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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“The Peace Problem”

bondarenko peace problem

In 1984, at the height of the Cold War, Ukrainian chess journalist F.S. Bondarenko dedicated this puzzle to English chess editor A.J. Roycroft in the spirit of peace and goodwill.

White’s army is arranged as above. Add Black’s army (the standard complement of 8 pieces and 8 pawns) so that no piece of either color is under attack.

All Hands on Deck?

A reader named Hamp Stevens sent this conundrum to Martin Gardner, who published it in his Mathematical Magic Show (1965). Can these 25 cards be arranged to form five poker hands, each of them a straight or better (that is, straight, flush, full house, four of a kind, straight flush, or royal flush)? If it’s possible, find the five hands; if it’s not, prove that it’s impossible.

“This ingenious puzzle is quickly solved if you go about it correctly,” Gardner wrote. “A single card is the key.”

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Hard Target

In a special football game, a team scores 7 points for a touchdown and 3 points for a field goal. What’s the largest mathematically unreachable number of points that a team can score (in an infinitely long game)?

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