Western Michigan University mathematician Allen J. Schwenk discovered this oddity in 2000: Consider three fair six-sided dice of different colors, marked with the following numbers:

• Red: 2, 2, 2, 11, 11, 14
• Blue: 0, 3, 3, 12, 12, 12
• Green: 1, 1, 1, 13, 13, 13

Now:

• The red die beats the green die 7/12 of the time.
• The blue die beats the red die 7/12 of the time.
• The green die beats the blue die 7/12 of the time.

We’ve seen that before. But look at this:

• A pair of green dice beats a pair of red dice 693/1296 of the time.
• A pair of red dice beats a pair of blue dice 675/1296 of the time.
• A pair of blue dice beats a pair of green dice 693/1296 of the time.

The favored color in each pairing has changed! Schwenk writes, “I call this a perverse reversal.”

(And a bonus: It turns out that a pair of Schwenk dice of any one color is an even match against a mixed pair of the other two colors.)

(Allen J. Schwenk, “Beware of Geeks Bearing Grifts,” Math Horizons 7:4 [April 2000], 10-13, via Jennifer Beineke and Lowell Beineke, “Some ABCs of Graphs and Games,” in Jennifer Beineke and Jason Rosenhouse, eds., The Mathematics of Various Entertaining Subjects, 2016.)

Shortly after conquering the Matterhorn on July 14, 1865, Edward Whymper watched four of his companions fall to their deaths down the mountain’s precipitous north face. Afterward he and his two Swiss guides, the Taugwalders, beheld a remarkable figure in the sky:

A mighty arch appeared, rising above the Lyskamm, high into the sky. Pale, colourless, and noiseless, but perfectly sharp and defined, except where it was lost in the clouds, this unearthly apparition seemed like a vision from another world; and, almost appalled, we watched with amazement the gradual development of two vast crosses, one on either side. If the Taugwalders had not been the first to perceive it, I should have doubted my senses. They thought it might have some connection with the accident, and I, after a while, that it might bear some relation to ourselves. But our movements had no effect on it. It was a fearful and wonderful sight; unique in my experience, and impressive beyond description, coming at such a moment.

What was this? Whymper later called it a fog bow, a bow that forms in fog rather than rain. Unfortunately, we have no photograph, only a sketch and a woodcut. In a 2002 simulation C.J. Hardwick tried to account for the features as Whymper had described them. “A fogbow and ice crystal arcs could have produced a circle and crosses in a direction consistent with the apparition,” he concluded in 2005. “However, while this simulation used a crystal type that can occur, it required an unusual alignment that would be very rare.”

(C.J. Hardwick, “Simulation of the Whymper Apparition,” Weather 57:12 [December 2002], 457-463; Cedric John Hardwick and Jason C. Knievel, “Speculations on the Possible Causes of the Whymper Apparition,” Applied Optics 44:27 [Sept. 20, 2005], 5637-5643.)

“Truel,” a mathematical romance by Tom Vaughan.

(This is based on a problem in game theory, but interestingly the hero is named Galois and one of his opponents is d’Herbinville — that’s the name of the man Alexandre Dumas identified as the opponent of Évariste Galois in his fatal duel of 1832.)

In 1917 Henry Dudeney asked: What’s the maximum number of lattice points that can be placed on an n × n grid so that no three points are collinear?

The answer can’t be more than 2n, since if we place one point more than this, we’re forced to put three into the same row or column. (The 10 × 10 grid above contains 20 points.)

For a grid of each size up to 52 × 52, it’s possible to place 2n points without making a triple. For larger grids it’s conjectured that fewer than 2n points are possible, but today, more than a century after Dudeney posed the question, a final answer has yet to be found.

In the 17th century the French mathematician Bernard Frénicle de Bessy described all 880 possible order-4 magic squares — that is, all the ways in which the numbers 1 to 16 can be arranged in a 4 × 4 array so that the long diagonals and all the rows and columns have the same sum.

These squares share a curious property: If we subtract 1 from each cell, to get a square of the numbers 0-15, then each of the rows and columns has a nim sum of 0. A nim sum is a binary sum in which 1 + 1 is evaluated as 0 rather than “0, carry 1.” For example, here’s one of Frénicle’s squares:

Translating each of these numbers into binary we get

And the binary sums of the four rows, evaluated without carry, are

0000 + 0101 + 1010 + 1111 = 0000
1110 + 1011 + 0100 + 0001 = 0000
1101 + 1000 + 0111 + 0010 = 0000
0011 + 0110 + 1001 + 1100 = 0000

The same is true of the columns. (The diagonals won’t necessarily sum to zero, but they will equal one another. And note that the property described above won’t necessarily work in a “submagic” square in which the diagonals don’t add to the magic constant … but it does work in all 880 of Frénicle’s “true” 4 × 4 squares.)

(John Conway, Simon Norton, and Alex Ryba, “Frenicle’s 880 Magic Squares,” in Jennifer Beineke and Jason Rosenhouse, eds., The Mathematics of Various Entertaining Subjects, Vol. 2, 2017.)

A card curiosity via Martin Gardner: Deal 10 cards from an ordinary deck and hold this packet face down in your left hand. Turn the top two cards face up and then cut the packet anywhere you like. Again, turn the top two cards and cut. Continue doing this for as long as you like, turning over the top two cards and cutting the packet.

When you’ve finished, deal the cards in a row on the table and turn over the cards at even positions in the row: the second, fourth, sixth, eighth, and tenth cards.

This will always leave five cards face up.

(Martin Gardner, “Curious Counts,” Math Horizons 10:3 [February 2003], 20-22.)