By A.F. Rockwell. White to mate in two moves.
By A.F. Rockwell. White to mate in two moves.
Artist Pierre Vivant performed a sort of typographical sleight of hand in an Oxfordshire field in 1990. In early summer oilseed rape changes from green to yellow as its flowers open. Vivant cut the words GREEN and YELLOW into the flowering field so that each word bore the color it named. Over the ensuing month, the flowers faded and the field reverted to green while the plants in the areas that Vivant had cut grew and flowered. The end result was the reverse of what you see here: a green field in which the word GREEN is yellow and the word YELLOW is green.
Editorial guidelines from Spicy Detective magazine, 1935:
“The idea is to have a very strong sex element in these stories without anything that might be intrepreted as being vulgar or obscene.”
(From Nicholas Parsons, The Book of Literary Lists, 1987.)
In this week’s episode of the Futility Closet podcast we’ll look at the strange phenomenon of poet doppelgängers — at least five notable poets have been seen by witnesses when their physical bodies were elsewhere. We’ll also share our readers’ research on Cervino, the Matterhorn-climbing pussycat, and puzzle over why a man traveling internationally would not be asked for his passport.
Sources for our feature on poet doppelgängers:
John Oxenford, trans., The Autobiography of Wolfgang von Goethe, 1969.
G. Wilson Knight, Byron and Shakespeare, 2002.
Julian Marshall, The Life & Letters of Mary Wollstonecraft Shelley, 1889.
Jon Stallworthy, Wilfred Owen, 2013.
W.E. Woodward, The Gift of Life, 1947.
Little House of Cats has a photo of Cervino, the (purported) Matterhorn-scaling kitty cat of 1950.
Further data on cat rambles:
BBC News, “Secret Life of the Cat: What Do Our Feline Companions Get Up To?”, June 12, 2013 (accessed March 26, 2015).
National Geographic, “Watch: How Far Do Your Cats Roam?”, Aug. 8, 2014 (accessed March 26, 2015).
This week’s lateral thinking puzzles are from Kyle Hendrickson’s 1998 book Mental Fitness Puzzles.
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Many thanks to Doug Ross for the music in this episode.
Suppose that two teams of equal ability are playing football. If goals are scored at regular intervals, it seems natural to expect that each team will be in the lead for half the playing time. Surprisingly, this isn’t so: If a total of n = 20 goals are scored, then the probability that Team A leads after the first 10 goals and Team B leads after the second 10 goals is only 6 percent, while the probability that one team leads throughout the entire game is about 35 percent. (When the scores are equal, the leading team is considered to be the one that was leading before the last goal.) And the chance that one team leads throughout the second half is 50 percent, no matter how large n is.
Such questions began with a study of ballot problems: In 1887 Joseph Bertrand found that if in an election Candidate P scores p votes and Candidate Q scores q votes, where p > q, then the probability that P leads throughout the voting is (p – q)/(p + q).
But pursuing them has led to “conclusions that play havoc with our intuition,” writes Princeton mathematician William Feller. If Peter and Paul toss a coin 20,000 times, we tend to think that each will lead about half the time. But in fact it is 88 times more probable that Peter leads in all 20,000 trials than that each player leads in 10,000 trials. No matter how long the series of coin tosses runs, the most probable number of changes of lead is zero.
“In short, if a modern educator or psychologist were to describe the long-run case histories of individual coin-tossing games, he would classify the majority of coins as maladjusted,” Feller writes. “If many coins are tossed n times each, a surprisingly large proportion of them will leave one player in the lead almost all the time; and in very few cases will the lead change sides and fluctuate in the manner that is generally expected of a well-behaved coin.”
(Gábor J. Székely, Paradoxes in Probability Theory and Mathematical Statistics, 2001; William Feller, An Introduction to Probability Theory and Its Applications, 1957.)
