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
Prove that the product of four consecutive positive integers cannot be a perfect square.
A curious effect produced by lightning is described to us by Dr. Enfield, writing from Jefferson, Iowa, U.S. A house which he visited was struck by lightning so that much damage was done. After the occurrence, a pile of dinner plates, twelve in number, was found to have every other plate broken. It would seem as if the plates constituted a condenser under the intensely electrified condition of the atmosphere. The particulars are, however, so meagre that it is difficult to decide whether the phenomenon was electrical or merely mechanical.
– Nature, June 12, 1902
In an anonymous letter to the London Times in 1825, Thomas Steele of Magdalen College, Cambridge, proposed enshrining Isaac Newton’s residence in a stepped stone pyramid surmounted by a vast stone globe. The physicist himself had died more than a century earlier, in 1727, and lay in Westminster Abbey, but Steele felt that preserving his home would produce a monument “not unworthy of the nation and of his memory”:
When travelling through Italy, I was powerfully struck by the unique situation and singular appearance of the Primitive Chapel at Assisi, founded by St. Francis.
As you enter the porch of the great Franciscan church, you view before you this small cottage-like chapel, standing directly under the dome, and perfectly isolated.
Now, Sir, among the many splendid improvements which are making in the capital, would it not be a noble, and perhaps the most appropriate, national monument which could be erected, if an azure hemispherical dome, or what would be better, a portion of a sphere greater than a hemisphere, supported on a massive base, were to be reared, like that of Assisi, over the house and observatory of the writer of the Principia?
The house might be fitted up in such a manner as to contain a council-chamber and library for the Royal Society; and it is perhaps not unworthy of being remarked, that it is not more than about two hundred yards distant from the University Club House.
Protected, by the means which I have described, from the dilapidating influence of rains and winds, the venerable edifice in which Newton studied, or was inspired, — that ‘palace of the soul,’ might stand fast for ages, a British monument more sublime than the Pyramids, though remote antiquity and vastness be combined to create their interest.
Steele wasn’t an architect, and he left the details to others, but he was imagining something enormous: In a subsequent letter to the Mechanics’ Magazine he wrote that “the base of my design [appears] to coincide with the base of St. Paul’s (a sort of crude coincidence of course, in consequence of the angle at the transept), and that the highest point of my designed building should, at the same time, appear to coincide with a point of the great tower of the cathedral, about 200 feet high — the height of the building which I propose to have erected.”
The plan never went forward, but the magazine endorsed the idea: “We need scarcely add, that there is no description of embellishment which might not be with ease introduced into the structure, so as to render it as perpetual a monument to the taste as it would be to the national spirit and gratitude of the British people.”
Future president Herbert Hoover published a surprising title in 1912: An English translation of the 16th-century mining textbook De Re Metallica, composed originally by Georg Bauer in 1556. Bauer’s book had remained a classic work in the field for two centuries, with some copies deemed so valuable that they were chained to church altars, but no one had translated the Latin into good modern English. Biographer David Burner wrote, “Hoover and his wife had the distinct advantage of combining linguistic ability with mineralogical knowledge.”
Hoover, a mining engineer, and his wife Lou, a linguist, spent five years on the project, visiting the areas in Saxony that Bauer had described, ordering translations of related mining books, and spending more than $20,000 for experimental help in investigating the chemical processes that the book described.
The Hoovers offered the 637-page work, complete with the original woodcuts, to “strengthen the traditions of one of the most important and least recognized of the world’s professions.” Of the 3,000 copies that were printed, Hoover gave away more than half to mining engineers and students.
All the statements below this one are false.
All the statements below this one are false.
All the statements below this one are false.
All the statements below this one are false.
All the statements below this one are false.
These statements can’t all be false, because that would make the first one true, a contradiction. But neither can any one of them be true, as a true statement would have to be followed by an infinity of false statements, and the falsity of any one of them implies the truth of some that follow. Thus there’s no consistent way to assign truth values to all the statements.
This is reminiscent of the well-known liar paradox (“This sentence is false”), except that none of the sentences above refers to itself. MIT philosopher Stephen Yablo uses it to show that circularity is not necessary to produce a paradox.
A remarkable spelling trick by American magician Howard Adams:
From a deck of cards choose five cards and their mates. A card’s mate is the card of the same value and color; for example, the mate of the five of clubs is the five of spades.
Arrange the cards in the order ABCDEabcde, where ABCDE are the chosen cards and abcde are the mates. Cut this packet as many times as you like, then deal five cards onto the table, reversing their order. Place the remaining five cards beside them in a second pile.
Now spell the phrase LAST TWO CARDS MATCH. As you say “L,” choose either pile at random and transfer a card from the top to the bottom. Do the same for A, S, and T. Now remove the top card from each pile and set them aside as a pair.
