This figure contains four “cliques” of four points each, with each of the four points in each clique connected to each of the others, and each pair of cliques intersecting at a single point. Four colors suffice to color all the points so that no two linked points share a color.

Is this always possible? If k cliques, each containing k points, are arranged in similar fashion, can the result always be colored properly with k colors? In 2021, half a century after Paul Erdős first posed the question, Dong Yeap Kang and his colleagues proved that, for sufficiently large k, the conjecture is true.

The Wallace–Bolyai–Gerwien theorem, first proven in 1807, states that any two polygons of equal area must have a common dissection. That is, there’s always a way to cut up the first one and assemble the pieces to form the second.

But what if the pieces must be connected by hinges? In his “haberdasher” puzzle of 1907, Henry Dudeney showed that it’s possible to convert a triangle into a square by cutting it in pieces and turning it “inside out”:

Is it always possible to arrange such a “hinged dissection” between two polygons of equal area? The question remained open until 2007, when Erik Demaine showed that the answer is yes — and provided an algorithm to find it.

The numbers 1-7 are disposed among the regions in this figure such that each of the circular sets yields the same sum. This makes it a “magic Venn diagram,” a concept that occurred to mathematician David Robinson while teaching a course in mathematical logic at the University of West Georgia. His article appears in the December 2025 issue of Recreational Mathematics Magazine.

(David Robinson and Anja Remshagen, “Magic Venn Diagrams,” Recreational Mathematics Magazine 12:21 [December 2025], 25-44.)

We tend to think that mirrors reverse left to right, but in fact they reverse “back to front,” along an axis perpendicular to the mirror’s surface. Our confusion arises when we misinterpret this.

“[I]magine a back-front reversal of yourself, with your nose, face, eyes, and so forth pushed through to the back of the head, and your back somehow oozed through to the front,” writes University of Auckland psychologist Michael C. Corballis. “You might then ‘feel’ your watch as having remained on the left wrist (say), while back and front have reversed. However it is likely that you will also experience a strong compulsion to recalibrate your internal axes, and then feel the watch to be on the right wrist. In short, a back-front reversal is reinterpreted as a left-right reversal.”

(Michael C. Corballis, “Much Ado About Mirrors,” Psychonomic Bulletin & Review 7:1 [2000], 163-169. More mirror puzzles.)

It’s been known since 1882 that it’s impossible to construct a square with the area of a given circle using only a compass and straightedge.

But in 1977 Thomas Elsner showed that the task becomes possible if the circle is allowed to roll:

From Lee Sallows:

“In his book Amusements in Mathematics, H.E. Dudeney presents a method of classifying 4×4 magic squares based on the distribution of their 8 complementary pairs 1 & 16, 2 & 15, .., 8 & 9. There are just 12 distinct such distributions or ‘graphic types’, which he labelled I to XII. The square above is an example of a type X square.”

(Thanks, Lee.)

The sum of the first n square numbers is n(n+1)(2n+1)/6.

These sums comprise the square pyramidal numbers — each corresponds to the number of oranges that can be stacked in a square pyramid whose base has side n.

This visual proof, for n=3, shows that six square pyramids with n steps fit in a cuboid of size n(n + 1)(2n + 1).

(By CMG Lee.)