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If a sphere is inscribed in a tetrahedron, and segments are drawn on each face connecting its vertices to the point of tangency, then the same three angles will appear on each face. And the 12 triangles produced will form congruent pairs across each edge of the tetrahedron. Discovered by A.S. Bang in 1897.

Into an empty vase drop balls numbered 1 to 10. Remove ball 1. Add balls numbered 11 to 20. Remove ball 2. Continue in this way, spending half an hour on the first transaction, 15 minutes on the next, and so on. After one hour all the transactions will be finished. Obviously, in the end the vase will contain infinitely many balls, since with...

He, then, who says that something true exists either only asserts that something true exists or proves it. And if he merely asserts it, he will be told the opposite of his mere assertion, namely, that nothing is true. But if he proves that something is true, he proves it either by a true proof or by one that is not true. But he...

From Bismarck’s Reflections and Reminiscences, 1898: At the time of my first stay at St. Petersburg, in 1859, I had an example of another Russian peculiarity. During the first spring days it was then the custom for every one connected with the Court to promenade in the Summer Garden between Paul’s Palace and the Neva. There the Emperor had noticed a sentry standing in...

The audience at … a play are spectators of a world they are not in. They see what they may well describe as, say, Othello in front of a certain palace in Venice … [b]ut they are not themselves at any specifiable distance from that palace. … I see Othello strangle Desdemona; but that will not entail that I, as part of my biography,...

A puzzle from the 18th century — punctuate this sentence so that it makes sense: King Charles the First walked and talked half an hour after his head was cut off.

Play with work blend, keep warmish feet, Away drive trouble, slowly eat; Air pure breathe, and early rise; Beware excess, take exercise. Exercise take, excess beware; Rise early and breathe pure air; Eat slowly; trouble drive away; Feet warmish keep, blend work with play. — “W.E.R.,” in Truth, Jan. 13, 1881

There are several simple little drawing tricks which the nurse may use to arouse the interest of her patient as she uses puzzles and catches. The oldest of these is by Hogarth and represents a soldier and his dog going through a doorway. As is seen by the diagram, it consists of three straight lines and one curved one. — William Rush Dunton, Occupation...

In March 1939, students at McGill University dictated this sentence to a dozen faculty members: Outside a cemetery sat a harassed cobbler and an embarrassed oculist, picknicking on a desiccated apple, and gazing at the symmetry of a lady’s ankle with unparalleled ecstasy. The participants included three English professors, the head of the journalism department, and a proofreading instructor. Only Esther Korstad, instructor of...

In the classic Indian rope trick, a rope rises into the sky, its end lost to view. A boy disappears up the rope, and when he fails to return the angry magician climbs up after him. Body parts fall to the ground, the magician descends and places the parts in a basket, and the boy reappears uninjured. This is all thought to be a...

Examination questions from the final Classical Honours School at Oxford University, 1899 — “to have passed through it was the hallmark of a superbly educated man, and its graduates went on to rule the nation and, in that heyday of British imperialism, half the world too”: Sketch the history of the Syracusan democracy between the fall of Thrasybulus in 466 B.C. and the accession...

“We went back to England together. When we arrived at the customs shed, Syrie said: ‘Always choose the oldest customs official. No chance of promotion.'” — Somerset Maugham, quoting his wife

In the King James Bible, Ezra 7:21 lacks only the letter J, and 1 Chronicles 12:40 lacks only Q. In 2016 Pharmacy Times judged talimogene laherparepvec the hardest drug name to pronounce. 34 × 72 × 875 = 3472875 (B.J. van der Zwaag) The Estonian word kuulilennuteetunneliluuk (“bullet tunnel hatch”) is a palindrome. “No one ever forgets where he buried the hatchet.” — Kin...

From Wikimedia user Cmglee, two digital arithmetic techniques: Above: To multiply a positive single-digit integer by 9, hold up your hands palm up, imagine the fingers numbered consecutively 1 to 10, and fold down the finger corresponding to the number to be multiplied (here, 8). The product is the two-digit number represented by the remaining two groups of fingers — here there are seven...

A duel was lately fought in Texas by Alexander Shott and John S. Nott. Nott was shot, and Shott was not. In this case it is better to be Shott than Nott. There was a rumor that Nott was not shot, and Shott avers that he shot Nott, which proves either that the shot Shott shot at Nott was not shot, or that Nott...

