If you erect equilateral triangles on two adjacent sides of a square and then connect the triangle vertices distant from the square to the square vertex distant from the triangles, you get a third equilateral triangle.
Pleasingly, this works whether the triangles are erected inside or outside the square. It was discovered by French mathematician Victor Thébault.
In 1949 English physician Zachary Cope published The Diagnosis of the Acute Abdomen in Rhyme:
The muscles of the bowel-wall are strong
And by their strength they force the food along;
In rhythmic waves contractions come and go
Making the intestinal contents flow;
So if through any kind of morbid state
A bit of bowell wall invaginate
The muscle-wall may force it on and on
Until far down the lumen it has gone.
The gut below Dilates for its reception
And thus you get what’s called intussusception.
He dedicated the 100-page work to his students. The preface to the fifth edition reads:
I thank those readers who oft write to me
Suggesting things with which I oft agree
But most of all I thank the youth who said
‘I always keep a copy by me bed.’
Albrecht Dürer’s 1514 engraving Melencolia I includes this famous magic square: The magic sum of 34 can be reached by adding the numbers in any row, column, diagonal, or quadrant; the four center squares; the four corner squares; the four numbers clockwise from the corners; or the four counterclockwise.
In Power Play (1997), University of Toronto mathematician Ed Barbeau points out that there’s even more magic when we consider squares and cubes. Take the numbers in the first two and the last two rows:
16 + 3 + 2 + 13 + 5 + 10 + 11 + 8 = 9 + 6 + 7 + 12 + 4 + 15 + 14 + 1
162 + 32 + 22 + 132 + 52 + 102 + 112 + 82 = 92 + 62 + 72 + 122 + 42 + 152 + 142 + 12
Or alternate columns:
16 + 5 + 9 + 4 + 2 + 11 + 7 + 14 = 3 + 10 + 6 + 15 + 13 + 8 + 12 + 1
162 + 52 + 92 + 42 + 22 + 112 + 72 + 142 = 32 + 102 + 62 + 152 + 132 + 82 + 122 + 12
Most amazingly, if you compare the numbers on and off the diagonals, this works with both squares and cubes:
16 + 10 + 7 + 1 + 13 + 11 + 6 + 4 = 2 + 3 + 5 + 8 + 9 + 12 + 14 + 15
162 + 102 + 72 + 12 + 132 + 112 + 62 + 42 = 22 + 32 + 52 + 82 + 92 + 122 + 142 + 152
163 + 103 + 73 + 13 + 133 + 113 + 63 + 43 = 23 + 33 + 53 + 83 + 93 + 123 + 143 + 153
In his 2014 book Describing Gods, Graham Oppy presents the “divine liar” paradox, by SUNY philosopher Patrick Grim:
1. X believes that (1) is not true.
If we suppose that (1) is true, then this tells us that X believes that (1) is not true. But if an omniscient being believes that (1) is not true, then it follows that (1) is not true. So the assumption that (1) is true leads to a contradiction.
Suppose instead that (1) is not true. That is, suppose that it’s not the case that X believes that (1) is not true. If an omniscient being fails to believe that (1) is not true, then it’s not true that (1) is not true. So this alternative also leads to a contradiction.
But, on the assumption that there is an omniscient being X, either it’s the case that (1) is true or it’s the case that (1) is not true.
“So, on pain of contradiction,” Oppy explains, “we seem driven to the conclusion that there is no omniscient being X.”
(Also: Patrick Grim, “Some Neglected Problems of Omniscience,” American Philosophical Quarterly 20:3 [July 1983], 265-276.)
Philosopher Mark Bedau points out that, even on a dead planet, the microscopic crystallites that make up clay and mud seem to have the flexibility to adapt and evolve by natural selection:
So a population of crystals can exhibit reproduction, variation, heredity, and adaptivity. “What is rather surprising is that, in the process, the planet remains entirely devoid of life. Thus, natural selection can take place in an entirely inorganic setting.”
(Mark Bedau, “Can Biological Teleology Be Naturalized?” Journal of Philosophy 88:11 [November 1991], 647-655. I think A.G. Cairns-Smith originated this idea in Seven Clues to the Origin of Life in 1985, and Richard Dawkins took it up in The Blind Watchmaker the following year.)
