Pillow Problem

From a 1921 essay by A.A. Milne:

TERALBAY is not a word which one uses much in ordinary life. Rearrange the letters, however, and it becomes such a word. A friend — no, I can call him a friend no longer — a person gave me this collection of letters as I was going to bed and challenged me to make a proper word of it. He added that Lord Melbourne — this, he alleged, is a well-known historical fact — Lord Melbourne had given this word to Queen Victoria once, and it had kept her awake the whole night. After this, one could not be so disloyal as to solve it at once. For two hours or so, therefore, I merely toyed with it. Whenever I seemed to be getting warm I hurriedly thought of something else. This quixotic loyalty has been the undoing of me; my chances of a solution have slipped by, and I am beginning to fear that they will never return. While this is the case, the only word I can write about is TERALBAY.

The answer is not RATEABLY, or BAT-EARLY, which “ought to mean something, but it doesn’t.” Rudolf Flesch notes that TRAYABLE is not a word, and that, though TEARABLY appears in small type in Webster’s Unabridged, “it obviously won’t do.”

What’s the answer? There’s no trick — it’s an ordinary English word.

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BETRAYAL.

December 8, 2012February 8, 2013 | Puzzles

Black and White

By Charles F. Stubbs. White to mate in two moves.

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The surprising 1. Bg1! forces Black to capture a rook, and White mates with the light-squared bishop. From Chess Problems, 1904.

December 7, 2012February 8, 2013 | Puzzles

Black and White

By James Pierce. White to mate in two moves.

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The subtle 1. Ne2 leaves Black with no options: 1. Ne2 Kc6 2. Rb3# 1. Ne2 Ke6 2. Rf6# 1. Ne2 Ke4 2. Nf6# 1. Ne2 Kc4 2. Rc3# From Chess Problems, 1873.

November 29, 2012February 8, 2013 | Puzzles

Black and White

By Francis Healey. White to mate in two moves.

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The absurd-looking 1. Re7! wins in every variation. From A Collection of Two Hundred Chess Problems, 1866.

November 23, 2012February 8, 2013 | Puzzles

Hole Hearted

gardner geometric vanish

Martin Gardner called this the proudest puzzle of his own devising. When the pieces on the left are rearranged as on the right, a hole appears in the center of the square. How is this possible?

“I haven’t the foggiest notion of how to succeed in inventing a good puzzle,” he told the College Mathematics Journal. “I don’t think psychologists understand much either about how mathematical discoveries are made. … The creative act is still a mystery.”

November 19, 2012November 18, 2012 | Puzzles

Black and White

By George E. Carpenter. White to mate in two moves.

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1. Qh3 Ke4 2. Rc4# From Will H. Lyons, Chess-Nut Burrs: How They Are Formed and How to Open Them, 1886.

November 15, 2012February 7, 2013 | Puzzles

Set Theory

A puzzle by Polish mathematician Paul Vaderlind:

Andre Agassi and Boris Becker are playing tennis. Agassi wins the first set 6-3. If there were 5 service breaks in the set, did Becker serve the first game?

(Service changes with each new game in the 9-game set. A service break is a game won by the non-server.)

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If there were 9 games in the set, then one player served 4 games and one served 5. Suppose Agassi served 4 games and Becker had k service breaks. That would mean that Agassi won 4-k of the games in which he served, and 5-k of the games in which Becker served (because there were 5 service breaks altogether). Then Agassi would have won a total of (4-k) + (5-k) = 9-2k games, an odd number. But we know that Agassi won 6 games, so this is a contradiction. Hence Agassi must really have served 5 rather than 4 games, and therefore he served the first game. From The Inquisitive Problem Solver, 2002.

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November 13, 2012February 7, 2013 | Puzzles

Black and White

keym chess puzzle

By Werner Keym, from Die Schwalbe, 1979. What were the last moves by White and Black?

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Black’s king is in check, so White must just have moved his knight from b4 to a2. But what was Black’s move before that? He cannot have moved his king, as there’s no square from which it can have come. The only other possibility is that Black moved a piece that’s not now on the board — a piece that White’s a2 knight has just captured. What can that have been? Not a knight or a pawn, as there’s no square that either of these can have come from. Nor a queen or a rook, as those could only have moved from a1, where they would have been checking the White king at the end of White’s last move. The only remaining possibility is a bishop: Black has just moved a bishop from b1 to a2, and White has responded by capturing it with a knight from b4.

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November 9, 2012February 7, 2013 | Puzzles

The Lock Key

Two adjoining lakes are connected by a lock. The lakes differ by 2 meters in elevation. A boat can pass from the lower lake to the upper by passing through the lock gate, which is closed behind it; then water is added to the lock chamber until its level matches that of the upper lake, and the boat can pass out through the upper gate.

Now suppose two boats do this in succession. The first boat weighs 50 tons, the second only 5 tons. How much more water must be used to raise the small boat than the large one?

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None. The same amount of water is necessary to raise any boat. One way to make this intuitive is to imagine that a 2-meter “slab” of water is inserted at the bottom of the lock chamber. This will raise a boat of any size to the level of the higher lake.

November 7, 2012February 7, 2013 | Puzzles

No Touching

This scale balances a cup of water with a certain weight. Will the balance be upset if you put your finger in the water, if you’re careful not to touch the glass?

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Yes. The water imparts an upward force on your finger as you immerse it, so you must exert a downward force to overcome it. That force is transferred to the glass and then to the scale, which sinks. From Yuri B. Chernyak, The Chicken From Minsk, 1995.

November 2, 2012February 7, 2013 | Puzzles