Perform the same procedure as you spell TWO, CARDS, and MATCH. When you’re finished, two cards will remain on the table. Not only do these cards match, but so do each of the other pairs!
Using a 7-quart and a 3-quart jug, how can you obtain exactly 5 quarts of water from a well?
That’s a water-fetching puzzle, a familiar task in puzzle books. Most such problems can be solved fairly easily using intuition or trial and error, but in Scripta Mathematica, March 1948, H.D. Grossman describes an ingenious way to generate a solution geometrically.
Let a and b be the sizes of the jugs, in quarts, and c be the number of quarts that we’re seeking. Here, a = 7, b = 3, and c = 5. (a and b must be positive integers, relatively prime, where a is greater than b and their sum is greater than c; otherwise the problem is unsolvable, trivial, or can be reduced to smaller integers.)
Using a field of lattice points (or an actual pegboard), let O be the point (0, 0) and P be the point (b, a) (here, 3, 7). Connect these with OP. Then draw a zigzag line Z to the right of OP, connecting lattice points and staying as close as possible to OP. Now “It may be proved that the horizontal distances from OP to the lattice-points on Z (except O and P) are in some order without repetition 1, 2, 3, …, a + b – 1, if we count each horizontal lattice-unit as the distance a.” In this example, if we take the distance between any two neighboring lattice points as 7, then each of the points on the zigzag line Z will be some unique integer distance horizontally from the diagonal line OP. Find the one whose distance is c (here, 5), the number of quarts that we want to retrieve.
Now we have a map showing how to conduct our pourings. Starting from O and following the zigzag line to C:
- Each horizontal unit means “Pour the contents of the a-quart jug, if any, into the b-quart jug; then fill the a-quart jug from the well.”
- Each vertical unit means “Fill the b-quart jug from the a-quart jug; then empty the b-quart jug.”
So, in our example, the map instructs us to:
- Fill the 7-quart jug.
- Fill the 3-quart jug twice from the 7-quart jug, each time emptying its contents into the well. This leaves 1 quart in the 7-quart jug.
- Pour this 1 quart into the 3-quart jug and fill the 7-quart jug again from the well.
- Fill the remainder of the 3-quart jug (2 quarts) from the 7-quart jug and empty the 3-quart jug. This leaves 5 quarts in the 7-quart jug, which was our goal.
You can find an alternate solution by drawing a second zigzag line to the left of OP. In reading this solution, we swap the roles of a and b given above, so the map tells us to fill the 3-quart jug three times successively and empty it each time into the 7-quart jug (leaving 2 quarts in the 3-quart jug the final time), then empty the 7-quart jug, transfer the remaining 2 quarts to it, and add a final 3 quarts. “There are always exactly two solutions which are in a sense complementary to each other.”
Grossman gives a rigorous algebraic solution in “A Generalization of the Water-Fetching Puzzle,” American Mathematical Monthly 47:6 (June-July 1940), pp. 374-375.
In 1991, David Collison sent this figure to Canadian magic-square expert John Henricks, with no explanation, and then died.
It’s believed to be the first odd-ordered bimagic square ever discovered. Each row, column, and diagonal produces a sum of 369. The square remains magic if each number is squared, with a magic sum of 20,049.
No one knows how Collison created it.
UPDATE: Wait, Collison’s wasn’t the first — G. Pfeffermann published a 9th-order bimagic square as a puzzle in Les Tablettes du Chercheur in 1891. (Thanks, Baz.)
On June 28, 2009, Stephen Hawking hosted a party for time travelers, but he sent out the invitations only afterward.
No one turned up.
He offered this as experimental evidence that time travel is not possible.
“I sat there a long time,” he said, “but no one came.”
- To frustrate eavesdroppers, Herbert Hoover and his wife used to converse in Chinese.
- Asteroids 30439, 30440, 30441, and 30444 are named Moe, Larry, Curly, and Shemp.
- COMMITTEES = COST ME TIME
- 15618 = 1 + 56 – 1 × 8
- How is it that time passes but space doesn’t?
Webster’s Third New International Dictionary gives no pronunciation for YHWH.
The tippe top is a round top that, when spun, tilts to one side and leaps up onto its stem. This is perplexing, as the toy appears to be gaining energy — its center of mass rises with the flip.
How is this possible? The geometrical center of the top is higher than its center of mass. As the toy begins to topple to one side, friction with the underlying surface produces a torque that kicks it up onto its stem. It does gain potential energy, but it loses kinetic energy — in fact, during the inversion it actually reverses its direction of rotation.
Entire treatises have been written on the underlying physics, and the toy has occupied at least two Nobel Prize winners — below, Wolfgang Pauli and Niels Bohr play with one at the inauguration of the Institute of Physics at Lund, Sweden, in July 1954.
