The 1968 Putnam Competition included a beautiful one-line proof that π is less than 22/7, its common Diophantine approximation:
The integral must be positive, because the integrand’s denominator is positive and its numerator is the product of two non-negative numbers. But it evaluates to 22/7 – π — and if that expression is positive, then 22/7 must be greater than π.
University of St Andrews mathematician G.M. Phillips wrote, “Who will say that mathematics is devoid of humour?”
In October 2005, Neil Armstrong received a letter from a social studies teacher charging that the moon landings had been faked. “[O]ver 30 years on from the pathetic TV broadcast when you fooled everyone by claiming to have walked upon the Moon,” he wrote, “I would like to point out that you, and the other astronauts, are making yourselfs a worldwide laughing stock … Perhaps you are totally unaware of all the evidence circulating the globe via the Internet. Everyone now knows the whole saga was faked, and the evidence is there for all to see.”
Armstrong replied:
Mr. Whitman,
Your letter expressing doubts based on the skeptics and conspiracy theorists mystifies me.
They would have you believe that the United States Government perpetrated a gigantic fraud on its citizenry. That the 400,000 Americans who worked on an unclassified program are all complicit in the deception, and none broke ranks and admitted their deceit.
If you believe that, why would you contact me, clearly one of those 400,000 liars?
I trust that you, as a teacher, are an educated person. You will know how to contact knowledgeable people who could not have been party to the scam.
The skeptics claim that the Apollo flights did not go to the moon. You could contact the experts from other countries who tracked the flights on radar (Jodrell Bank in England or even the Russian Academicians).
You should contact the Astronomers at Lick Observatory who bounced their laser beam off the Lunar Ranging Reflector minutes after I installed it. Or, if you don’t find them persuasive, you could contact the astronomers at the Pic du Midi observatory in France. They can tell you about all the other astronomers in other countries who are still making measurements from these same mirrors — and you can contact them.
Or you could get on the net and find the researchers in university laboratories around the world who are studying the lunar samples returned on Apollo, some of which have never been found on earth.
But you shouldn’t be asking me, because I am clearly suspect and not believable.
Neil Armstrong
(From James R. Hansen, A Reluctant Icon: Letters to Neil Armstrong, 2020.)
In a 1961 contribution to Analog, Maurice Price explained that, while a new engineer might arrive to a clean desk, a “cycle of confusion” quickly begins as papers pile up so deeply that the engineer can do little more than fish old technical journals from the pile and read them (this is known as “keeping up with the state of the art”). Eventually the engineer resolves to clean the desk, but that only returns the system to the start of the cycle. The key equation is
C = K1 exp (K2t).
“In this equation K1 is the constant of confusion and K2 is the coefficient of chaos. These may vary from desk to desk and from engineer to engineer, but the general form of the curve is not altered. Note that the amount of work to be done does not affect the curve at all. This is because the amount of work expands to overflow the available desk area.”
The only way to break the cycle is to promote the engineer, but that only restarts the cycle at a new desk. And productivity and concentration actually increase with clutter. So the best solution is for the engineer to “bring his desk to the state of stagnation as soon as possible. From then on, the desk should be ignored completely. All work must be carried out by telephone.”
(Maurice Price, “An Introduction to the Calculus of Desk-Clearing,” Analog Science Fact & Fiction, British edition, 1961, 68-72.)
Reader Éric Angelini points out that the regular continued fraction of 
A simpler specimen: 9 9 9 9 … is NINE NINE NINE NINE …
(Thanks, Éric.)
On a standard calculator keypad like the one shown here, any four-digit number that is typed in a rectangular shape is evenly divisible by 11. Some examples: 7964, 6523, 1793, 7128. (The numbers must not include 0.)
Parallelograms work too: 1562, 6875, 2783, etc.
Discussion and some proofs here.
09/20/2023 More: Take three digits in order from any row, column, or main diagonal and append the same three digits in reverse order (e.g., 951159). The resulting number will always be evenly divisible by 37 (and, indeed, by 1221). Mathematical Gazette, December 1986 and June 1987. See also A Keypad Oddity.
