A pencil-and-paper game guarantees an ending — but why, and after how many moves?
In Sprouts, two players take turns joining dots with a curved line (never crossing another line, and no dot may ever get more than three line-ends), adding a fresh dot on every line they draw. Whoever makes the last legal move wins. The game can never go on forever, but why not, and within what range of move counts must it end?
Reveal the answer
Every move uses up 'connection slots' at dots while adding back at most one new slot, so for n starting dots the game must last between 2n and 3n-1 moves, a topological argument, not luck, that forces it to end. Sprouts was invented by mathematicians John Conway and Michael Paterson at Cambridge in 1967, and Martin Gardner introduced it to a wide audience that same year in his Scientific American column.
— Martin Gardner, Scientific American (Mathematical Games column) — July 1967 column, later collected in 'Mathematical Carnival' (1975)
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