The Paradox of Goals

https://commons.wikimedia.org/wiki/File:U20-WorldCup2007-Okotie-Onka_edit2.jpg
Image: Wikimedia Commons

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.)

| Science & Math

New Tropes for Old

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

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.)

| Literature

Chin Up

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“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).

(Signature only)

(Date)

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.”

(Thanks, Garrett.)

| History · Society

The Long Way

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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.

(Thanks, Derek.)

| Trivia

Curves of Constant Width

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:

http://commons.wikimedia.org/wiki/File:Rouleaux_triangle_Animation.gif
Image: Wikimedia Commons

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.)

| Science & Math

Rain, Rain

https://www.google.com/patents/US273115

Torontonian John Maguire wasn’t satisfied with the standard raincoat in 1883, so he added a gutter:

The object of the invention is to provide a water-proof coat which can be worn in rainy weather without the wearer’s leg being made wet from water dripping off the skirt of the coat; and it consists of a water-proof coat having the bottom edge of its skirt turned up, forming a trough or channel to receive the water flowing on the surface of the coat, suitable provision being made to carry off the water away from the legs of the wearer of the garment.

“Although the coat is specially designed for gentlemen’s use, it will of course be understood that ladies’ coats may be similarly made.”

| Technology

Duty Calls

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In May 1905 British MP Sir Gilbert Parker insisted that he had seen the astral body of Sir Crane Rasch in the House of Commons while Rasch was ill at home.

Sir Arthur Hayter supported him: “I beg to say that I not only saw Sir Carne Rasch myself sitting below the gangway but I called him to the attention of Sir Henry Campbell-Bannerman, with whom I was talking on the front opposition bench, saying I wondered why all the papers had inserted notices of Sir Carne’s illness while he was sitting opposite, apparently quite well. Sir Henry replied that he hoped his illness was not catching.”

Rasch declared later that he had never left his room.

“It seems that this is not the first instance of the sort that has occurred in the House,” noted the New York Sun. “In 1897 Mr. O’Connor, an Irish member, went to Ireland to be present at the deathbed of one of his parents. Swift McNeill saw his wraith in his usual seat on the third opposition bench. It was also seen from the press gallery.”

| Oddities

Unfolding Hopes

Albert Szent-Györgyi, who knew a lot about maps
according to which life is on its way somewhere or other,
told us this story from the war
due to which history is on its way somewhere or other:

The young lieutenant of a small Hungarian detachment in the Alps
sent a reconnaissance unit out into the icy wasteland.
It began to snow
immediately, snowed for two days and the unit
did not return. The lieutenant suffered: he had dispatched
his own people to death.

But the third day the unit came back.
Where had they been? How had they made their way?
Yes, they said, we considered ourselves
lost and waited for the end. And then one of us
found a map in his pocket. That calmed us down.
We pitched camp, lasted out the snowstorm and then with the map
we discovered our bearings.
And here we are.

The lieutenant borrowed this remarkable map
and had a good look at it. It was not a map of the Alps
but of the Pyrenees.

Goodbye now.

— From Miroslav Holub, Notes of a Clay Pigeon, reprinted in G.Y. Craig and E.J. Jones, A Geological Miscellany, 1982.

| Poems

Animal Spirits

Football fans found an unlikely oracle during the 2008 European championship: an octopus named Paul. Before each match his keepers at the Sea Life Centre in Oberhausen, Germany, would lower two boxes of food into his tank, each bearing the flag of an upcoming competitor. Surprisingly, Paul correctly chose the winner in four of Germany’s six games.

When some observers expressed skepticism, Paul went on to pick the winners of all seven of Germany’s World Cup games in 2010, as well as the final between Spain and the Netherlands, giving him an overall success rate of 85 percent.

Competitors sprang up around the world, including a Singaporean parakeet, a German parrot, and a saltwater crocodile named Dirty Harry, who predicted the result of Australia’s general election by snatching a chicken carcass dangling beneath a caricature of Prime Minister Julia Gillard. Maybe we should quit while we’re ahead.

(Thanks, Lauren.)

| Oddities