A brainteaser from the Soviet science magazine Kvant, via Quantum, January/February 1991:

Bobby found the sum of three consecutive integers, then of the next three consecutive integers, then multiplied these two sums together. Could the product have been 111,111,111?

Click for Answer

Tale Spinner

William Wallace Cook (1867-1933) claimed to have worn out 25 typewriters in as many years turning out hundreds of nickel and dime novels, all of them written in the same format, 40,000 words divided into 16 chapters of five single-spaced pages each. At the end of his career he published his system for generating plots, billed as “Plotto, an invention which reduces literature to an exact science.”

The “invention” is really a list of story ideas, all molded to Cook’s basic notion of a plot: “Purpose, opposed by Obstacle, yields Conflict.” The protagonist wants to find happiness in love and courtship, married life, or enterprise; he encounters a conflict and must reach a resolution. What makes the book fun is the absurd specificity of some of the ideas. Here’s an example:

(b) (1083)(1287)
A has invented a life preserver for the use of shipwrecked persons*
A, in order to prove the value of the life preserver he has invented, dons the rubber suit, inflates it and secretly, by night, drops overboard from a steamer on the high seas.** (1414b) (1419b)

The numbers refer to elements that might be varied, to related plots, and to character types that might figure in the story. Varying the combinations might produce several million different stories. This is certainly formulaic, but, Cook said, “There are any number of highbrow authors who will ridicule this invention in public and use it in private.” (In fact both Alfred Hitchcock and Erle Stanley Gardner admitted in interviews that they’d read the book, which went through multiple editions.)

The numbered master list gives 1,462 plots, all linked with character symbols and apparently all thought up by the author. The full text is on the Internet Archive.

More Magic

Albrecht Dürer’s 1514 engraving Melencolia I includes this famous magic square: The magic sum of 34 can be reached by adding the numbers in any row, column, diagonal, or quadrant; the four center squares; the four corner squares; the four numbers clockwise from the corners; or the four counterclockwise.

In Power Play (1997), University of Toronto mathematician Ed Barbeau points out that there’s even more magic when we consider squares and cubes. Take the numbers in the first two and the last two rows:

16 + 3 + 2 + 13 + 5 + 10 + 11 + 8 = 9 + 6 + 7 + 12 + 4 + 15 + 14 + 1

162 + 32 + 22 + 132 + 52 + 102 + 112 + 82 = 92 + 62 + 72 + 122 + 42 + 152 + 142 + 12

Or alternate columns:

16 + 5 + 9 + 4 + 2 + 11 + 7 + 14 = 3 + 10 + 6 + 15 + 13 + 8 + 12 + 1

162 + 52 + 92 + 42 + 22 + 112 + 72 + 142 = 32 + 102 + 62 + 152 + 132 + 82 + 122 + 12

Most amazingly, if you compare the numbers on and off the diagonals, this works with both squares and cubes:

16 + 10 + 7 + 1 + 13 + 11 + 6 + 4 = 2 + 3 + 5 + 8 + 9 + 12 + 14 + 15

162 + 102 + 72 + 12 + 132 + 112 + 62 + 42 = 22 + 32 + 52 + 82 + 92 + 122 + 142 + 152

163 + 103 + 73 + 13 + 133 + 113 + 63 + 43 = 23 + 33 + 53 + 83 + 93 + 123 + 143 + 153


In his 2014 book Describing Gods, Graham Oppy presents the “divine liar” paradox, by SUNY philosopher Patrick Grim:

1. X believes that (1) is not true.

If we suppose that (1) is true, then this tells us that X believes that (1) is not true. But if an omniscient being believes that (1) is not true, then it follows that (1) is not true. So the assumption that (1) is true leads to a contradiction.

Suppose instead that (1) is not true. That is, suppose that it’s not the case that X believes that (1) is not true. If an omniscient being fails to believe that (1) is not true, then it’s not true that (1) is not true. So this alternative also leads to a contradiction.

But, on the assumption that there is an omniscient being X, either it’s the case that (1) is true or it’s the case that (1) is not true.

