A problem proposed by Ashay Burungale of Satara, Maharashtra, India, in the November 2008 issue of American Mathematical Monthly: In a certain town of population 2n + 1, all relations are reciprocal: If Person 1 knows Person 2, then Person 2 knows Person 1. For any set A that consists of n citizens, there’s some person among the remaining n + 1 who knows everyone in A. Prove that there’s some citizen of the town who knows all the others.
Ups and Downs
An Edinburgh startup called Gravitricity is hoping to create a “virtual battery” by hoisting and dropping weights in disused mine shafts. If the weights are hoisted when renewable energy is plentiful, and dropped when it’s expensive, then they can help to balance the energy grid with an efficient source of “gravity energy.”
Managing director Charlie Blair told the Guardian, “The beauty of this is that this can be done multiple times a day for many years, without any loss of performance. This makes it very competitive against other forms of energy storage — including lithium-ion batteries.”
Dropping 12,000 tonnes to a depth of 800 meters would produce enough electricity to power 63,000 homes for more than an hour. Oliver Schmidt of Imperial College London said, “I don’t expect Gravitricity to displace all lithium batteries on grids, but it certainly looks like a compelling proposition.”
(Via Tom Whitwell’s “52 Things I Learned in 2019.”)
Turn, Turn, Turn
Austin game developer Frank Force has won the Neural Correlate Society’s 2019 Best Illusion of the Year Contest with this “dual axis illusion”:
“This spinning shape appears to defy logic by rotating around both the horizontal and vertical axis at the same time! To make things even more confusing, the direction of rotation is also ambiguous. Some visual cues in the video will help viewers change their perception.”
The top 10 finalists are here.
12/18/2019 UPDATE: Good heavens, people are making these things. Five years ago Bill Gosper created the Lissajoke ambiguous roller, which rolls in a perplexing way:
And reader Joe Kisenwether has rendered Force’s illusion in reality — here the shape on the left spins about a horizontal axis while the one on the right spins about the vertical one:
The same thing rendered more slowly:
(Thanks, Joe.)
A New Find
From Lee Sallows, a remarkable new self-inventorying list:
ONE A, ONE B, ONE C, ONE D, TWENTYEIGHT E, SEVEN F, FIVE G, FIVE H, EIGHT I, ONE J, ONE K, ONE L, ONE M, EIGHTEEN N, EIGHTEEN O, ONE P, ONE Q, FOUR R, TWO S, TEN T, FOUR U, FIVE V, FOUR W, ONE X, TWO Y, ONE Z
“I may be wrong, but I believe it to be the most concise self-descriptive (or ‘self-enumerating’) English pangram yet discovered, with as many as 12 of its 26 letters occurring just once.”
(Thanks, Lee!)
12/18/2019 UPDATE: We’ve learned that the same self-descriptive pangram had already been found in 1998 by Gilles Esposito-Farese, in collaboration with Éric Angelini and Nicolas Graner.
Now Showing
This is neat — Eric Harshbarger finds that a list of the 12 top-performing movies in the U.S. last weekend (Dec. 13-15, 2019) contains all 26 letters of the alphabet:
JUMANJI: THE NEXT LEVEL
FROZEN II
KNIVES OUT
RICHARD JEWELL
BLACK CHRISTMAS
FORD V FERRARI
QUEEN & SLIM
A BEAUTIFUL DAY IN THE NEIGHBORHOOD
DARK WATERS
21 BRIDGES
MIDWAY
PLAYING WITH FIRE
The top 8 alone contain all the letters but P.
(Thanks, Eric.)
Podcast Episode 277: The Mad Trapper of Rat River
In the winter of 1931, a dramatic manhunt unfolded in northern Canada when a reclusive trapper shot a constable and fled across the frigid landscape. In the chase that followed the mysterious fugitive amazed his pursuers with his almost superhuman abilities. In this week’s episode of the Futility Closet podcast we’ll describe the hunt for the “Mad Trapper of Rat River.”
We’ll also visit a forgotten windbreak and puzzle over a father’s age.
“A Rebus-Letter”
Mark Twain sent this letter to his wife and daughters from Montreal on Nov. 27, 1881. What does it mean?
