Three in One
Image: Wikimedia Commons

A stunning geometric alphamagic square by Lee Sallows. The 3 × 3 grid is a familiar magic square in which each number is spelled out: The first cell contains the number 25, the second 2, and so on. Interpreted in this way, each row, column, and long diagonal sums to 45.

But there’s more: The English name of the number in each cell has been arranged onto a distinctive tile, such that the three tiles in any row, column, or long diagonal can be combined to form the same 21-cell figure, as shown. (Shapes with dotted outlines have been turned over.)

And yet more: Count the number of letters in each of the number names (or, equivalently, count the number of cells that make up each tile). So, for example, TWENTY-FIVE has 10 letters, so replace the TWENTYFIVE tile with the number 10. Similarly, replace TWO with 3, EIGHTEEN with 8, and so on. This produces another magic square:

10  3  8
 5  7  9
 6 11  4

Each row, column, and long diagonal totals 21.

Image: Wikimedia Commons

Composer Bohuslav Martinu had a remarkably inauspicious start in life: He was born in a church tower in Bohemia, which the town had granted to his father, a sexton. A sickly child, he often had to be carried up the 143 stairs on his father’s back. He was shy, reticent, and physically uncoordinated. “It would be hard to imagine a more unpromising environment for a child composer, for there was no significant musical tradition within the family and for several years he hardly ever ventured from the tower,” writes Barry Cooper in Child Composers and Their Works.

He excelled in violin, but since manuscript paper wasn’t available to him it’s not clear how many of his early compositions have been lost. When he ventured to the Prague Conservatory in 1906, he offered a three-movement string quartet that he’d written at age 10. He hadn’t even learned the alto clef, and used the treble clef instead for the viola part.

His mother said that the director “was so impressed by Bohus’ composition that at first he doubted whether the score was my son’s own work, and asked who had helped him.” But no one in his village would have been skillful enough to do so.

They accepted him, and he went on to a distinguished international career, writing six symphonies, 15 operas, 14 ballet scores, and a wide range of orchestral, chamber, vocal, and instrumental works.

Image: Wikimedia Commons

I’m just sharing this because I think it’s pretty — it’s the smallest arrangement of identical non-crossing matchsticks that one can make on a tabletop in which each match-end touches three others.

Presented by German mathematician Heiko Harborth in 1986, it’s known as the Harborth graph.

Turning Back Time

A watch for left-handed people has been invented by a Kalamazoo jeweler, who believes that the left-handed look at things in a ‘left-handed’ fashion. The left-handed watch runs backward. The dial is arranged so that the numeral 1 is on the left hand of 12 instead of on the right as in the case of the ordinary watch. The hands also run from right to left instead of in the usual fashion. Mechanically, with the exceptions given, the left-handed watch differs very slightly from the ordinary time-piece.

The inventor constructed the unusual watch for the benefit of his daughter, who is left-handed.

Popular Science Monthly, January 1916

Crowd Control

A knight’s tour is a series of moves by a chess knight such that it visits each square on the chessboard once. The example above is a “closed” tour because it ends on the square where it started.

This inspired a puzzle posed by Martin Gardner. If we filled a standard chessboard with knights, one on each square, could all 64 of them move simultaneously? The closed knight’s tour shows that they could — they form a long conga line, with each knight vacating a square for the knight behind it to occupy.

Gardner asks: Could the same feat be accomplished on a 5 × 5 chessboard?

Click for Answer


In a letter written home from the Western Front in May 1915, 2nd Lt. Alexander Gillespie of the Argyll and Sutherland Highlanders describes a striking midnight experience:

Presently a misty moon came up, a nightingale began to sing … It was strange to stand there and listen, for the song seemed to come all the more sweetly and clearly in the quiet intervals between the bursts of firing. There was something infinitely sweet and sad about it, as if the countryside were singing gently to itself, in the midst of all our noise and confusion and muddy work; so that you felt the nightingale’s song was the only real thing which would remain when all the rest was long past and forgotten. …

So I stood there, and thought of all the men and women who had listened to that song, just as for the first few weeks after Tom was killed I found myself thinking perpetually of all the men who had been killed in battle — Hector and Achilles and all the heroes of long ago, who were once so strong and active, and now are so quiet. Gradually the night wore on, until day began to break, and I could see clearly the daisies and buttercups in the long grass about my feet. Then I gathered my platoon together, and marched back past the silent farms to our billets.

Another front-line soldier, J.C. Faraday, noticed the same effect. He wrote to the Times in July 1917, “You will have a terrific tearing and roaring noise of artillery and shot in the dead of night; then there will be a temporary cessation of the duel, with great quietness, when lo! and behold and hear! Hearken to his song! Out come the nightingales, right about the guns … And another kind of love music is introduced to our ears and souls, which does us good. Think? It makes you think — and beautiful thoughts come along to relieve you from the devilment of war and the men who cause it.”

(From Stephen Moss, A Bird in the Bush: A Social History of Birdwatching, 2013.)


The Hurwitz Singularity, an anamorphic sculpture by artist Jonty Hurwitz, began with a scan of the artist’s own head.

“I wanted to capture my physical being in as much detail as technology allowed,” he writes. “It felt appropriate to be able to analyse myself at the highest resolution that modern science could record spacetime.”

“This sculpture evolved when I was deep in Freudian Therapy. Four days a week on the sofa blazing new trails from the road that Sigmund Freud first mapped. To my analyst I dedicate this piece. Dr Sanchez Bernal this is the Hurwitz Singularity!”

There’s more at the artist’s website.

Intangible Assets

A perplexing story from logician Raymond Smullyan:

Oh, one other thing. I must tell you of a certain great Sage in the East who was reputed to be the wisest man in the world. A philosopher heard about him and was anxious to meet him. It took him fifteen years to find him, but when he finally did, he asked him: ‘What is the best question that can be asked, and what is the best answer that can be given?’ The great Sage replied: ‘The best question that can be asked is the question you have asked, and the best answer that can be given is the answer I am now giving.’

It’s at the very end of his last book, A Mixed Bag, from 2016.

The Other Half
Image: Wikimedia Commons

Friday, September 19. [1777], after breakfast, Dr. Johnson and I set out in Dr. Taylor’s chaise to go to Derby. The day was fine, and we resolved to go by Keddlestone, the seat of Lord Scarsdale, that I might see his lordship’s fine house. I was struck with the magnificence of the building; and the extensive park, with the finest verdure, covered with deer, and cattle, and sheep, delighted me. The number of old oaks, of an immense size, filled me with a sort of respectful admiration; for one of them sixty pounds was offered. The excellent smooth gravel roads; the large piece of water formed by his lordship from some small brooks, with a handsome barge upon it; the venerable Gothic church, now the family chapel, just by the house; in short, the grand group of objects agitated and distended my mind in a most agreeable manner. ‘One should think,’ said I, ‘that the proprietor of all this must be happy.’

‘Nay, Sir,’ said Johnson, ‘all this excludes but one evil — poverty.’

— James Boswell, Life of Samuel Johnson, 1791