File:Rumble_Seat_(PSF).png

A puzzle from Oswald Jacoby’s Mathematics for Pleasure (1962):

The MacDonalds are planning a long car journey of 27,000 miles. If they use tires that last 12,000 miles each, how many tires will they need, and how can they make the best use of them?

Click for Answer

Larghissimo

John Cage indicated that his 1987 piece Organ2/ASLSP should be played “as slow as possible,” but he declined to say how slow that is. Because a pipe organ can be rebuilt piecemeal as it plays, in principle there’s no limit to how long a performance can last.

In 1997 a conference of musicians and philosophers decided to take Cage’s instruction seriously and arranged a performance that would last 639 years. Fed by a bellows, a custom-built organ in the St. Burchardi church in Halberstadt, Germany, has been playing the piece since Sept. 5, 2001; it began with a contemplative 17-month pause, then played the first chord (A4-C5-F#5) for two years. Since then it’s got through only 12 changes; the next won’t occur until Sept. 5, 2020.

This will go on for another 620 years, ending on September 5, 2640. By that time someone somewhere will probably be playing it even more slowly.

In a Word

http://commons.wikimedia.org/wiki/File:BitterkaelderSoed.jpg

diffidation
n. a severing of peaceful relations

clarigation
n. a recital of wrongs before declaring war

Sad Magic

sallows tragic square

The magic square at upper left arranges the numbers 3-11 so that each row, column, and long diagonal totals 21.

Lee Sallows found nine tragic words that vary in length from 3 to 11 letters and arranged them into the same square — and he found a unique shape for each word so that every triplet can be assembled into the same 3×7 shape, shown in the border.

Unquote

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“It has been said, not truly, but with a possible approximation to truth, that in 1802 every hereditary monarch was insane.” — Walter Bagehot

Bright Idea

https://www.google.com/patents/US273074

Robert Heath thought we should all wear luminous hats. Confronted with the resounding question Why?, he offered this poetic paragraph:

Among the advantages of the invention are, the facility of seeing and finding the hat, &c., in closets and dark rooms and places, the presentation of a hat, &c., of different shades during day and night, the beautiful appearance of the article when worn at night, and the provision of distinguishing or indicating the localities of those who may wear the hats, &c., whose occupations are dangerous, such as miners, mariners, &c. For persons who are exposed to weather, sea, &c., the head-wear will be suitably waterproofed, so that the self-luminous nature thereof will not be injured by water.

Simple enough. His patent was granted on Feb. 27, 1883.

Portions of Pi

A quickie submitted by John Astolfi to MIT Technology Review’s Puzzle Corner, July/August 2013:

Consider the expansion of π (3.14159 …) in base 2. Does it contain more 0s than 1s, more 1s than 0s, or an equal number of both? Or is it impossible to tell?

Click for Answer

A Rising Profile

http://commons.wikimedia.org/wiki/File:Gustav_Mahler_1909_3.jpg

One characteristic incident of his fearlessness occurred when friends of Mahler recommended the Berlin Royal Opera to engage him just before he had signed to go to Hamburg. The intendant at the German capital, who was said to be anti-Semitic, is reported to have replied, ‘We cannot engage Mahler here, as we do not like the shape of his nose.’ When in 1897 Vienna offered Mahler the directorial and managerial control of its opera, Berlin suddenly awoke to the importance of the artist who was leaving Germany, and made him a proposition financially better than the one from Vienna. Mahler at once signed the contract to go to the banks of the Danube and telegraphed Berlin: ‘Regret that I cannot accept. My nose still the same shape.’

Musical Courier, quoted in Current Literature, July 1911

See Late Acceptance.

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