Dueling Pennies

A certain strange casino offers only one game. The casino posts a positive integer n on the wall, and the customer flips a fair coin repeatedly until it falls tails. If he has tossed n – 1 times, he pays the house 8n – 1 dollars; if he’s tossed n + 1 times, the house pays him 8n dollars; and in all other cases the payoff is zero.

The probability of tossing the coin exactly n times is 1/2n, so the customer’s expected winnings are 8n/2n + 1 – 8n – 1/2n – 1 = 4n – 1 for n > 1, and 2 for n = 1. So his expected gain is positive.

But suppose it turns out that the casino arrived at the number n by tossing the same fair coin and counting the tosses, up to and including the first tails. This presents a puzzle: “You and the house are behaving in a completely symmetric manner,” writes David Gale in Tracking the Automatic ANT (1998). “Each of you tosses the coin, and if the number of tosses happens to be the consecutive integers n and n + 1, then the n-tosser pays the (n + 1)-tosser 8n dollars. But we have just seen that the game is to your advantage as measured by expectation no matter what number the house announces. How can there be this asymmetry in a completely symmetric game?”

| Science & Math

Unquote

https://commons.wikimedia.org/wiki/File:Emmanuel_Benner_-_Prehistoric_Man_Hunting_Bears.jpg

“What leapings of the heart must there not have been throughout that long warfare! What moments of terror and triumph! What acts of devotion and desperate wonders of courage!” — H.G. Wells, of prehistoric man

| Quotations

Sizing Up

https://commons.wikimedia.org/wiki/File:Star_polygon_3-1.svg

Two lines divide this equilateral triangle into four sections. The shaded sections have the same area. What is the measure of the obtuse angle between the lines?

Click for Answer
| Puzzles

Good Boy

In 1794 Haydn visited the singer Venanzio Rauzzini at Bath. In the garden of Rauzzini’s villa he noticed a monument to a much-loved dog named Turk, with the inscription TURK WAS A FAITHFUL DOG AND NOT A MAN. As a tribute he turned the text into a four-part canon:

http://books.google.com/books?id=88IJAQAAMAAJ&pg=PA6&lpg=PA6

Rauzzini was so pleased that he had the music added to Turk’s memorial stone.

| Art

“Curious Sculpture”

http://books.google.com/books?id=FNA6AQAAMAAJ

A letter from W.C. Trevelyan to John Adamson, secretary of the Antiquarian Society of Newcastle, Jan. 20, 1825:

In the autumn of 1823, I visited the interesting Church at Bridlington [Yorkshire] (founded about 1114, by Gilbert de Gant). On examining a tomb stone with an inscription and date of 1587, standing on two low pillars of masonry near the font, I found some appearances of sculpture on the under side of it, and having obtained leave to turn it over, the curious sculpture represented in the etching herewith sent, was discovered.

Its meaning, or date, I cannot attempt to explain. Can it have any reference to the building of the church? You will perceive both the circular and pointed arch (though the latter is probably only accidental, the space being limited).

The roof, I think, resembles some of the Roman buildings of the lower empire of which I have seen engravings.

The tiles, in shape, correspond exactly with those which were found among the remains of a Roman villa discovered a few years since at Stonesfield, near Oxford. The upper figures are very like some on Bridekirk Font (of the 10th century).

The figures of the Fox and Dove remind one of Æsop’s fable of the Fox and the Stork.

The society published the plate in its Archæologia Æliana. The best guess I can find is that it’s a 12th-century coffin lid that had been appropriated as a tombstone in 1587. But the meaning of the figures is unclear.

| Oddities

Bartitsu

https://commons.wikimedia.org/wiki/File:Copy_of_Montage.jpg

When engineer E.W. Barton-Wright returned to England after three years in Japan, he brought with him a new discipline: Bartitsu, a martial art of his own devising that combined jujutsu, judo, boxing, and stick fighting. He listed its essential principles in an article in Pearson’s Magazine in March 1899:

1. To disturb the equilibrium of your assailant.
2. To surprise him before he has time to regain his balance and use his strength.
3. If necessary to subject the joints of any part of his body, whether neck, shoulder, elbow, wrist, back, knee, ankle, etc. to strain which they are anatomically and mechanically unable to resist.