German astronomer Karl Reinmuth discovered and named more than 400 asteroids. Among them are these eight:
Their initials spell G. STRACKE, for Gustav Stracke, a fellow astronomer who had asked that no planet be named after him. In this way Reinmuth could honor his colleague without contradicting his wish.
University of Michigan mathematician Norman Anning offered this “non-commutative soliloquy of an introspective epistemologist” in Scripta Mathematica in 1948:
[(N + H)ow + (T + W)hat](I know).
Expand the expression and you get Now I know how I know that I know what I know.
Astronaut John Young smuggled a corned beef sandwich into space. As Gemini 3 was circling Earth in March 1965, Young pulled the sandwich out of his pocket and offered it to Gus Grissom:
Grissom: What is it?
Young: Corned beef sandwich.
Grissom: Where did that come from?
Young: I brought it with me. Let’s see how it tastes. Smells, doesn’t it?
Grissom: Yes, it’s breaking up. I’m going to stick it in my pocket.
Young: Is it? It was a thought, anyway.
“Wally Schirra had the sandwich made up at a restaurant at Cocoa Beach a couple of days before, and I hid it in a pocket of my space suit,” Young explained later. “Gus had been bored by the official menus we’d practiced with in training, and it seemed like a fun idea at the time.”
Grissom wrote, “After the flight our superiors at NASA let us know in no uncertain terms that non-man-rated corned beef sandwiches were out for future space missions. But John’s deadpan offer of this strictly non-regulation goodie remains one of the highlights of our flight for me.”
- Colombia is the only South American country that borders both the Atlantic and the Pacific.
- GRAVITATIONAL LENS = STELLAR NAVIGATION
- 28671 = (2 / 8)-6 × 7 – 1
- Can a man released from prison be called a freeee?
- “Nature uses as little as possible of anything.” — Johannes Kepler
Sergei Prokofiev died on the same day that Joseph Stalin’s death was announced. Moscow was so thronged with mourners that three days passed before the composer’s body could be removed for a funeral service.
In 1894, Indiana physician Edwin J. Goodwin published a one-page article in the American Mathematical Monthly claiming to have found a method of squaring the circle — that is, of constructing a square with the same area as a given circle using only a compass and straightedge, a task known to be impossible. He proposed a bill to state representative Taylor I. Record, laying out the “new mathematical truth” and offering it “as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the legislature of 1897.”
Apparently flummoxed, the House referred the bill to its Committee on Swamp Lands, which transferred it to the Committee on Education … which approved it. Whereupon the whole house passed it unanimously.
The bill, which the Indianapolis Journal was already calling “the strangest bill that has ever passed an Indiana Assembly,” moved on to the senate, which referred it the Committee on Temperance. (Chronicler Will E. Edington writes, “One wonders whether this was done intentionally, for certainly the bill could have been referred to no committee more appropriately named.”) Equally flummoxed, the committee recommended that it pass.
The bill might have achieved full passage had not Purdue mathematics professor C.A. Waldo happened to be visiting the House that day. “A member … showed the writer a copy of the bill just passed and asked him if he would like an introduction to the learned doctor,” Waldo later recalled in the Proceedings of the Indiana Academy of Science. “He declined the courtesy with thanks, remarking that he was acquainted with as many crazy people as he cared to know.”
That did it. “Representative Record’s mathematical bill legalizing a formula for squaring the circle was brought up and made fun of,” reported by Indianapolis News on Feb. 13. “The Senators made bad puns about it, ridiculed it and laughed over it. The fun lasted half an hour. Senator Hubbell said that it was not meet for the Senate, which was costing the State $250 a day, to waste its time in such frivolity.”
“Senator Hubbell characterized the bill as utter folly,” added the Indianapolis Journal. “The Senate might as well try to legislate water to run up hill as to establish mathematical truth by law.”
In describing a large water basin, 2 Chronicles 4:2 reads, “Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.” A similar verse appears at 1 Kings 7:23.
Critics point out that this implies that π is 3, and in 1983 about 100 professors and students at Emporia State University in Kansas founded an Institute of Pi Research to lobby (wryly) for adopting this new value in place of the awkward 3.14159 …
“To think that God in his infinite wisdom would create something as messy as this is a monstrous thought,” medieval historian Samuel Dicks told the Kansas City Times. “I think we deserve to be taken as seriously as the creationists.”
“If the Bible is right in biology, it’s right in math,” added economic historian Loren Pennington.
But writing in the Mathematical Gazette in 1985, M.D. Stern of Manchester Polytechnic noted (also wryly) that the word translated as line above is transliterated qwh but read qw. Further, the ancient Greeks and Jews used letters to denote numbers, with the letters q, w, and h taking the numerical values 100, 6, and 5.
