From Henry Dudeney’s Amusements in Mathematics, 1917. Without removing these checks from their ring, divide them into three groups so that the first group multiplied by the second makes the third. For example, one valid try might be 28, 907, 15463, except that 28 × 907 doesn’t equal 15463.
“Of course, you may have as many of the checks as you like in any group. The puzzle calls for some ingenuity, unless you have the luck to hit on the answer by chance.”
The dashed disk here has radius 1. Suppose we want to cover it entirely with n smaller disks. How small can those disks be?
Pleasingly, no one has yet found a general answer to this question. If we have only a single covering disk, then obviously it will need to be fully as large as the target. But if we’re allowed six discs, they can do the job even if each has a radius of only 0.555905…, as shown here.
Similar configurations work up to n = 10. But if we’re allowed 11 disks then some creative thinking again becomes necessary to find the best solution. No one has yet found a general strategy that reliably finds the minimum successful size.
“Reason misleads us more often than nature.” — Vauvenargues
This diagram appears in a 1673 Portuguese composition inscribed to the Conde de Villaflor. The title explains, “Each circle is a verse, each verse two anagrams. The letters are composed by the numbers and the numbers by the letter, on the periphery of this globe.”
Following the instructions we can read around the outer circle the words ‘DOM SANCHO MANOEL.’ To each of the letters of this name a number is attributed, so that we have the numbers from 1 to 15, corresponding to the letters over which they are placed. In the inner circles those numbers are to be retranslated into letters and, if the reader does so, he will decipher the riddle and end up with the announced sonnet, in which the name DOM SANCHO MANOEL is found in an acrostic and in the twenty-eight anagrams (two in each line) formed by the combination of letters in those words.
Hatherly, a professor of Baroque literature at UC Berkeley, discovered the solution in an 18th-century manuscript:
D O Onde nam macho o sol o sol manchandome; M mancha nem dolo so nem sol mo achando: S sol como de manhan nam escolho, mando: A achem. Mando no sol Solon chamandome N Nome mancha do sol no cham. Sol andome C chamando sol nem o encham o sol. Mando H homem os do cannal nos mostre chamando O oh do mesmo cannal com al sonhandome, M Mancha medo no sol, sol nam, chamo onde A achem damno no sol, nem sol chamando N nam ilho escondam o sol, nome dam ancho O Onde o sol mancham, mal o sol ham conde E echo nam dam no sol em sol manchando L lem coando sonham no Leam Dom Sancho.
(From Merald E. Wrolstad and Dick Higgins, Visible Language, 1986.)
A recent survey of London school children has shown that youngsters between the ages of five and seven have forgotten so many of the attributes of peacetime living that they will have a hard time adjusting themselves to normal conditions again.
Most of the children, when questioned about such things as street lights or foods like bananas, stared suspiciously at the teacher and indicated plainly that they did not believe in their existence.
— New York Times, June 1943
(Shown a seashell, one boy replied, “That’s no shell. Shells come out of guns.”)
Electrical terms that Benjamin Franklin appears to have been the first to use, at least in print in English:
This list is from Carl Van Doren’s 1938 biography. “Though he never lost sight of what was being done in electricity during his whole lifetime, he was perfectly willing to have his contributions to it absorbed in the enlarging science. They were absorbed, and it is now difficult to trace the details of his influence.”
One might conjecture that there is an interesting fact concerning each of the positive integers. Here is a ‘proof by induction’ that such is the case. Certainly, 1, which is a factor of each positive integer, qualifies, as do 2, the smallest prime; 3, the smallest odd prime; 4, Bieberbach’s number; etc. Suppose the set S of positive integers concerning each of which there is no interesting fact is not vacuous, and let k be the smallest member of S. But this is a most interesting fact concerning k! Hence S has no smallest member and therefore is vacuous. Is the proof valid?
— Edwin F. Beckenbach, “Interesting Integers,” American Mathematical Monthly, April 1945
From the Fall 1983 issue of Pi Mu Epsilon Journal, a wordless proof by Purdue University freshman Hao-Nhen Qua. Vu:
The Royal Museums of Fine Arts of Belgium hold two anthropomorphic landscapes painted in the 16th century, one masculine (above) and one feminine (below).
The artist is unknown.
This unusual poetic form made its first appearance in William Browne’s 1625 verse collection Britannia’s Pastorals.
The text traces a single continuous intertwined line, involved with itself and never ending.
