Nontransitive Dice

Label the faces of a fair set of dice with these numbers:

Die A: 3, 3, 3, 3, 3, 6
Die B: 2, 2, 2, 5, 5, 5
Die C: 1, 4, 4, 4, 4, 4

This gives them a curious property. In the long run Die A will tend to beat Die B, Die B will tend to beat Die C, and Die C will tend to beat Die A. The three dice form a ring in which each die beats its successor. No matter which die our opponent chooses, we can select another that is likely to beat it.

Business magnate Warren Buffet once challenged Bill Gates to such a game using four nontransitive dice. “Buffett suggested that each of them choose one of the dice, then discard the other two,” wrote Janet Lowe in her 1998 book Bill Gates Speaks. “They would bet on who would roll the highest number most often. Buffett offered to let Gates pick his die first. This suggestion instantly aroused Gates’s curiosity. He asked to examine the dice.”

“It wasn’t immediately evident that because of the clever selection of numbers for the dice, they were nontransitive,” Gates said. “Assuming the dice were rerolled, each of the four dice could be beaten by one of the others.” He invited Buffett to choose first.

| Science & Math

Rest Easy

https://www.google.com/patents/US1596

Henry Bourne patented this combination mattress and life preserver in 1840. Essentially it’s an ordinary berth mattress split in two; each half is filled with broken cork and waterproofed, and then the two are reattached and fitted with leggings, an oar, and two shoulder straps.

If your ship is sinking, you leap out of bed, stuff the mattress with “papers, moneys, clothes, and provisions for many days,” pull the straps over your shoulders, and jump overboard. When you’re in the water the buoyant mattress keeps you afloat, the waterproofing keeps your valuables dry, and you can navigate using the oar. Thanks to the leggings, when you reach the shore you can waddle up the beach “beyond the reach of the returning wave.”

| Technology

Special Projects

Facilities suggested by Lewis Carroll for a school of mathematics at Oxford, 1868:

  1. A very large room for calculating Greatest Common Measure. To this a small one might be attached for Least Common Multiple: this, however, might be dispensed with.
  2. A piece of open ground for keeping Roots and practising their extraction: it would be advisable to keep Square Roots by themselves, as their corners are apt to damage others.
  3. A room for reducing Fractions to their Lowest Terms. This should be provided with a cellar for keeping the Lowest Terms when found, which might also be available to the general body of Undergraduates, for the purpose of “keeping Terms.”
  4. A large room, which might be darkened, and fitted up with a magic lantern for the purpose of exhibiting Circulating Decimals in the act of circulation. This might also contain cupboards, fitted with glass-doors, for keeping the various Scales of Notation.
  5. A narrow strip of ground, railed off and carefully levelled, for investigating the properties of Asymptotes, and testing practically whether Parallel Lines meet or not: for this purpose it should reach, to use the expressive language of Euclid, “ever so far.”

He introduced this topic with an administrator by writing, “Dear Senior Censor,–In a desultory conversation on a point connected with the dinner at our high table, you incidentally remarked to me that lobster-sauce, ‘though a necessary adjunct to turbot, was not entirely wholesome.’ It is entirely unwholesome. I never ask for it without reluctance: I never take a second spoonful without a feeling of apprehension on the subject of possible nightmare. This naturally brings me to the subject of Mathematics …”

| Humor · Science & Math

Round and Round

Since demolishing 78 traffic signals and installing 80 roundabouts, the northern Indiana city of Carmel has reduced the number of accidents by 40 percent and the number of accidents with injuries by 78 percent.

“It’s nearly impossible to have a head-on or T-bone collision when using the roadways, and collisions that do happen tend to occur at much lower speeds,” noted Governing magazine. “Other benefits of roundabouts include reduced fuel consumption, due to a lack of idling, and a construction cost that is at least $150,000 less than installing traffic lights.”

