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.

The Rod of Asclepius, left, with a single snake, is the symbol of medicine. Unfortunately, a large number of commercial American medical organizations instead use the caduceus, right, which has two snakes. Asclepius was the Greek god of healing, but the caduceus was wielded by Hermes and connotes commerce, negotiation, and trickery.

The confusion began when the American military began using the caduceus in the late 19th century, and it persists today. In a survey of 242 healthcare logos (reported in his 1992 book The Golden Wand of Medicine), Walter Friedlander found that 62 percent of professional associations used the rod of Asclepius, while 76 percent of commercial organizations used the caduceus.

“If it’s got wings on it, it’s not really the symbol of medicine,” the communications director of the Minnesota Medical Association told author Robert Taylor. “Some may find it hard to believe, but it’s true. It’s something like using the logo for the National Rifle Association when referring to the Audubon Society.”

In February 1962 John Glenn circled Earth three times on Friendship 7.

When he landed, he received a card from the International Flat Earth Research Society.

It said, “OK wise guy.”

In 1945, Oxford University’s Museum of the History of Science realized that 14 astrolabes were missing from its collection. Curator Robert T. Gunther had arranged for storage of the museum’s objects during the war, but both he and the janitor who had helped him had died in 1940. The missing instruments, the finest of the museum’s ancient and medieval astrolabes, were irreplaceable, the only examples of their kind. Where had Gunther hidden them?

The museum consulted the Oxford city police and Scotland Yard, who searched basements and storerooms throughout the city. The Times, the Daily Mail, and the Thames Gazette publicized the story. Inquiries were extended to local taxi drivers and 108 country houses. At Folly Bridge, Gunther’s house, walls were inspected, flagstones lifted, and wainscoting prised away. A medium and a sensitive were even consulted, to no avail. Finally the detective inspector in charge of the case reviewed the evidence and composed a psychological profile of Gunther, a man he had never met:

Clever professor type, a bit irascible, who didn’t get on too well with his colleagues. Single minded. Lived for the Museum. Hobby in Who’s Who ‘… founding a Museum’. Used to gloat over the exhibits and looked upon them as his own creation. Never allowed anyone else to handle them. Reticent, even secretive. Never told anyone what he what he was going to do. Didn’t trust them, perhaps. Not even his friends the Rumens, who would have offered their car to move the things. Had original ideas though. Safe from blast below street level. Germans would never bomb Oxford. Why, its total war damage was £100 and that from one of our own shells. How right he was. He never expected to die then. Believed he’d live to 90. Hadn’t made any plans; like most of us he thought he might get bumped off when the war started. That’s what he was telling his son in those letters. There was only one conclusion with a man like that anyhow: he’d never let the things out of his reach if he could have helped it. Didn’t even take the trouble to pack his own treasures away in Folly Bridge.

In 1948 the new curator found the missing instruments — they were right “within reach” in the museum’s basement. Gunther had disguised their crate with a label reading “Eighteenth-Century Sundials,” and it had evaded detection throughout the searches.

From A.E. Gunther, Early Science in Oxford, vol. XV, 1967, 303-309.

(Thanks, Diane.)

If we stand immediately below a painting in a gallery, it appears foreshortened. But if we stand on the other side of the room, it appears small. Somewhere between these two points must be the optimum viewing position, where the painting fills the widest possible angle in our vision. How can we find it?

The German mathematician Regiomontanus posed this question in 1471. We can solve it using calculus, but it also yields to simple geometry: Draw a circle defined by the top and bottom of the painting and our eye level. That’s the point we want — any other point at eye level will define a larger circle, in which the picture makes a smaller chord and subtends a smaller angle.

(Thanks, David.)

The Pythagorean theorem works for any similar shapes, not just squares.

In the figure above, A + B = C.

If the three sides of a right triangle are made the diameters of three circles, then the combined area of the two smaller circles equals that of the largest. That’s also the area of the circumcircle, since a right triangle’s hypotenuse forms the diameter of its circumscribing circle.