Town and Country

More chess masters reside in New York City than in the rest of the United States combined. We’re planning a chess tournament that all American masters are expected to attend, and we want to minimize the total intercity traveling done by the players. The New York players argue that, by this criterion, the tournament should be held in their city. The West Coast players argue that a city should be chosen near the center of the gravity of the players. Where should we hold the tournament?

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Mass Transit

A city has 10 bus routes. Is it possible to arrange the routes and bus stops so that if one route is closed it’s still possible to get from any one stop to any other (possibly changing buses along the way), but if any two routes are closed, there will be at least two stops that become inaccessible to one another?

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Pip Squeak

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

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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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.”