Once I read a book of 100 numbered pages with one sentence on each page. Page 1: ‘The sentence on page 2 is true.’ Page 2: ‘The sentence on page 3 is true.’ And so on to page 100: ‘The sentence on page 1 is false.’
On the second reading, page 100 changes the entire book. If page 1 is false, then the truth is ‘The sentence on page 2 is false.’ Likewise, page 2 becomes ‘The sentence on page 3 is false.’ And so on to page 100, which now should be read as ‘The sentence on page 1 is true.’
What happens on the third reading?
— David Morice, “Kickshaws,” Word Ways 26:1 (February 1993), 44-55. See Yablo’s Paradox.
Can a chess knight visit every square on a board with 4 rows by a series of successive moves?
No, it can’t, and Hungarian mathematics prodigy Louis Pósa proved this while still in his early teens. Suppose that such a tour is possible. Then, on any board bearing the standard checkerboard pattern, the knight will land alternately on white and black squares. But now imagine that the board’s top and bottom rows have been colored red and the middle two rows are blue. Now a knight on any red square must jump to a blue square, and because the board has an equal number of red and blue squares, any knight on a blue square must jump to a red one (if it visits two blue squares in a row, it won’t be able to make up for this later by visiting two red ones in a row). So the knight’s tour on any 4 × n board must alternate strictly between red and blue squares.
“But this is impossible,” notes Colorado College mathematician John J. Watkins. “The same knight’s tour alternated between white and black squares in the one coloring, and between red and blue squares in the other coloring, which would mean the two color patterns must be the same, which of course they aren’t. Isn’t that a clever proof, especially for a teenager to discover?”
(John J. Watkins, Across the Board, 2004. Pósa’s proof is given more rigorously here, and it’s also presented in Ross Honsberger’s 1973 book Mathematical Gems.)
UPDATE: It’s important to note that it’s a “closed” knight’s tour that’s impossible — that’s one that ends where it began. An open tour, which can end anywhere, is possible — it breaks Pósa’s proof because it need not alternate strictly between red and blue squares. Thanks to reader Marjan Milanović for pointing this out.
When the 15th-century Benedictine abbot Johannes Trithemius died in 1516, he left behind a three-volume work that was ostensibly about magic — specifically, how to use spirits to send secret messages over distances. Only when the Steganographia and its key were published in 1606 did it become clear that it was really a book of ciphers — the “incantations” were encrypted instructions for concealing secret messages in letters sent between correspondents.
Books I and II were now plain enough, but Book III remained mysterious — it was shorter than the first two books, and its workings were not mentioned in the key that explained the ciphers in those volumes. Scholars began to conclude that it was simply what it appeared to be, a book on the occult, with no hidden content. Amazingly, nearly 400 years would go by before Book III gave up its secrets — Jim Reeds of AT&T Labs finally deciphered the mysterious codes in the third volume in 1998.
It turned out to be still more material on cryptography. But it’s still not clear why Trithemius had couched this third book in magical language. Did he think that his subject was inherently magical, or was he simply trying to enliven a tedious subject? We’ll probably never know. “Trithemius’s use of angel language might … be a rhetorical strategy to engage the reader’s interest,” Reeds writes. “If so, he was vastly successful, even if he completely miscalculated how his book would be received.”
(Jim Reeds, “Solved: The Ciphers in Book III of Trithemius’s Steganographia,” Cryptologia 22:4 (October 1998), 291-317.)
arrident
adj. pleasant
agrised
adj. terrified
presentific
adj. causing something to be present in the mind
cacology
n. a bad choice of words
Shortly after physicist Anthony French joined the MIT faculty in 1962, he was asked to teach an introductory mechanics course to hundreds of freshmen.
“I wanted to be cautious about giving it a name,” he said. “So I called it, blandly, ‘Physics: A New Introductory Course.’
“I couldn’t imagine how I could have been so stupid. The students read that as ‘PANIC’ … it was known forever afterwards as the PANIC course.”