In an 1810 satire, C.L. Pitt noted that “a novel may be made out of a romance, or a romance out of a novel with the greatest ease, by scratching out a few terms, and inserting others.” The steps below will, “like machinery in factories,” convert a Gothic romance into a sentimental novel:
Where you find: Put: A castle An house A cavern A bower A groan A sigh A giant A father A bloodstained dagger A fan Howling blasts Zephyrs A knight A gentleman without whiskers A lady who is the heroine Need not be changed, being versatile Assassins Telling glances A monk An old steward Skeletons, skulls, etc. Compliments, sentiments etc. A gliding ghost A usurer, or an attorney A witch An old housekeeper A wound A kiss A midnight murder A marriage
“The same table of course answers for transmuting a novel into a romance.”
(From a footnote in Pitt’s The Age: A Poem, Moral, Political, and Metaphysical, With Illustrative Annotations, 1810.)
“I adore war. It is like a big picnic without the objectlessness of a picnic. I’ve never been so well or so happy. No one grumbles at one for being dirty.” So wrote professional soldier and poet Julian Grenfell in October 1914, shortly after arriving at the western front.
The unparalleled horrors of the First World War seemed to call forth untapped reserves of mannerly British sang-froid, a “stoical reticence” that artillery officer P.H. Pilditch traced to training in the public schools: “Everything is toned down. … Nothing is ‘horrible.’ That word is never used in public. Things are ‘darned unpleasant,’ ‘Rather nasty,’ or, if very bad, simply ‘damnable.'”
General James Jack reported, “On my usual afternoon walk today a shrapnel shell scattered a shower of bullets around me in an unpleasant manner.” When Private R.W. Mitchell moved to trenches in Hebuterne in June 1916, he complained of “strafing and a certain dampness.”
This unreality reached its peak in the Field Service Post Card, which soldiers were required to complete to reassure next of kin after a particularly dangerous engagement:
I am quite well.
I have been admitted into hospital (sick) (wounded) (and am going on well) (and hope to be discharged soon).
I am being sent down to base.
I have received your (letter dated ____) (telegram dated ____) (parcel dated ____)
Letter follows at first opportunity.
I have received no letter from you (lately) (for a long time).
A soldier would cross out any text that did not apply, perhaps leaving only the line “I am quite well.” “The implicit optimism of the post card is worth noting,” writes Paul Fussell in The Great War and Modern Memory (1975), “the way it offers no provision for transmitting news like ‘I have lost my left leg’ or ‘I have been admitted into hospital wounded and do not expect to recover.’ Because it provided no way of saying ‘I am going up the line again’ its users have to improvise. Wilfred Owen had an understanding with his mother that when he used a double line to cross out ‘I am being sent down to the base,’ he meant he was at the front again.”
It’s possible to sail in a straight line from Pakistan to Siberia — a carefully plotted great-circle route will thread a line between Madascar and the African mainland, between Tierra del Fuego and Antarctica, and through the Aleutian Islands to arrive at the Kamchatka Peninsula, a total distance of nearly 20,000 miles, about 80 percent of the Earth’s circumference. You can reverse course to get back to Karachi.
n. the action of lying down
n. the manner or posture of lying in bed
Trap a circle inside a square and it can turn happily in its prison — a circle has the same breadth in any orientation.
Perhaps surprisingly, circles are not the only shapes with this property. The Reuleaux triangle has the same width in any orientation, so it can perform the same trick:
In fact any square can accommodate a whole range of “curves of constant width,” all of which have the same perimeter (πd, like the circle). Some of these are surprisingly familiar: The heptagonal British 20p and 50p coins and the 11-sided Canadian dollar coin have constant widths so that vending machines can recognize them. What other applications are possible? In the June 2014 issue of the Mathematical Intelligencer, Monash University mathematician Burkard Polster notes that a curve of constant width can produce a bit that drills square holes:
… and a unicycle with bewitching wheels:
The self-accommodating nature of such shapes permits them to take part in fascinating “dances,” such as this one among seven triangles:
This inspired Kenichi Miura to propose a water wheel whose buckets are Reuleaux triangles. As the wheel turns, each pair of adjacent buckets touch at a single point, so that no water is lost:
Here’s an immediately practical application: Retired Chinese military officer Guan Baihua has designed a bicycle with non-circular wheels of constant width — the rider’s weight rests on top of the wheels and the suspension accommodates the shifting axles:
(Burkard Polster, “Kenichi Miura’s Water Wheel, or the Dance of the Shapes of Constant Width,” Mathematical Intelligencer, June 2014.)