Black and White

maximov chess problem

By Nikolai Maximov. White to mate in two moves.

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The Pythagoras Tree

Draw a square and perch two smaller squares above it, forming a right triangle:

Now perch still smaller squares upon these, and continue the pattern recursively:

Charmingly, if you keep this up you’ll grow a tree:

It was dubbed the Pythagoras tree by Albert Bosman, the Dutch mathematics teacher who discovered the figure in 1942. (Each trio of squares demonstrates the Pythagorean theorem.)

At first it looks as though the tree must grow without bound, but in fact it’s admirably tidy: Because the squares eventually begin to overlap one another, a tree sprouted from a unit square will confine itself to a rectangle measuring 6 by 4.


The epilogue of The Time Machine contains this strange passage:

One cannot choose but wonder. Will he ever return? It may be that he swept back into the past, and fell among the blood-drinking, hairy savages of the Age of Unpolished Stone; into the abysses of the Cretaceous Sea; or among the grotesque saurians, the huge reptilian brutes of the Jurassic times. He may even now — if I may use the phrase — be wandering on some plesiosaurus-haunted Oolitic coral reef, or beside the lonely saline lakes of the Triassic Age.

What indeed can “now” mean in this context? If the Time Traveller’s life ended on a prehistoric beach, argues philosopher Donald C. Williams, then surely this became an established fact on the day that it happened. If the concept of time is to have any coherence, then history is a tapestry that is eternal and unchanging; to say that it can be changed “at” some future moment seems to be a flat contradiction. “At” where?

“Time travel,” Williams writes, “is analyzable either as the banality that at each different moment we occupy a different moment from the one we occupied before, or the contradiction that at each different moment we occupy a different moment from the one which we are then occupying — that five minutes from now, for example, I may be a hundred years from now.”

(Donald C. Williams, “The Myth of Passage,” Journal of Philosophy, July 1951.)


In order to restore Shakespeare to popularity in the 1930s, the theater critic and satirist A.E. Wilson suggested getting Noël Coward to rewrite Romeo and Juliet:

Julia (sweetly): O, Ro, must you be going? It isn’t four o’clock yet. Another cocktail, darling?
Romeo: Thanks.
Julia: And anyway, don’t be stupid, darling. That wasn’t the lark, silly. It was the thingummyjig, believe me.
Romeo: Rot; it was the lark. The beastly thing’s always singing at this devastating hour of the morning. And it’s getting light and I’d rather leave and live than be caught by your beastly husband and kicked out.
Julia (yawning): Oh, very well, then. Have it your own way, darling.
Romeo: Beastly fag getting up. I’ll stay. Give me another cocktail.
Julia: Sweetest.
Romeo (drinking cocktail): Angel face. (A pause.) But it wasn’t a nightingale.
Julia: It was.
Romeo: Oh, do shut up talking about it. You make me sick.
Julia (sweetly insistent): But dearest, it was the nightingale.
Romeo: Oh, what does it matter, you ass. Let’s get back to bed and forget it. (They go.)

(From Gordon Snell, The Book of Theatre Quotes, 1982.)

Dueling Pennies

A certain strange casino offers only one game. The casino posts a positive integer n on the wall, and the customer flips a fair coin repeatedly until it falls tails. If he has tossed n – 1 times, he pays the house 8n – 1 dollars; if he’s tossed n + 1 times, the house pays him 8n dollars; and in all other cases the payoff is zero.

The probability of tossing the coin exactly n times is 1/2n, so the customer’s expected winnings are 8n/2n + 1 – 8n – 1/2n – 1 = 4n – 1 for n > 1, and 2 for n = 1. So his expected gain is positive.

But suppose it turns out that the casino arrived at the number n by tossing the same fair coin and counting the tosses, up to and including the first tails. This presents a puzzle: “You and the house are behaving in a completely symmetric manner,” writes David Gale in Tracking the Automatic ANT (1998). “Each of you tosses the coin, and if the number of tosses happens to be the consecutive integers n and n + 1, then the n-tosser pays the (n + 1)-tosser 8n dollars. But we have just seen that the game is to your advantage as measured by expectation no matter what number the house announces. How can there be this asymmetry in a completely symmetric game?”


“What leapings of the heart must there not have been throughout that long warfare! What moments of terror and triumph! What acts of devotion and desperate wonders of courage!” — H.G. Wells, of prehistoric man

Sizing Up

Two lines divide this equilateral triangle into four sections. The shaded sections have the same area. What is the measure of the obtuse angle between the lines?

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Podcast Episode 35: Lateral Thinking Puzzles

For this Thanksgiving episode of the Futility Closet podcast, enjoy seven lateral thinking puzzles that didn’t make it onto our regular shows. Solve along with us as we explore some strange scenarios using only yes-or-no questions. Happy Thanksgiving!

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at

If you have any questions or comments you can reach us at Thanks for listening!

Good Boy

In 1794 Haydn visited the singer Venanzio Rauzzini at Bath. In the garden of Rauzzini’s villa he noticed a monument to a much-loved dog named Turk, with the inscription TURK WAS A FAITHFUL DOG AND NOT A MAN. As a tribute he turned the text into a four-part canon:

Rauzzini was so pleased that he had the music added to Turk’s memorial stone.

“Curious Sculpture”

A letter from W.C. Trevelyan to John Adamson, secretary of the Antiquarian Society of Newcastle, Jan. 20, 1825:

In the autumn of 1823, I visited the interesting Church at Bridlington [Yorkshire] (founded about 1114, by Gilbert de Gant). On examining a tomb stone with an inscription and date of 1587, standing on two low pillars of masonry near the font, I found some appearances of sculpture on the under side of it, and having obtained leave to turn it over, the curious sculpture represented in the etching herewith sent, was discovered.

Its meaning, or date, I cannot attempt to explain. Can it have any reference to the building of the church? You will perceive both the circular and pointed arch (though the latter is probably only accidental, the space being limited).

The roof, I think, resembles some of the Roman buildings of the lower empire of which I have seen engravings.

The tiles, in shape, correspond exactly with those which were found among the remains of a Roman villa discovered a few years since at Stonesfield, near Oxford. The upper figures are very like some on Bridekirk Font (of the 10th century).

The figures of the Fox and Dove remind one of Æsop’s fable of the Fox and the Stork.

The society published the plate in its Archæologia Æliana. The best guess I can find is that it’s a 12th-century coffin lid that had been appropriated as a tombstone in 1587. But the meaning of the figures is unclear.

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