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The Graph of Positions for the Game of SIM *

The game of SIM was introducted by Simmons [1]. Two players, black and red, alternately color the 15 edges of the complete graph K 6. A player loses if he is forced to complete a triangle in his own color. It is well-known that K 6 cannot be colored in two colors without a monochromatic triangle; thus a tie is impossible. It is easily deduced that one of the players must have a winning strategy and it was shown by Mead, Rosa and Huang [2] that the second player (whom we always take to be red) is so blessed. Rounds and Yau [3] reached the same conclusion by means of a statistical approach.