“Thus the word translated line in its written form has numerical value 111 whereas as read the value is 106. If we take the ratio of these numbers as a correcting factor for the apparent value of π as 3 and calculate 3 × (111/106), we obtain 3.141509 to 7 significant figures. This differs from the true value of π by less than 10-4 which is remarkable. In view of this, it might be suggested that this peculiar spelling is of more significance than a cursory reading might have suggested.”
Remove any nine cards from an ordinary deck, shuffle them, and deal them face down into three piles. Choose any pile and note its bottom card. Then assemble the three piles into one, being sure to place the chosen pile on top.
Suppose the card you chose is the three of spades. Spell T-H-R-E-E, dealing one card face down onto the table with each letter. Place the remaining cards on top of these five and take up the whole packet. Now spell O-F, and again place the remaining cards on top of these two. Then spell S-P-A-D-E-S and place the remaining cards on top.
Now pick up the packet and spell M-A-G-I-C, dealing the final card face up. It’s the three of spades.
Remarkably, this trick will produce any card, from the 10-letter ace of clubs to the 15-letter queen of diamonds. It was invented by California magician Jim Steinmeyer and appears in his 2002 book Impuzzibilities (used by permission).
- Mississippi didn’t ratify the 13th Amendment, abolishing slavery, until 2013.
- To protect its ecosystem, the location of Hyperion, the world’s tallest living tree, is kept secret.
- 34425 = 34 × 425
- CIRCUMSTANTIAL EVIDENCE = ACTUAL CRIME ISN’T EVINCED
- “Well, if I called the wrong number, why did you answer the phone?” — James Thurber
Lewis Carroll on the perils of physics:
Suppose a solid held above the surface of a liquid and partially immersed: a portion of the liquid is displaced, and the level of the liquid rises. But, by this rise of level, a little bit more of the solid is of course immersed, and so there is a new displacement of a second portion of the liquid, and a consequent rise of level. Again, this second rise of level causes a yet further immersion, and by consequence another displacement of liquid and another rise. It is self-evident that this process must continue till the entire solid is immersed, and that the liquid will then begin to immerse whatever holds the solid, which, being connected with it, must for the time be considered a part of it. If you hold a stick, six feet long, with its end in a tumbler of water, and wait long enough, you must eventually be immersed. The question as to the source from which the water is supplied — which belongs to a high branch of mathematics, and is therefore beyond our present scope — does not apply to the sea. Let us therefore take the familiar instance of a man standing at the edge of the sea, at ebb-tide, with a solid in his hand, which he partially immerses: he remains steadfast and unmoved, and we all know that he must be drowned.
“The multitudes who daily perish in this manner to attest a philosophical truth, and whose bodies the unreasoning wave casts sullenly upon our thankless shores, have a truer claim to be called the martyrs of science than a Galileo or a Kepler.”
Frustrated in trying to describe higher topology abstractly to students, Xian Wang invented a model train that can hug either side of a track:
It is therefore a primary object of the present invention to provide an electrically-operated ornament travelling on a rail which can be used to explain the Mobius Theorem. … In general textbooks, this advanced mathematic rule is usually explained by demonstrating a body circularly moving on a front and a reverse side of a twisted two-ends-connected belt. Most people can not understand and imagine the theorem from such explanation and demonstration.
Of course, once you’ve built one you can put it to other uses:
This is the opening page of “The Metamorphosis,” from Vladimir Nabokov’s teaching copy. Kafka’s novella held a special interest for Nabokov, who was a trained entomologist. From his lecture notes at Cornell:
Now, what exactly is the ‘vermin’ into which poor Gregor, the seedy commercial traveler, is so suddenly transformed? It obviously belongs to the branch of ‘jointed leggers’ (Arthropoda), to which insects, and spiders, and centipedes, and crustaceans belong. … Next question: What insect? Commentators say cockroach, which of course does not make sense. A cockroach is an insect that is flat in shape with large legs, and Gregor is anything but flat: he is convex on both sides, belly and back, and his legs are small. He approaches a cockroach in only one respect: his coloration is brown. That is all. Apart from this he has a tremendous convex belly divided into segments and a hard rounded back suggestive of wing cases. In beetles these cases conceal flimsy little wings that can be expanded and then may carry the beetle for miles and miles in a blundering flight. … Further, he has strong mandibles. He uses these organs to turn the key in a lock while standing erect on his hind legs, on his third pair of legs (a strong little pair), and this gives us the length of his body, which is about three feet long. … In the original German text the old charwoman calls him Mistkafer, a ‘dung beetle.’ It is obvious that the good woman is adding the epithet only to be friendly. He is not, technically, a dung beetle. He is merely a big beetle.
“Curiously enough,” he added, “Gregor the beetle never found out that he had wings under the hard covering of his back. This is a very nice observation on my part to be treasured all your lives. Some Gregors, some Joes and Janes, do not know that they have wings.”