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.
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
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?
https://www.youtube.com/watch?v=UIR22SgrOz4
Meiji Institute mathematician Kokichi Sugihara (he of the reverse-gravity ramps) devised this separate illusion in 2018 — an arrow that points in only one direction.
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.
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.
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
Writing in the San Francisco journal The Californian in 1865, Mark Twain answered this inquiry from a reader:
I loved and still love, the beautiful Edwitha Howard, and intended to marry her. Yet during my temporary absence at Benicia, last week, alas! she married Jones. Is my happiness to be thus blasted for life? Have I no redress?
“Of course you have,” Twain answered. He argued that intention is everything in the law — if you call your friend a fool, this is not an insult if you intended it playfully. And killing a man by accident does not constitute murder.
Ergo, if you had married Edwitha accidentally, and without really intending to do it, you would not actually be married to her at all, because the act of marriage could not be complete without the intention. And, ergo, in the strict spirit of the law, since you deliberately intended to marry Edwitha, and didn’t do it, you are married to her all the same — because, as I said before, the intention constitutes the crime. It is as clear as day that Edwitha is your wife, and your redress lies in taking a club and mutilating Jones with it as much as you can. Any man has a right to protect his own wife from the advances of other men.
But you have another alternative — you were married to Edwitha first, because of your deliberate intention, and now you can prosecute her for bigamy, in subsequently marrying Jones.
But there is another phase in this complicated case: You intended to marry Edwitha, and consequently, according to law, she is your wife — there is no getting around that — but she didn’t marry you, and if she never intended to marry you you are not her husband, of course. Ergo, in marrying Jones, she was guilty of bigamy, because she was the wife of another man at the time — which is all very well as far as it goes — but then, don’t you see, she had no other husband when she married Jones, and consequently she was not guilty of bigamy.
Now according to this view of the case, Jones married a spinster, who was a widow at the same time and another man’s wife at the same time, and yet who had no husband and never had one, and never had any intention of getting married, and therefore, of course, never had been married; and by the same reasoning you are a bachelor, because you have never been any one’s husband, and a married man because you have a wife living, and to all intents and purposes a widower, because you have been deprived of that wife, and a consummate ass for going off to Benicia in the first place, while things were so mixed.
“And by this time I have got myself so tangled up in the intricacies of this extraordinary case that I shall have to give up any further attempt to advise you,” he added. “I might get confused and fail to make myself understood. I think I could take up the argument where I left off, and by following it closely awhile, perhaps I could prove to your satisfaction, either that you never existed at all, or that you are dead, now, and consequently don’t need the faithless Edwitha — I think I could do that, if it would afford you any comfort.”
In 1994 bushwalker David Noble abseiled off a cliff 150 kilometers northwest of Sydney and found himself in a very deep canyon surrounded by trees with strange serrated leaves and curious bubbly bark.
He took a sample back to his colleagues at the New South Wales National Parks and Wildlife Service and discovered that he’d found one of the greatest living fossils of the 20th century, with roots in the age of dinosaurs 110-120 million years ago. Somehow the species had survived raging brushfires and 17 ice ages, apparently by retreating to a single canyon in a national park.
The location of that canyon has been kept secret to protect the survivors, which numbered only 100 adult trees in three or four patches, but a special auction in 2005 raised more than a million dollars from bidders eager to receive the first trees cultivated from the rare conifers.
Tim Entwistle, executive director of Sydney’s Botanic Gardens Trust, said that when he learned about Noble’s discovery he went through the classic stages of botanical shock: disbelief, amazement, and excitement. “The Wollemi Pine is a unique reminder that the world is full of undiscovered wonders, that there is a lot more to know about our planet and a lot to protect.”
(Richard Allen and Kimbal Baker, Australia’s Remarkable Trees, 2009.)
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