Sim (Pencil Game)
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This article is about a pencil game called Sim. For other meanings of the term "sim", see Sim (disambiguation).
The game of Sim is played by two players, Red and Blue, on a board consisting of six dots ('vertices'). Each dot is connected to each other with a line.
Complete_graph_K6.png
Players alternate coloring any uncolored line in their own color. Players try to avoid making triangles of their color; the player who completes a triangle of their color loses immediately. (A triangle is three dots, each connected to the other two with lines of the same color.) The other player is the winner.
A simple theorem of Ramsey theory shows that no game of Sim can end in a tie; one player must lose by the end. Specifically, since R(3,3;2)=6, any coloring of the complete graph on 6 vertices must contain a monochromatic triangle, and therefore is not a tied position.