False Features
Just before his death in 1702, butterfly collector William Charlton delivered an unusual specimen to London entomologist James Petiver. Petiver wrote, “It exactly resembles our English Brimstone Butterfly (R. Rhamni), were it not for those black spots and apparent blue moons on the lower wings. This is the only one I have seen.” Carl Linnaeus named it Papilio ecclipsis and included it in the 12th edition of his Systema Naturae in 1767.
It wasn’t until 1793 that Danish zoologist Johan Christian Fabricius discovered that the dark patches had been painted on — it was only an ordinary brimstone butterfly after all. The curator at the British Museum “indignantly stamped the specimen to pieces” at this news, but entomologist William Jones created two new replicas to commemorate the “Charlton Brimstones.”
Never Mind
On April 8, 1885, the St. Louis Evening Chronicle published an astonishing story — coal miner Tim Collins had broken into a forgotten cavern hewn by subterranean giants:
A large room, probably 65 by 100 feet in extent, showed itself dimly by the light of our tapers. It was about 20 or 25 feet from floor to ceiling, and had evidently been lighted from the top, though there were openings in the walls where, from appearances, great oaken blinds or doors had once been. These doors had rotted, and only small portions of them remained, small bits of which we chipped off with our knives as souvenirs of our visit. Further examination showed that this room had been used as a workshop but mechanics who had been at work long before Huram’s artificers hewed the architecture for Solomon’s temple. On each side near the walls, and also in the center, were found tables or benches where they had fashioned the work of their hands. These benches were of stone, and there were but few evidences of the character of work done. The wood that had been employed was damp, rotten and so covered with mold as to be almost indistinguishable in shape, and when touched, crumbled to dust. Tools were found on the benches, the handles of which had long since rotted away. But the tools themselves were in a good state of preservation and show that they were fashioned by master mechanics. A number of them were brought to the surface and are now exposed to the gaze of the curious.
The story drew worldwide attention for three days before it was revealed to be false — reporter J.W. Estes of the Moberly Daily Monitor had concocted it as a late April Fools’ Day hoax.
Collins was real enough — to keep off curiosity seekers he was forced to mount a sign at his mine’s entrance: NO BURRYIED SITY LUNATICKS ALOUD ON THESE PREMISES.
(From Kenneth L. Feder, Archaeological Oddities, 2019.)
Breaking the News
A letter from William Cullen Bryant to his mother, Jan. 16, 1821:
Dear Mother:
I hasten to communicate to you the melancholy intelligence of what has lately happened to me.
Early in the evening of the eleventh day of the present month, I was at a neighbouring house in this village. Several people of both sexes assembled in one of the apartments — three or four others, together with myself were in another. At last came a little elderly gentleman, pale, thin, with a solemn countenance, a pleuritic voice, hooked nose, and hollow eyes. Presently we were summoned to attend in the room where he and the rest of the company were assembled. We went in, and took our seats; the little elderly gentleman with the hooked nose then prayed, and we all stood up. When he had finished, most of us sat down. The little elderly gentleman with the hooked nose then asked those who remained standing for something which he called a certificate … A paper was accordingly produced, inscribed with certain significant characters, upon which having mused a little while, he turned to us and pronounced several cabalistical expressions, which I was too much frightened to remember — but I recollect very well, that, at the conclusion, I was given to understand that I was married to a young lady of the name of Frances Fairchild, whom I perceived standing by my side, and whom I hope, in the course of the next few months, to have the pleasure of introducing to you as your daughter-in-law; which is a matter of some interest to the poor girl who has neither father nor mother in the world. …
I looked only for goodness of heart, an ingenuous and affectionate disposition, a good understanding &c. &c., and the character of my wife is too frank and single-hearted to suffer me to suppose the possibility of my being disappointed. — I misstate the matter — I did not look for these, nor any qualities — but they trapped me before I was aware, and now I am married in spite of myself. When we shall begin to keep house will depend, as everything else does, altogether upon circumstances.
Thus the current of destiny carries us all along. None but a madman would swim against the stream, and none but a fool would exert himself to swim with it. The best way is to float idly with the tide. …
Your affectionate son
W.C. Bryant