Knock Down Pins One or Two at a Time — Who Wins?
Set up a row of bowling pins. Two players take turns: on each turn, knock down either a single pin, or two pins that happen to be standing right next to each other. Whoever knocks down the very last pin wins. For a row of a given starting length, can you tell in advance which player has the winning strategy?
Reveal the answer
Yes — every starting length can be analyzed exactly using Sprague-Grundy (nimber) theory, which assigns each row a number that predicts the outcome under perfect play; the full analysis was worked out by Richard Guy and Cedric Smith in the 1950s. The game is called Kayles, invented by Henry Dudeney in 1908 and introduced in his book The Canterbury Puzzles, and it remains a standard teaching example in combinatorial game theory.
— Henry Ernest Dudeney, The Canterbury Puzzles — First published 1907; the Kayles game was analysed via Sprague-Grundy theory by R. K. Guy and C. A. B. Smith (1956)
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