In Christopher Marlowe’s play Edward II, one of the king’s gaolers receives a message reading Edwardum occidere nolite timere bonum est. This can be read either as Edwardum occidere nolite; timere bonum est (“Do not kill Edward; it is good to be afraid [to do so]”) or as Edwardum occidere nolite timere; bonum est (“Do not be afraid to kill Edward; [to do so]...

“Stilpo having escaped the burning of his city, in which he had lost wife, children, and property, Demetrius Poliorcetes, seeing him unperturbed in expression amid the great ruin of his country, asked him if he had not suffered loss. He replied No, that thanks to God he had lost nothing of his own.” — Montaigne “In general, the greatest reverses of fortune are the...

According to the Guardian (March 1872), Lord Palmerston once dictated this sentence to 11 British cabinet ministers, “not one of whom, it is said, spelled it correctly”: It is disagreeable to witness the embarrassment of a harassed peddler gauging the symmetry of a peeled potato. “And Lord R. Cecil, in the House of Commons, some time ago, quoted the following lines which he said...

An episode from P.T. Barnum’s childhood in Bethel, Connecticut: ‘What is the price of razor strops?’ inquired my grandfather of a peddler, whose wagon, loaded with Yankee notions, stood in front of our store. ‘A dollar each for Pomeroy’s strops,’ responded the itinerant merchant. ‘A dollar apiece!’ exclaimed my grandfather; ‘they’ll be sold for half the money before the year is out.’ ‘If one...

Punctuate this expression so that it makes sense: Time flies you cannot they pass at such irregular intervals.

If Socrates was born, Socrates became either when Socrates existed not or when Socrates already existed; but if he shall be said to have become when he already existed, he will have become twice; and if when he did not exist, Socrates was both existent and non-existent at the same time — existent through having become, non-existent by hypothesis. And if Socrates died, he...

A square can be divided into an even number of triangles of equal area — but not into an odd number.

A puzzle by Virginia McCarthy: A 20-story office building has one busy elevator. On a recent trip upward, the elevator transported seven passengers. Each of these went up at least two floors, but no two of them went up the same number of floors. On this one upward journey, no more than one passenger crossed the elevator’s threshold on any given floor; no passenger...

Every unjust act is inexpedient; No unjust act is expedient; No expedient act is unjust; Therefore every expedient act is just. — Ralph L. Woods, How to Torture Your Mind, 1969

He was prettily and fantastically troubled, who, having used to put his trust in dreams, one night dreamed that all dreams were vain; for he considered, if so, then this was vain, and then dreams might be true for all this: but if they might be true, then this dream might be so upon equal reason: and then dreams were vain, because this dream...

This curious epitaph is found at Martham Church in Norfolk: Here Lyeth the Body of Christr. Burraway, who departed this Life ye 18 day of October, Anno Domini 1730. Aged 59 years. And there Lyes ☞ Alice who by his Life Was my Sister, my mistress My mother and my wife. Dyed Feb. ye 12. 1729 Aged 76 years. According to Thomas Joseph Pettigrew...

The Moscow Radio announced that five million Russians filed past Josef Stalin’s bier in 72 hours. That means, according to the calculations of Frank Baker, a Mangum, Okla., accountant, that the mourners, two abreast, three and one third feet apart, ran past the bier at 22 miles an hour. Twenty-two miles an hour is 9.3 seconds a hundred yards, which is the world’s record...

“If the northern hemisphere were land, and all the southern hemisphere water, ought we to call the northern hemisphere an island, or the southern hemisphere a lake?” — Augustus De Morgan

A “simple but pretty little puzzle” by Henry Dudeney: If this flag measures 4 feet by 3 feet and presents equal areas of red and white bunting, how wide are the cross’s arms?

The French phrase ouate de phoque (“seal’s cotton wool”) sounds like what the fuck in English. For whatever that’s worth. Listen to the pronunciation file here.

Because it depends partly on the phase of the moon, the date of Easter Sunday can vary by as much as 35 days. In 1928, the Parliament of the United Kingdom passed an act that would establish its date as the Sunday following the second Saturday in April (and thus always falling between April 9 and 15). Though passed and enacted, the act has...

A puzzle by Russian mathematician Sergei Berlov: Here are two decagons. Suppose that a positive integer is written at each of the 20 vertices such that the sum of the numbers around each decagon is 99. Prove that it’s possible to mark some number of successive vertices on each decagon so that the two sums of marked numbers are equal. (The marked segment on...

