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.)
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.’
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.”
This is interesting — in order for a simple arch to stand, its shape when inverted must fit into the limits described by a hanging chain (a catenary).
Spanish Catalan architect Antoni Gaudí followed this principle in designing some of his buildings — he created inverted models and let the shapes of hanging chains or weighted strings determine the shapes of the arches.
In the early 1940s a curious question began to circulate among the members of the Princeton physics department. An ordinary lawn sprinkler like the one shown here would turn clockwise (in the direction of the long arrow) as its jets ejected water (short arrows). If you reversed this — that is, if you submerged the sprinkler in a tank of water and induced the jets to suck in the fluid — would the sprinkler turn in the opposite direction?
The problem is associated with Richard Feynman, who was a grad student at the time (and who destroyed a glass container in the university’s cyclotron laboratory trying to find the answer).
In fact Ernst Mach had first asked the question in an 1883 textbook. The answer, briefly, is no: The submerged sprinkler doesn’t turn counterclockwise because counterbalancing forces at the back of the nozzle result in no net torque. Experiments tend to bear this out, although in some cases the sprinkler turns slightly counterclockwise, perhaps due to the formation of a vortex within the sprinkler body.
I don’t know whether this is contrived or whether a student offered it on an actual exam — Ed Barbeau presented “this little beauty of a howler” in the January 2002 College Mathematics Journal, citing Ross Honsberger of the University of Waterloo in Ontario.
In his Harmonices Mundi of 1619, Johannes Kepler wrote, “The heavenly motions are nothing but a continuous song for several voices, to be perceived by the intellect, not by the ear; a music which, through discordant tensions, through syncopations and cadenzas as it were, progresses toward certain pre-designed six-voiced cadences, and thereby sets landmarks in the immeasurable flow of time.” In 1979 Yale geologist John Rodgers and musician Willie Ruff scaled up the frequencies of the planetary orbits into the range of human hearing so that Kepler’s “harmony of the world” could become audible:
Mercury, as the innermost planet, is the fastest and the highest pitched. It has a very eccentric orbit (as planets go), which it traverses in 88 days; its song is therefore a fast whistle, going from the E above the piano (e′′′′′) down more than an octave to about C# (c#′′′′) and back, in a little over a second. Venus and Earth, in contrast, have nearly circular orbits. Venus’s range is only about a quarter tone, near the E next above the treble staff (e′′′); Earth’s is about a half tone, from G (g′′) to G# at the top of that staff. … Next out from Earth is Mars, again with an eccentric orbit … it ranges from the C above middle C (c′′) down to about F# (f#′) and back, in nearly 10 seconds. The distance from Mars to Jupiter is much greater than that between the inner planets … and Jupiter’s song is much deeper, in the baritone or bass, and much slower. It covers a minor third, from D to B (D to B1) just below the bass staff. Still farther out and still lower is Saturn, only a little more than a deep growl, in which a good ear can sometimes hear the individual vibrations. Its range is a major third, from B to G (B2 to G2), the B at the top being just an octave below the B at the bottom of Jupiter’s range. Thus the two planets together define a major triad, and it may well have been this concord … that made Kepler certain he had cracked the code and discovered the secret of the celestial harmony.
(The outer planets, discovered after Kepler’s time, are represented here with rhythmic beats.) “The Earth sings Mi, Fa, Mi,” Kepler wrote. “You may infer even from the syllables that in this our home misery and famine hold sway.”
(John Rodgers and Willie Ruff, “Kepler’s Harmony of the World: A Realization for the Ear,” American Scientist 67:3 [May-June 1979], 286-292.)