“So, on pain of contradiction,” Oppy explains, “we seem driven to the conclusion that there is no omniscient being X.”

(Also: Patrick Grim, “Some Neglected Problems of Omniscience,” American Philosophical Quarterly 20:3 [July 1983], 265-276.)

Say It With Flowers

The lost art of floriography assigned meanings to flowers so that lovers could exchange messages with “talking bouquets.” In his 1839 Language of Flowers, English journalist Frederic Shoberl rendered an entire verse by French poet Évariste de Parny as the combination of 16 flowers:

Aimer est un destin charmant,
C’est un bonheur qui nous enivre,
Et qui produit l’enchantement.
Avoir aimé, c’est ne plus vivre,
Hélas! c’est avoir acheté
Cette accablante vérité,
Que les serments sont un mensonge,
Que l’amour trompe tôt ou tard,
Que l’innocence n’est qu’un art,
Et que le bonheur n’est qu’un songe.

“It may be thus rendered: ‘To love is a pleasure, a happiness, which intoxicates; to love no longer, is to live no longer; it is to have bought this sad truth, that innocence is falsehood, that love is an art, and that happiness is a dream.'”

Pillow Verse

English textile designer William Morris always said he wanted to dream a poem. When he finally did and was asked whether he could remember it, he said, “Only the first line, and it went like this: The moonlight slept on a treacle sea.

Archbishop Edward Benson told Edmund Gosse that he dreamed he had been appointed poet laureate and found himself reciting this couplet to the queen:

Your latest atmosphere device
Is all composed of dust and lice.

And Sir John Squire confessed that when he dreamed the following lines they seemed impressive until he woke up:

There was a boy grew twenty inch, yes,
Twenty inch a year,
It might have made his mother flinch, but
She was quite a dear;
Yes, she was excellent,
And she was well content
To watch her offspring forge ahead in his
Peculiar sphere.

(From Stephen Brook, ed., The Oxford Book of Dreams, 1983. See Night Work.)

Line Work

bilinski problem

Robert Bilinski proposed this problem in the April 2006 issue of Crux Mathematicorum. On square ABCD, two equilateral triangles are constructed, ABE internally and BCF externally, as shown. Prove that D, E, and F are collinear.

Click for Answer


“It was the irony. It was the same irony that caused me to think, pause, and just inwardly chuckle, just momentarily, that, God, here are two guys further away from home … than two guys had ever been, but there are more people watching us than anybody else has ever watched two people before in history.” — Buzz Aldrin

Evolution Without Life

Philosopher Mark Bedau points out that, even on a dead planet, the microscopic crystallites that make up clay and mud seem to have the flexibility to adapt and evolve by natural selection:

  • Crystals reproduce in the sense that when they become large enough they cleave and pieces break off, becoming seeds for new crystals. A population of reproducing crystals can become the setting for crystal “evolution.”
  • All crystals have flaws, which are reproduced randomly and can become a source of novel information that gets expressed in “phenotypic” traits such as shape, growth rate, and the conditions that cause cleaving.
  • Each new layer of crystals copies the geometrical arrangement of atoms in the layer below, and defects can be copied in the same way. Grain boundaries and dislocations in one crystal tend to be copied to its “offspring” and subsequent generations. Over time these “mutations” can produce “species” among which nature selects.
  • A crystal’s shape, growth rate, and cleaving conditions all affect the rate at which it proliferates, and crystals with different properties will disperse and diffuse differently, which affects the rate at which they reproduce.

So a population of crystals can exhibit reproduction, variation, heredity, and adaptivity. “What is rather surprising is that, in the process, the planet remains entirely devoid of life. Thus, natural selection can take place in an entirely inorganic setting.”

(Mark Bedau, “Can Biological Teleology Be Naturalized?” Journal of Philosophy 88:11 [November 1991], 647-655. I think A.G. Cairns-Smith originated this idea in Seven Clues to the Origin of Life in 1985, and Richard Dawkins took it up in The Blind Watchmaker the following year.)