I just found this in a 1950 issue of Pi Mu Epsilon Journal: Question: How many legs has a horse? Answer: Twelve; two in front, two behind, two on each side, and one in each corner. Later in the same volume, Marlow Sholander shows that in fact every horse has an infinite number of legs. That’s to say nothing of color. And see Chasing...

Suppose your dog eats the ace of spades and you’re forced to play poker with a 51-card deck. Does this make it more or less likely that you’ll be dealt a two-pair hand? The answer is not immediately clear. The loss of the ace means that fewer two-pair hands are possible, but the total universe of possible five-card hands is also reduced. Surprisingly, the...

Demetri Martin composed this palindromic poem as part of a project for a fractal geometry class at Yale in spring 1993. The first two and last two lines are palindromes, the middle line (“Be still if I fill its ebb”) minus its last letter is a palindrome, and the entire poem is a palindrome: Dammit I’m mad Evil is a deed as I live....

An odd incident from the memoir of Lt. John Worsham of the 21st Virginia Infantry — during the Battle of Saunders Field in May 1864, two opposing soldiers found themselves in a gully and agreed to a fistfight to decide which had captured the other: Then they decided that they would go into the road and have a regular fist and skull fight, the...

These verses can be interpreted to support either the Stuarts or the Hanovers, according as they’re read. If each is addressed separately, from top to bottom, they’ll seem to support the Hanoverian regime; read together, right across the page, they declare for the Stuarts: From Reuben Percy, Relics of Literature, 1823.

In this conundrum by Mitsunobu Matsuyama, when four congruent quadrilaterals are rotated about their centers, their area seems to increase. What has become of the red square? The answer is that each side of the large square is slightly shorter after the rotation. If θ is the angle between two opposing sides in each quadrilateral, then the ratio of the areas of the two...

J. Bryan III used to challenge his friends to think of an English word that contains either of the following strings of letters: ILIL ILIW “I’ve never played the game with anyone who could solve either of these combinations.” Can you?

“Improprieties in pronunciation” among the people of New England in 1808, collected by Caleb Bingham and published in The Child’s Companion: afraid: afeard apron: apun audacious: outdacious bonfire: burnfire brittle: brickle cards: cairds caught: cotch coin: quine cucumber: cowcumber dictionary: dixonary drain: dreen earth: airth fanciful: fancical five pence: fippence gown: gound grasshopper: hoppergrass jaundice: janders musician: musicianer poplar: popple quart: quairt quotient: coshun...

How can we make sense of this passage, composed by Willard R. Espy? The Optic Reed has a Haste Harm. It Hose, one assumes, not to be Lever; it neither Heats the Ripple, nor Hides the Razed sinner for his Rude Rime; it is not Old. Then Leave to it; ere thy grave Loses, thou mayst find here the Hart to the Rest thou...

A circle is divided into six sectors, into which are written (say, counterclockwise) these numbers: 1, 0, 1, 0, 0, 0 You can increase any two neighboring numbers by 1. By doing this repeatedly, is there a way to make all six numbers equal?

A problem by Nikolai Rozov: “If the brainteaser you solved before you solved this one was harder than the brainteaser you solved after you solved the brainteaser that you solved before you solved this one, was the brainteaser you solved before you solved this one harder than this one?”

This remarkable phenomenon was discovered by Cambridge mathematician James Grime. Number five six-sided dice as follows: A: 2, 2, 2, 7, 7, 7 B: 1, 1, 6, 6, 6, 6 C: 0, 5, 5, 5, 5, 5 D: 4, 4, 4, 4, 4, 9 E: 3, 3, 3, 3, 8, 8 Now, on average: A beats B beats C beats D beats E beats...

Excerpts from letters received by the British pensions office, quoted in George Lyttelton’s Commonplace Book, 2002: Mrs R. has no clothes, has not had any for many years. The clergy have been visiting her. In reply to your letter, I have already cohabited with your officers, so far without any result. You have changed my little boy to a little girl. Will this make...

By Wikimedia user Efbrazil. Fill in four squares to make a continuous dark path connecting the stars. No diagonal moves are allowed.

When I went out to throw the discus I went and sprained some little viscus. Since that disruption of my viscera I don’t go out and throw the discera. — Don Laycock

subagitate v. to have sex with verecund adj. bashful; modest reme v. to cry or call out cacoëpistic adj. badly pronounced [Sir Walter Raleigh] loved a wench well; and one time getting up one of the Mayds of Honour up against a tree in a Wood (’twas his first Lady) who seemed at first boarding to be something fearfull of her Honour, and modest,...