The difference of any pair of the numbers {1873432, 2288168, 2399057, 6560657} is a perfect square (of 644, 725, 2165, 333, 2067, or 2040).
A Prose Maze
Max Beerbohm wrote two parodies of Henry James’ impenetrable style. The first, “The Mote in the Middle Distance,” appeared in The Saturday Review in 1906:
It was with the sense of a, for him, very memorable something that he peered now into the immediate future, and tried, not without compunction, to take that period up where he had, prospectively, left it. But just where the deuce had he left it? The consciousness of dubiety was, for our friend, not, this morning, quite yet clean-cut enough to outline the figures on what she had called his ‘horizon,’ between which and himself the twilight was indeed of a quality somewhat intimidating. He had run up, in the course of time, against a good number of ‘teasers;’ and the function of teasing them back — of, as it were, giving them, every now and then, “what for” — was in him so much a habit that he would have been at a loss had there been, on the face of it, nothing to lose.
He wrote the second, “The Guerdon,” when he learned that James was about to receive the Order of Merit:
That it hardly was, that it all bleakly and unbeguilingly wasn’t for “the likes” of him — poor decent Stamfordham — to rap out queries about the owner of the to him unknown and unsuggestive name that had, in these days, been thrust on him with such a wealth of commendatory gesture, was precisely what now, as he took, with his prepared list of New Year colifichets and whatever, his way to the great gaudy palace, fairly flicked his cheek with the sense of his having never before so let himself in, as he ruefully phrased it, without letting anything, by the same token, out.
He wasn’t alone. “It’s not that he bites off more than he can chaw,” Clover Adams once wrote of James, “but he chaws more than he bites off.”
Perspective
“If you think of this world as a place intended simply for our happiness, you find it quite intolerable: think of it as a place of training and correction and it’s not so bad.” — C.S. Lewis
The Bingo Paradox
Surprisingly, when a large number of people play bingo, it’s much more likely that the winning play occupies a row on its card rather than a column.
The standard bingo card is a 5 × 5 square in which the columns are headed B-I-N-G-O. The columns are filled successively with numbers drawn at random from the intervals 1-15, 16-30, 31-45, 46-60, and 61-75. And it turns out that, during play, it’s very likely that at least one number from each column group will be called (enabling a horizontal win) before some five numbers are called that occupy a single column (enabling a vertical win). In fact it’s more than three times as likely.
The math is laid out rigorously in the article below. If a free space appears in the middle of the board, as is common, the effect still obtains — Joseph Kisenwether and Dick Hess found that the chance of a horizontal win is still 73.73 percent.
(Arthur Benjamin, Joseph Kisenwether, and Ben Weiss, “The BINGO Paradox,” Math Horizons 25:1 [2017], 18-21.)
Podcast Episode 268: The Great Impostor
Ferdinand Demara earned his reputation as the Great Impostor: For over 22 years he criss-crossed the country, posing as everything from an auditor to a zoologist and stealing a succession of identities to fool his employers. In this week’s episode of the Futility Closet podcast we’ll review Demara’s motivation, morality, and techniques — and the charismatic spell he seemed to cast over others.
We’ll also make Big Ben strike 13 and puzzle over a movie watcher’s cat.
The 36 Officers Problem
Suppose we have a group of officers in six regiments, each regiment consisting of the same six ranks (say, a colonel, a lieutenant colonel, a major, a captain, a first lieutenant, and a second lieutenant). Is it possible to arrange these 36 officers into a 6 × 6 square so that no rank or regiment is repeated in any row or column? That is, each row and column must contain an officer of each regiment and of each rank.
In 1782 Leonhard Euler wrote, “After we have put a lot of thought into finding a solution, we have to admit that such an arrangement is impossible, though we can’t give a rigorous demonstration of this.” He saw that the equivalent problem is impossible in a 2 × 2 square and surmised that it’s impossible in every case where the side of the square contains 4k + 2 cells.
It wasn’t until 1901 that French mathematician Gaston Terry proved that the 6 × 6 square has no solution, and it wasn’t until 1960 that Euler’s conjecture about the pattern of impossible squares was proven wrong: In fact, the task is impossible only in these two cases, 2 × 2 and 6 × 6.
Oh
Nathan Bailey’s Universal Etymological Dictionary of 1764 defines thunder as “a noise known by persons not deaf.”
Fair Enough
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Butterflies in the genus Diaethria are commonly called “eighty-eights” because their wings bear a pattern that resembles the number 88 or 89.
The Australian ringneck parrot has four subspecies, one of which is known as the 28 parrot for its triple-noted call, which sounds like “twentee-eight.”
W Hour
Each year on August 1 the city of Warsaw comes to a voluntary standstill for one minute at 5 p.m.
It’s done to honor those who fought for freedom during the Warsaw Uprising, which began at that hour on August 1, 1944.
Actual Size
This is neat: Figure 1 in this physics paper is a 1:1 scale image of a primordial black hole whose mass is 5 times that of Earth.
A black hole of 10 Earth masses “is roughly the size of a ten pin bowling ball.”
(Jakub Scholtz and James Unwin, “What If Planet 9 Is a Primordial Black Hole?”, arXiv preprint arXiv:1909.11090 [2019].)