The combination of systems, he wrote, “can be very terrible in the hands of a quick and confident exponent. One of its greatest advantages is that the exponent need not necessarily be a strong man, or in training, or even a specially active man in order to paralyse a very formidable opponent, and it is equally applicable to a man who attacks you with a knife, or a stick, or against a boxer; in fact, it can be considered a class of self-defence designed to meet every possible kind of attack, whether armed or otherwise.”

In 1899 Barton-Wright established an academy in London to promote the new art, but he proved an indifferent promoter and the school closed its doors within three years. His eccentric fighting style might have been forgotten entirely but for one immortal mention: In The Adventure of the Empty House, when asked how he defeated Professor Moriarty in their climactic struggle at the Reichenbach Falls, Sherlock Holmes credits “baritsu, or the Japanese system of wrestling, which has more than once been very useful to me.”

| Oddities

Double Entendre

The Exeter Book, an anthology of Anglo-Saxon poetry from the 10th century, contains three riddles that seem shockingly risqué until you see the answers:

I’m a strange creature, for I satisfy women,
a service to the neighbors! No one suffers
at my hands except for my slayer.
I grow very tall, erect in a bed,
I’m hairy underneath. From time to time
a good-looking girl, the doughty daughter
of some churl dares to hold me,
grips my russet skin, robs me of my head
and puts me in the pantry. At once that girl
with plaited hair who has confined me
remembers our meeting. Her eye moistens.

(An onion.)

A strange thing hangs by a man’s thigh,
hidden by a garment. It has a hole
in its head. It is stiff and strong
and its firm bearing reaps a reward.
When the man hitches his clothing high
above his knee, he wants the head
of that hanging thing to poke the old hole
(of fitting length) it has often filled before.

(A key.)

A young man made for the corner where he knew
she was standing; this strapping youth
had come some way — with his own hands
he whipped up her dress, and under her girdle
(as she stood there) thrust something stiff,
worked his will; they both shook.
This fellow quickened: one moment he was
forceful, a first-rate servant, so strenuous
that the next he was knocked up, quite
blown by his exertion. Beneath the girdle
a thing began to grow that upstanding men
often think of, tenderly, and acquire.

(A churn.)

| Literature

Visual Calculus

https://commons.wikimedia.org/wiki/File:Cycloid_(PSF).png

As a circle rolls along a line, a point on its circumference traces an arch called a cycloid. The arch encloses an area three times that of the circle, a result commonly proven using calculus. Now Armenian mathematician Mamikon Mnatsakanian has devised a “sweeping-tangent theorem” that accomplishes the same proof using intuition:

https://commons.wikimedia.org/wiki/File:Mamikon_Cycloid.svg
Image: Wikimedia Commons

Imagine a tangent to the rolling circle. As the circle rolls, the tangent sweeps out a series of vectors (approximated here using colors). If these vectors are then gathered to a common point while preserving their length and orientation, they form a sort of bouquet whose size and shape turn out to match exactly those of the original circle. Because the enclosing rectangle has four times the area of the rolling circle (2πr × 2r = 4πr2), this shows that the area under the arch has three times the circle’s area.

All this is proven rigorously in Mnatsakanian’s 2012 book New Horizons in Geometry, written with his Caltech colleague Tom Apostol. The two have now collaborated on some 30 papers showing that many surprising and useful results that heretofore had required integration can now be obtained using intuitive methods that can appeal even to a young student.

That’s a welcome outcome for Mnatsakanian, who found himself stranded in the United States when the Armenian government collapsed in 1990. Apostol writes, “When young Mamikon showed his method to Soviet mathematicians they dismissed it out of hand and said ‘It can’t be right. You can’t solve calculus problems that easily.'”

| Science & Math