Sad Magic

sallows tragic square

The magic square at upper left arranges the numbers 3-11 so that each row, column, and long diagonal totals 21.

Lee Sallows found nine tragic words that vary in length from 3 to 11 letters and arranged them into the same square — and he found a unique shape for each word so that every triplet can be assembled into the same 3×7 shape, shown in the border.


lee sallows triangle theorem

A pretty new theorem by Lee Sallows: Connect each vertex of a triangle to the midpoint of the opposite side, and place a hinge at that point. Now rotate the smaller triangles about these hinges and you’ll produce three congruent triangles.

If the original triangle is isosceles (or equilateral), then the three resulting triangles will be too.

The theorem appears in the December 2014 issue of Mathematics Magazine.

Stamps and Math

Lee Sallows tells me that the postal system of Macau is releasing a new series of stamps based on magic squares. The full set will touch on everything from the Roman SATOR square to Dürer’s Melencolia. Details are here.

Charmingly, the values of the stamps will be 1, 2, …, 9 Macau patacas, so that the sheet of the nine stamps will itself form a classic Lo Shu magic square. Lee’s contribution, above, is a Nasik 2D geomagic square of order 3 — not only are all the rows and columns magic, but so are all six diagonals, including the four “broken” diagonals.

Somewhat related: In 2000 Finland issued seven stamps in classic tangram shapes, featuring images of science and education. (One of the small triangles, barely visible here, is a Sierpinski gasket.) Only three of the seven shapes are denominated postage, but I should think the temptation is overwhelming to arrange all seven on an envelope in the shape of a little man or a fish or something. I wonder what the post office makes of that.


In 1948 Melvin Wellman discovered this pretty anagram:


And Dave Morice found this:


Lee Sallows discovered two similar specimens in Spanish:



These can be combined to make more:



Self-Replicating Resistors

From Lee Sallows:

self-replicating resistors

In an electrical network, if resistors x and y are placed in series their total resistance is x + y; if they’re placed in parallel it’s 1/(1/x + 1/y).

This offers an intriguing opportunity for self-reference. Each of the networks above contains four resistors with values 1, 2, 3, and 4, and the total resistances of the networks themselves are 1, 2, 3, and 4. So any one of the numbered resistors in these networks can be replaced by one of the networks themselves.

The challenge was posed by Sallows and Stan Wagon as a Macalester College “problem of the week”; these examples were discovered by Brian Trial, an automotive electronics engineer from Ferndale, Mich. Sallows points out that any such solution has a dual that results from changing series connections to parallel, and vice versa, and then replacing all resistors values by their reciprocals.

This leads to a further idea: The two sets of resistors below are “co-replicating” — the four networks on the left can be used to replace the four resistors in any of the networks on the right, and vice versa.

co-replicating resistors

(Thanks, Lee.)

Saving Face
Image: Wikimedia Commons

“The Joker,” a picture-preserving geomagic square by Lee Sallows. The 16 pieces can be assembled in varying groups of 4 to produce the same picture in 16 different ways, without rotation or reflection.

The outline need not be a joker — it can take almost any shape.

Self-Tiling Tile Sets

These tiles have a remarkable property — by working together, the four can impersonate any one of their number (click to enlarge):

sallows rep-tiles 1

The larger versions could then perform the same trick, and so on. Here’s another set:

sallows rep-tiles 2

In this set, each of the six pieces is paved by some four of them:

sallows rep-tiles 3

By the fathomlessly imaginative Lee Sallows. There’s more in his article “More on Self-Tiling Tile Sets” in last month’s issue of Mathematics Magazine.


A self-reproducing sentence by Lee Sallows — “Doing what it tells you to do yields a replica of itself”:

sallows palingram

This reminds me of a short short story by Fredric Brown:


Professor Jones had been working on time theory for many years.

“And I have found the key equation,” he told his daughter one day. “Time is a field. This machine I have made can manipulate, even reverse, that field.”

Pushing a button as he spoke, he said, “This should make time run backward run time make should this,” said he, spoke he as button a pushing.

“Field that, reverse even, manipulate can made have I machine this. Field a is time.” Day one daughter his told he, “Equation key the found have I and.”

Years many for theory time on working been had Jones Professor.