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Some Investigations into the Game of SIM

The game of SIM [1] is played by two people upon the six vertices of a regular hexagon. Each player takes a turn connecting two previously unconnected vertices of the hexagon with a straight line using a colored pencil so that the lines made by the two players are readily distinguishable. The object of the game is to force your opponent to form a triangle without forming one yourself. The player who first forms a triangle from three of his lines loses. (Only those triangles whose endpoints are the original vertices are considered, as is shown in Figure 1.) Valid losing triangle.