Pip Squeak

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A fair die bearing the numbers 1, 2, 3, 4, 5, 6 is repeatedly thrown until the running total first exceeds 12. What’s the most likely total that will be obtained?

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Rock Steady

If we’re given 32 stones, each a different weight, how can we find both the heaviest and the second heaviest stone in 35 weighings with an equal-arm balance?

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The Dark Side

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In a certain chess position, each row and each column contains an odd number of pieces. Prove that the total number of pieces on black squares is an even number.

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Amnesia

langstaff chess puzzle

W. Langstaff offered this conundrum in Chess Amateur in 1922. White is to mate in two moves. He tries playing 1. Ke6, intending 2. Rd8#, but Black castles and no mate is possible. But by castling Black shows that his last move must have been g7-g5. Knowing this, White chooses 1. hxg6 e.p. rather than 1. Ke6. Now if Black castles he can play 2. h7#.

“Not so fast!” Black protests. “My last move was Rh7-h8, not g7-g5, so you can’t capture en passant.”

“Very well,” says White. “If you can’t castle, then I play 1. Ke6.” And we’re back where we started.

“What was really Black’s last move?” asks Burt Hochberg in Chess Braintwisters (1999). “If a position has a history, it can have only a single history, and Black would not be able to choose what his last move was any more than I can choose today what I had for dinner last night.”

“This is not a real game, however, but a problem in chess logic. The position’s history does not exist in actuality but only as a logical construct.”

Hidden Sum

A problem from the 1973 American High School Mathematics Examination:

In this equation, each of the letters represents uniquely a different digit in base 10:

YE × ME = TTT.

What is E + M + T + Y?

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Words and Numbers

If you write out the numbers from 1 to 5000 in American English (e.g., THREE THOUSAND EIGHT HUNDRED SEVENTY-THREE), it turns out that only one of them has a unique number of characters. Which is it? Spaces and hyphens count as characters.

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