From Richard F. Mould’s Introductory Medical Statistics — this graph plots the population of Oldenburg, Germany, at the end of each year 1930-1936 against the number of storks observed in that year.
Does this explain the storks’ presence? Not necessarily: In 1888 J.J. Sprenger noted, “In Oldenburg there is a curious theory that the autumnal gatherings of the storks are in reality Freemasons’ meetings.”
A useful system of finger reckoning from the Middle Ages:
To multiply 6 x 9, hold up one finger, to represent the difference between the 5 fingers on that hand and the first number to be multiplied, 6.
On the other hand, hold up four fingers, the difference between 5 and 9.
Now add the number of extended fingers on each hand to get the first digit of the answer (1 + 4 = 5), and multiply the number of closed fingers on each hand to get the second (4 × 1 = 4). This gives the answer, 54.
In this way one can multiply numbers between 6 and 9 while knowing the multiplication table only up to 5 × 5.
A similar system could be used to multiply numbers between 10 and 15. To multiply 14 by 12, extend 4 fingers on one hand and 2 on the other. Add them to get 6; add 10 times that sum to 100, giving 160; and then add the product of the extended fingers, 4 × 2, to get 168.
This system reflects the fact that xy = 10 [(x – 10) + (y – 10)] + 100 + (x – 10)(y – 10).
(From J.T. Rogers, The Story of Mathematics, 1968.)
Take an ordinary magic square and imagine that the number in each cell denotes its altitude above some common underlying plane. And now suppose that it begins to rain, with an equal amount of water falling onto each cell. What happens? In the square at left, the water cascades from square 25 down to square 21, and thence down to 10, 7, 2, and into space; because there are no “lowlands” on this landscape, no water is retained. (Water flows orthogonally, not diagonally, and it pours freely over the edges of the square.)
By contrast, in the square on the right a “pond” forms that contains 69 cubic units of water — as it happens, the largest possible pond on a 5×5 square.
With the aid of computers, these imaginary landscapes can be “terraformed” into surprisingly detailed shapes. Craig Knecht, who proposed this area of study in 2007, created this 25×25 square in 2012:
Next year will mark the 500th anniversary of the famously fertile magic square in Albrecht Dürer’s 1514 engraving Melancholia — a fact that Knecht has commemorated in the shape of the ponds on the 14×14 square at right.
On his 36th birthday, feeling that his most fertile years were behind him, mathematician Abram Besicovitch said, “I have had four-fifths of my life.”
At age 59 he was elected to the Rouse Ball Chair of Mathematics at Cambridge.
When J.C. Burkill reminded him of his earlier remark, he said, “Numerator was correct.”
Twinkle, twinkle, little star,
How I wonder what you are.
Up above the world so high,
Like a diamond in the sky.
Twinkle, twinkle, little star,
How I wonder what you are.
Choose any word in the first two lines, count its letters, and count forward that number of words. For example, if you choose STAR, which has four letters, you’d count ahead four words, beginning with HOW, to reach WHAT. Count the number of letters in that word and count ahead as before. Continue until you can’t go any further. You’ll always land on YOU in the last line.
In 1966, asked to describe the person least likely to develop atherosclerosis, Cambridge research fellow Alan N. Howard answered, “A hypotensive, bicycling, unemployed, hypo-beta-lipoproteinic, hyper-alpha-lipoproteinic, non-smoking, hypolipaemic, underweight, premenopausal female dwarf living in a crowded room on the island of Crete before 1925 and subsisting on a diet of uncoated cereals, safflower oil, and water.”
Oxford physician Alan Norton added that her male counterpart was an ectomorphic Bantu who worked as a London bus conductor, had spent the war in a Norwegian prison camp, never ate refined sugar, never drank coffee, always ate five or more small meals a day, and was taking large doses of estrogen to check the growth of his prostate cancer.
“All these phrases mark correlations established in the last few years in a field of medical research which, in volume at least, is unsurpassed,” noted Richard Mould in Mould’s Medical Anecdotes. “The conflict of evidence is unequalled as well.”
If this sentence is true, then Santa Claus exists.
If that sentence is true, then it’s the case that Santa Claus exists. But wait — in making this observation, we seem to have confirmed the truth of the original sentence. And if that sentence is true, then Santa Claus exists! Where is the error?
(By Raymond Smullyan.)
Cihan Altay’s Rotator typeface presents the digits 0-9 whether it’s right side up or upside down.
So this equation:
… can be inverted to make this one:
Both are valid.
(Thanks, Lorenzo.)
Every now and again one comes across an astounding result that closely relates two foreign objects which seem to have nothing in common. Who would suspect, for example, that on the average, the number of ways of expressing a positive integer n as a sum of two integral squares, x2 + y2 = n, is π?
— Ross Honsberger, Mathematical Gems III, 1977
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