During World War II, Hungarian mathematician Pál Turán was forced to work in a brick factory. His job was to push a wagonload of bricks along a track from a kiln to storage site. The factory contained several kilns and storage sites, with tracks criss-crossing the floor among them. Turán found it difficult to push the wagon across a track crossing, and in his mind he began to consider how the factory might be redesigned to minimize these crossings.

After the war, Turán mentioned the problem in talks in Poland, and mathematicians Kazimierz Zarankiewicz and Kazimierz Urbanik both took it up. They showed that it’s always possible to complete the layout as shown above, with the kilns along one axis and the storage sites along the other, each group arranged as evenly as possible around the origin, with the tracks running as straight lines between each possible pair. The number of crossings, then, is

where m and n are the number of kilns and storage sites and denotes the floor function, which just means that we take the greatest integer less than the value in brackets. In the case of 4 kilns and 7 storage sites, that gives us

which is the number of crossings in the diagram above.

Is that the best we can do? No one knows. Zarankiewicz and Urbanik thought that their formula gave the fewest possible crossings, but their proof was found to be erroneous 11 years later. Whether a factory can be designed whose layout contains fewer crossings remains an open problem.

Today it’s possible to describe a color quantitatively, but how did people make such fine distinctions in the 18th century? German geologist Abraham Gottlob Werner proposed a solution in 1774: His Von den äußerlichen Kennzeichen der Foßilien included a “color dictionary” that located each hue in the natural world. Updated by Scottish painter Patrick Syme, it describes 110 colors, telling where each might be found in animal, vegetable, and mineral form: Number 35, for example, “bluish lilac purple,” is the shade of the male of the dragonfly Libellula depressa, the blue lilac, and the mineral lepidolite. Number 82, “tile red,” may be found in the breast of the cock bullfinch, in the shrubby pimpernel, and in porcelain jasper.

This common language gave naturalists an objective way to communicate what they were seeing. Off Brazil aboard the H.M.S. Beagle in 1832, Charles Darwin wrote, “I had been struck by the beautiful color of the sea when seen through the chinks of a straw hat. It was according to Werner nomenclature ‘Indigo with a little azure blue’. The sky at the time was ‘Berlin [blue] with little Ultra marine’.”

The Internet Archive has Syme’s full text.

The angle cos^-1(-1/3) = 109.47°, familiar from soap films and tetrahedral molecular geometry, can be produced with an ordinary piece of A4 paper: Because it has a width:length ratio of , folding it corner to corner as shown yields a shape with precisely that angle.

(Nick Lord, “A ‘Maths Bite’: How to Impress a Chemist,” Mathematical Gazette 80:489 [1996], 584-584.)

How many watchmen are needed to guard the art gallery at left, so that every part of it is under surveillance? The answer in this case is 4; four guards stationed as shown will be able to watch every part of the gallery.

In 1973 University of Montreal mathematician Václav Chvátal showed that, in a gallery with n vertices, n/3 guards will always be enough to do the job. (If n/3 is not an integer, you can dispense with the fractional guard.) And Bowdoin College mathematician Steve Fisk found a beautifully simple proof of Chvátal’s result.

The figure at right shows another art gallery. Cut its floor plan into triangles, and color the vertices of each triangle with the same three colors. The full area of any triangle is visible from any of its vertices, and that means that the whole gallery can be guarded by stationing watchmen at the points indicated by any of the three colors. Choosing the color with the fewest vertices will give us n/3 guards (again discarding fractional guards).

The Chvátal and Fisk proofs both give an answer that’s sufficient but sometimes not necessary. In this case, the gallery has 12 vertices, and 12/3 guards (say, the four green ones) will certainly do the job, but here as few as two will be enough.

(Steve Fisk, “A Short Proof of Chvátal’s Watchman Theorem,” Journal of Combinatorial Theory, Series B 24:3 [1978], 374.)

Mathematician John Horton Conway invented this game in 1961. A line divides an infinite checkerboard into two territories. An army of soldiers occupies the lower territory, one per cell. They want to deliver a man as far as possible into the upper territory, but they can proceed only as in peg solitaire: One soldier jumps orthogonally over another soldier and lands on an empty square immediately beyond him, whereupon the “jumped” man is removed.

It’s immediately obvious how the soldiers can get a man into the upper territory, and it’s fairly clear how they can get one as far as the fourth row above the line. But, surprisingly, Conway proved that that’s the limit: No matter how they arrange their efforts, the soldiers cannot get a man beyond that row in a finite number of moves.

Christopher, the 15-year-old hero of Mark Haddon’s 2003 novel The Curious Incident of the Dog in the Night-Time, says that Conway’s Soldiers is “a good maths problem to do in your head when you don’t want to think about something else because you can make it as complicated as you need to fill your brain by making the board as big as you want and the moves as complicated as you want.”

You can find any number of proofs online, but the most convincing way to see that the task is impossible is to try it yourself.

The Coudersport Ice Mine is a cave near Sweden Township, Pa., that bears icicles in spring and summer but not in winter. A shaft about 12 feet long is located at the base of a steep hill. In winter, when the shaft is relatively dry, it fills with cold air. In the spring, snow begins to melt, and water accumulates at the bottom of the shaft. At the same time, cold air descends through rock crevices from higher in the hill, which focus it on this spot and freeze the water. By September this fund of cold air has been depleted, the ice melts, and the shaft is dry again when cold weather arrives.

“The general skepticism regarding the existence of this phenomenon has been illustrated many times of late and has furnished the people of Coudersport with an endless source of amusement,” noted the Popular Science Monthly in 1913.

In 1911 a Detroit man offered to bet anyone $100 or more that the story was true. “A millionaire ice manufacturer took the bet and eight other business men of Detroit followed suit. Two newspaper men were selected as stake-holders to decide the bets. They visited the mine and, of course, verified the newspaper story, much to the disgust of the nine losers.”