Cut the decimal expansion of π into parcels of 10 digits:
Mathematician John Conway points out that the chance is only about 1 in 40,000 that all of the digits 1234567890 will appear in any given parcel … and yet there they all are in the seventh.
(Via Alfred Posamentier and Ingmar Lehmann, π: A Biography of the World’s Most Mysterious Number, 2004.)
This baffling message illustrates a cipher adopted by the Union Army in 1862:
TO GEORGE C. MAYNARD, Washington
Regulars ordered of my to public out suspending received 1862 spoiled thirty I dispatch command of continue of best otherwise worst Arabia my command discharge duty of my last for Lincoln September period your from sense shall duties the until Seward ability to the I a removal evening Adam herald tribune.
PHILIP BRUNER
The address and signature are “covers” that don’t enter into the cipher. The first word, Regulars, is a code indicating that the original message had been written in five columns of nine words each. Tribune, herald, spoiled, Seward, for, and worst are null words; Lincoln is code for Louisville, Kentucky; Adam means General Henry Wager Halleck; and Arabia is code for Major General Don Carlos Buell. The word Period indicates a full stop. This had been the original message:
Louisville, Kentucky September thirty 1862 General Halleck: (Adam) (period) I received last evening your dispatch suspending my removal from command. Out of a sense of public duty, I shall continue to discharge the duties of my command to the best of my ability until otherwise ordered. D.C. Buell, Major General
This message had been enciphered by reading up the fourth column, down the third, up the fifth, down the second, and up the first; inserting the null words; and encoding the most sensitive particulars. The system worked well until July 1864, when Union cipher operator Stephen L. Robinson was captured by Confederate guerrillas and the key seized.
(John Laffin, Codes and Ciphers Secret Writing Through the Ages, 1964.)
When a container of granular material is shaken, we might expect the largest particles to make their way to the bottom. Instead the opposite often happens: Vibrating a container of mixed nuts, muesli, or raisin bran often brings the largest (and presumably heaviest) items to the top.
Precisely why this happens is unclear. An irregularly shaped Brazil nut might “shoulder” its way above smaller nuts as it turns among them; the rising of large particles might help to lower the center of mass of the aggregate; or perhaps the size of the largest particles prevents them from descending in a container’s natural convection flow once they reach the surface. For now it’s an unsolved problem in physics.
J.M. Roberts’ 1987 Hutchinson History of the World contains this arresting sentence:
At one site in Spain the mind of what one scholar called a ‘primitive Archimedes’ has been seen at work three hundred thousand years ago, directing the removal and use of the tusks of slaughtered elephants as levers to shift the carcasses for cutting up.
The scholar seems to be archaeologist François Bordes, who had written in his 1968 book The Old Stone Age that the Acheuleans of Torralba-Ambrona had killed elephants half engulfed in mud, “and that a primitive Archimedes had the idea of using their tusks as levers for shifting their enormous bulk and making it easier to cut them up.”
From what I can understand, the evidence for butchery at these sites is now thought to be ambiguous, but it’s a striking image nonetheless.
Completely unrelated, but similarly notable: In Days With Bernard Shaw, his 1948 memoir of his friendship with George Bernard Shaw, Stephen Winsten remembers Shaw remarking, “Leonardo da Vinci ruled his notebooks in columns headed fox, wolf, bear and monkey and made notes of human faces by ticking them off in these columns.” I can’t confirm this either, but it seems worth recording.
In this network of 9 points, any two points that are linked have 1 linked point in common, forming a triangle. Any two points that aren’t linked have 2 linked points in common, forming a quadrilateral. Is such a pattern possible in a network of 99 points? In 2014 Princeton mathematician John Horton Conway offered $1000 for the answer to this question; so far the prize is unclaimed.
This is surprising: When water is poured from one container into another, floating particles can climb upstream, like inanimate salmon, into the higher container. Argentine physicist Sebastian Bianchini first noticed the phenomenon while making tea during his studies at the University of Havana. His paper inspired further work at Rutgers, which has confirmed the effect, but exactly what’s happening isn’t fully understood.
Take any triangle and divide it into sub-triangles as shown. Inscribe a circle in each of these smaller triangles.
Rearrange the order of the circles and adjust the intervening lines so that each line touches two circles:
No matter how this is done, each circle will always fit perfectly in its triangle. Here are some proofs.
“There exist only two kinds of modern mathematics books: ones which you cannot read beyond the first page and ones which you cannot read beyond the first sentence.” — Physics Nobelist Yang Chen-Ning
A self-inventorying crossword from Lee Sallows:
This one even tallies the number of spaces it contains!
(Thanks, Lee!)
In 1798, Adrien-Marie Legendre claimed that 6 is not the sum of two rational cubes.
In 1865, Gabriel Lamé observed that 6 = (37/21)3 + (17/21)3.
