The Game of Sim

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Six dots arranged in a hexagon — every pair connected by a line. Two players take turns colouring a line red or blue. This is a misère game: the first player to complete a triangle in their own colour loses. Click two numbered dots, or click a grey line directly. Toggle hints to see which moves would lose on the spot.

The Game of Sim was invented in 1969 by the American mathematician Gustavus Simmons as a playable illustration of Ramsey theory — the branch of combinatorics that studies how large a structure must become before a pattern is forced to appear. The Ramsey number R(3, 3) = 6 means that any two-colouring of the edges of a complete graph on six vertices must contain a monochromatic triangle, so on this board a draw is mathematically impossible. The related theorem on friends and strangers puts it in everyday terms: among any six people, there are always either three mutual acquaintances or three mutual strangers. With only five vertices, clever colouring can avoid triangles entirely — which is why R(3, 3) is exactly 6, not 5.