One April Fool’s Day, when logician Raymond Smullyan was 10 years old, his brother told him, “Today I am going to trick you like you have never been tricked before.”
“Little Raymond waited, and waited, and waited, and nothing happened,” writes Ron Aharoni in Circularity. “To this very day, he is not sure whether his brother tricked him or not.”
Imprisoned during the French Revolution, zoologist Pierre André Latreille found a beetle on the floor of his dungeon. He pointed out to the prison doctor that it was the rare Necrobia ruficollis, described by Johan Christian Fabricius in 1775. Impressed, the doctor sent it to a local naturalist, who knew Latreille’s work and managed to secure the release of Latreille and a cellmate.
All the other inmates were executed within a month.
In 1884, as engineers sawed brownstone out of a quarry near Manchester, Conn., amateur paleontologist Charles Owen realized that one block contained the hind part of a skeleton. He alerted professor Othniel Charles Marsh, who managed to acquire the block, but the corresponding stone containing the skeleton’s fore part had already been carted off for use in a new highway bridge.
In the ensuing years the identity of that bridge was forgotten, but in 1967, as a new highway was being constructed, Yale paleontologist John Ostrom saw an opportunity. He surveyed 60 bridges in the Manchester area and identified one 40-foot span over Hop Creek as the likeliest candidate. Then, in 1969, as that bridge was replaced, he and his colleagues examined more than 300 likely blocks at the site.
They found two 500-pound blocks that showed distinct fossil markings, and in New Haven Ostrom determined that one of the visible bones matched a thigh bone that Marsh had recovered 85 years earlier. The complete skeleton had belonged to a member of Ammosaurus, a genus that roamed the northeastern United States 200 million years ago.
(Thanks, Glenn.)
What does this mean?
PMVEB DWXZA XKKHQ RNFMJ VATAD YRJON FGRKD TSVWF TCRWC RLKRW ZCNBC FCONW FNOEZ QLEJB HUVLY OPFIN ZMHWC RZULG BGXLA GLZCZ GWXAH RITNW ZCQYR KFWVL CYGZE NQRNI JFEPS RWCZV TIZAQ LVEYI QVZMO RWQHL CBWZL HBPEF PROVE ZFWGZ RWLJG RANKZ ECVAW TRLBW URVSP KXWFR DOHAR RSRJJ NFJRT AXIJU RCRCP EVPGR ORAXA EFIQV QNIRV CNMTE LKHDC RXISG RGNLE RAFXO VBOBU CUXGT UEVBR ZSZSO RZIHE FVWCN OBPED ZGRAN IFIZD MFZEZ OVCJS DPRJH HVCRG IPCIF WHUKB NHKTV IVONS TNADX UNQDY PERRB PNSOR ZCLRE MLZKR YZNMN PJMQB RMJZL IKEFV CDRRN RHENC TKAXZ ESKDR GZCXD SQFGD CXSTE ZCZNI GFHGN ESUNR LYKDA AVAVX QYVEQ FMWET ZODJY RMLZJ QOBQ-
No one knows. Cryptologist Louis Kruh discovered it in the New York Public Library’s rare book room in 1993 among some old material from the U.S. Army Signal School. In 1915 first lieutenant Joseph O. Mauborgne had created what he believed was a more secure cipher than the ones currently in use, and had offered this challenge to see if his colleagues could break it. Kruh found no solution in the archive, and he published it in both The Cryptogram and Cryptologia, inviting their readers to try their hands at it. As far as I know, none succeeded.
Mauborgne described it as a “a simple, single-letter substitution cipher adapted to military use.” He invited the director of the Army Signal School to place it on a bulletin board and allow the officers there to work on it for three months and then to post the solution “to show why the standard method of attacking a substitution cipher fails in this case.” “If any attack upon this cipher is successful, I shall be glad to hear of it,” he wrote.
Kruh, who died in 2010, noted that “it was probably solved or otherwise deemed unsuitable for use because there is no knowledge of a new cipher being adopted by the Army around that time.” If a solution was found, I don’t believe anyone alive today knows what it is.
(Louis Kruh, “A 77-Year-Old Challenge Cipher,” Cryptologia 17:2 [April 1993], 172-174.)
How can you arrange seven points in a plane so that if any three are chosen, at least two of them will be a unit distance apart?
The solution is shown here — it’s known as a Moser spindle.
