In the 17th century, Prince Rupert of the Rhine wondered whether one cube might pass through another of the same size. John Wallis showed that the answer is yes, and, perversely, Pieter Nieuwland showed a century later that one cube can even accept another larger than itself — fully 6 percent larger in the optimal case. The diagram above shows the dimensions (blue) of a square tunnel through a unit cube that will accommodate a second unit cube (green) with room to spare.

Remarkably, all five Platonic solids have the “Rupert property” — a regular tetrahedron, for example, will fit through an identical tetrahedron if the hole is contrived cleverly enough. Whether every convex polyhedron can perform this unlikely feat is an open question.

From Lee Sallows:

(Thanks, Lee!)

The Pythagorean theorem has a reciprocal variant:

In combination with the inverse-square law, this means that identical lamps placed at A and B will produce the same light intensity at C as a single lamp at D.

Suppose that each country on Earth has a colony on the moon and that we want to draw maps on which each nation’s territory receives a consistent color. How many colors would we need?

In 1980 Thom Sulanke showed that we might need as many as nine (above), but it’s possible that a particularly challenging map would require more than that. The problem remains unsolved.

In 2022, amateur mathematician David Smith discovered a remarkable tile that will cover an infinite plane but only in a nonperiodic way.

This solves an open problem in mathematics — for years researchers had been seeking an aperiodic monotile, or “einstein,” from the German for “one stone.”

Technically Smith’s tile, known as the “hat,” must be used in combination with its mirror image. But last year his team found another nonperiodic tile, known as the spectre, which is strictly chiral — that is, not only will it tile the plane without its mirror image, but it must be used in that way.