“We have more than any other city in the U.S.,” says mayor James Brainard. “It’s a trend now in the United States. There are more and more roundabouts being built every day because of the expense saved and, more importantly, the safety.”

| Science & Math · Society

Shadow Play

Rensselaer Polytechnic Institute art professor Larry Kagan “hides” two-dimensional images in seemingly chaotic three-dimensional sculptures. The images are revealed when light is applied from the right angle.

“The shadows are a condensation of something that exists in more dimensions,” he says. “Behind them, there can be an awful lot going on.”

A few more playful sculptures from Japanese artist Shigeo Fukuda:

| Art

The Solway Firth Spaceman

http://en.wikipedia.org/wiki/File:SolwayfirthSpaceman.jpg

On May 23, 1964, Cumberland firefighter Jim Templeton was visiting Burgh Marsh in Cumbria, England, when he snapped three photos of his 5-year-old daughter, Elizabeth. When the pictures were developed, he was surprised to see what looked like a spaceman in the background of one of them.

Templeton told reporters, “I took the picture to the police in Carlisle, who, after many doubts, examined it and stated there was nothing suspicious about it. The local newspaper, the Cumberland News, picked up the story, and within hours it was all over the world. The picture is certainly not a fake, and I am as bemused as anyone else as to how this figure appeared in the background. Over the four decades the photo has been in the public domain, I have had many thousands of letters from all over the world with various ideas or possibilities — most of which make little sense to me.”

The best guess seems to be that the figure is Templeton’s wife, Annie, who had dark bobbed hair and was wearing a pale blue dress that appeared white in other photos taken that day. The camera’s viewfinder obscured part of the image area, so it would have been possible for him to take the photo without realizing that Annie was in the shot. But who knows?

| Oddities

A Perpetual Pussycat

jock v at chartwell

When Winston Churchill died in 1965, his family made a curious request: His country home Chartwell must always maintain “in comfortable residence” a marmalade cat named Jock.

The original Jock had been given to Churchill two years before his death by his private secretary, Sir John “Jock” Colville, and quickly became a favorite. When the cat died in 1975, 10 years after the prime minister, he was replaced with a Jock II, and the line has continued.

The current resident is Jock VI, who is “not at all fond of heights, lightning or opera.”

| Oddities

A Timely Escape

While lying on his deathbed in 1816, Welsh judge George Hardinge received a bill from his London stationers, Tripeaux and Co. It was addressed to “Mr. Justice Hardinge, if living; or his executors, if dead.” He wrote back:

Messrs. Tripeaux, what is fear’d by you,
Alas! the melancholy circumstance is true,
That I am dead; and more afflicting still,
My legal assets cannot pay your bill.
To think of this, I am almost broken hearted,
Insolvent I, this earthly life departed;
Dear Messrs. T., I am yours without a farthing,
For executors and self,
George Hardinge.

He died three hours later.

| Death

Bear Facts

blackmore bear

The Veterinary Record of April 1, 1972, contained a curious article: “Some Observations on the Diseases of Brunus edwardii.” Veterinarian D.K. Blackmore and his colleagues examined 1,598 specimens of this species, which they said is “commonly kept in homes in the United Kingdom and other countries in Europe and North America.”

“Commonly-found syndromes included coagulation and clumping of stuffing, resulting in conditions similar to those described as bumble foot and ventral (rupture in the pig and cow respectively) alopecia, and ocular conditions which varied from mild squint to intermittent nystagmus and luxation of the eyeball. Micropthalmus and macropthalmus were frequently recorded in animals which had received unsuitable ocular prostheses.”

They found that diseases could be either traumatic or emotional. Acute traumatic conditions were characterized by loss of appendages, often the result of disputed ownership, and emotional disturbances seemed to be related to neglect. “Few adults (except perhaps the present authors) have any real affection for the species,” and as children mature, they tend to relegate these animals to an attic or cupboard, “where severe emotional disturbances develop.”

The authors urged their colleagues to take a greater interest in the clinical problems of the species. “It is hoped that this contribution will make the profession aware of its responsibilities, and it is suggested that veterinary students be given appropriate instruction and that postgraduate courses be established without delay.”

| Humor · Science & Math