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.”

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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Podcast Episode 86: Lateral Thinking Puzzles

Here are six new lateral thinking puzzles to test your wits and stump your friends — play along with us as we try to untangle some strange situations using only yes-or-no questions.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

This episode’s puzzles were contributed by listeners David White and Sean Gilbertson and drawn from the following books:

Edward J. Harshman, Fantastic Lateral Thinking Puzzles, 1996.

Kyle Hendrickson, Mental Fitness Puzzles, 1998.

Paul Sloane and Des MacHale, Intriguing Lateral Thinking Puzzles, 1996.

David White sent two links to corroborate the third puzzle — these contain spoilers, so listen to the episode before clicking.

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