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.
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.”
“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.”
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).
The sea urchin Coelopleurus exquisitus was discovered on eBay. Marine biologist Simon Coppard was directed to a listing on the site in 2004 and realized that the species had not previously been described. When it was properly named and introduced in Zootaxa two years later, the value of specimens on eBay shot up from $8 to $138.
In 2008 a fossilized aphid on eBay was similarly found to be unidentified. Eventually it was named Mindarus harringtoni, after the buyer.
The dark polyomino at the center of this figure, devised by Craig S. Kaplan, has an unusual property: It can be surrounded snugly with copies of itself, leaving no overlaps or gaps. In this case, the “corona” (red) can be surrounded with a second corona (amber), itself also composed of copies of the initial shape. But that’s as far as we can get — there’s no way to create a third corona using the same shape.
That gives the initial shape a “Heesch number” of 2 — the designation is named for German geometer Heinrich Heesch, who had proposed this line of study in 1968.
Shapes needn’t be polyominos: Heesch himself devised the example below, the union of a square, an equilateral triangle, and a 30-60-90 triangle:
It earns a Heesch number of 1, as it can bear only the single corona shown.
Can all positive integers be Heesch numbers? That’s unknown. The Heesch number of the square is infinite, and that of the circle is zero. The highest finite number reached so far is 6.
