There are 8 ways to fold a 2 × 2 map along its creases:
But no one knows how to calculate the number of ways that a larger map might be folded. Amazingly, the number of ways to fold an n × n map has been worked out only as high as 7 × 7:
1, 8, 1368, 300608, 186086600, 123912532224, 129950723279272
Via Fermat’s Library.
“Humans feel affection for animals with juvenile features,” noted Konrad Lorenz. “Large eyes, bulging craniums, retreating chins. Small-eyed, long-snouted animals do not elicit the same response.”
This induces people to care for small, cuddly animals. “And this has led some experts to argue that the entire phenomenon of pet-keeping is nothing more nor less than an elaborate case of social parasitism,” writes zoologist James Serpell. “Needless to say, this idea has done little to promote a positive view of pets or their owners. Rather, it creates the impression that pet-owners are the victims of some kind of bizarre affliction, and that dogs, cats and budgerigars are little different from body lice, fleas or tapeworms or, indeed, any other sort of parasitic organism.”
(From James Serpell, In the Company of Animals, 1986.)
“Everyone believes in the normal law, the experimenters because they imagine that it is a mathematical theorem, and the mathematicians because they think it is an experimental fact.” — Gabriel Lippmann
(Thanks, Tom.)
I just noticed this last night in Joseph Wood Krutch’s Treasury of Bird Lore — in 1832 ornithologist Alexander Wilson encountered a flock of passenger pigeons near Frankfort, Kentucky, that he estimated at 2,230,270,000 birds. If each bird ate only a pint of beech nuts in the course of a day, the flock would consume nearly 35 million bushels a day. A century and a half earlier, in 1687, Louis Armand, Baron de Lahontan, reported that pigeons had “so swarmed and ravaged the colonists’ crop near Montreal that a bishop was constrained to exorcise them with holy-water, as if they had been demons.”
Yet by 1914 human rapacity had reduced the species to a single bird, Martha, who died that year at the Cincinnati Zoo.
See The Eighth Plague.
In 2012 computer scientist Andrew Adamatzky set a plasmodium of the slime mold Physarum polycephalum into a maze with an oat flake at its center. By following a gradient of chemo-attractants given off by the flake, the plasmodium was able to solve the maze in one pass, extending a protoplasmic tube to the target.
The original recording was made at one frame per five minutes; this playback is 25-30 frames per second.
(Andrew Adamatzky, “Slime Mold Solves Maze in One Pass, Assisted by Gradient of Chemo-Attractants,” IEEE Transactions on Nanobioscience 11:2 [2012], 131-134.)
“It is interesting that most of the human race has a reserve of the enzyme necessary to render alcohol harmless to the body — as if nature meant us to drink alcohol, unlike animals to which alcohol is a poison.”
— BUPA News, 1982, quoted in Richard Gordon, Great Medical Mysteries, 2014
Number eight cells:
Now suppose we want to color each cell red or blue such that no three cells are in arithmetic progression — for example, we don’t want cells 1, 2, and 3 to be the same color, or 4, 6, and 8. With eight cells it’s possible to accomplish this:
But if we want to add a ninth cell we can’t avoid an arithmetic progression: If the ninth cell is blue then cells 1, 5, and 9 are evenly spaced, and if it’s red then cells 3, 6, and 9 are. Dutch mathematician B.L. van der Waerden found that there’s always such a limit: For any given positive integers r and k, there’s some number N such that if the integers {1, 2, …, N} are colored, each with one of r different colors, then there will be at least k integers in arithmetic progression whose elements are of the same color. Determining what this limit is (in this example it’s 9) is an open problem.
(Bonus: Alexej Kanel-Belov found this pretty theorem concerning divisibility of integer sums within an infinite grid — Martin J. Erickson, in Beautiful Mathematics, calls it a two-dimensional version of van der Waerden’s theorem.)
A stick is broken at random into 3 pieces. It is possible to put them together into the shape of a triangle provided the length of the longest piece is less than the sum of the other 2 pieces; that is, provided the length of the longest piece is less than half the length of the stick. But the probability that a fragment of a stick shall be half the original length of the stick is 1/2. Hence the probability that a triangle can be constructed out of the 3 pieces into which the stick is broken is 1/2.
— Samuel Isaac Jones, Mathematical Wrinkles, 1912
Pick a positive integer, list the positive integers that will divide it evenly, add these up, and subtract the number itself:
Now do the same with that number, and continue:
Many of these sequences arrive at some resolution — they terminate in a constant, or an alternating pair, or some regular cycle. But it’s an open question whether all of them do this. The fate of the aliquot sequence of 276 is not known; by step 469 it’s reached 149384846598254844243905695992651412919855640, but possibly it reaches some apex and then descends again and finds some conclusion (the sequence for the number 138 reaches a peak of 179931895322 but eventually returns to 1). Do all numbers eventually reach a resolution? For now, no one knows.
Another contribution from Lee Sallows:
“The smallest, oldest and most famous magic square of all is the specimen of Chinese origin known as the Lo shu. In this, the numbers from 1 to 9 are so placed that their sum taken in any row, column or diagonal is 15. This is another way of saying that the sum of any three of them lying in a straight line is 15. Less well known is the ‘Egyptian’ Lo shu (seen below) in which the same numbers are rearranged in a triangular formation that exhibits the same property.”
(From his book Geometric Magic Squares, 2013.) (Thanks, Lee.)
