Sim (pencil game)
The game of Sim is played by two players on a board consisting of six dots ('vertices'). Each dot is connected to every other dot by a line.
Two players take turns coloring any uncolored lines. One player colors in one color, and the other colors in another color, with each player trying to avoid the creation of a triangle made solely of their color; the player who completes such a triangle loses immediately.
Ramsey theory shows that no game of Sim can end in a tie. Specifically, since the Ramsey number R(3,3)=6, any two-coloring of the complete graph on 6 vertices (K6) must contain a monochromatic triangle, and therefore is not a tied position. This will also apply to any super-graph of K6.
Computer search has verified that the second player can win Sim with perfect play, but finding a perfect strategy that humans can easily memorize is an open problem.
A Java applet is available for online play against a computer program. A technical report by Wolfgang Slany is also available online, with many references to literature on Sim, going back to the game's introduction by Gustavus Simmons in 1969.
This game of Sim is one example of a Ramsey game. Other Ramsey games are possible. For instance, according to Ramsey theory any three-coloring of the complete graph on 17 vertices must contain a monochromatic triangle. A corresponding Ramsey game uses pencils of three colors. One approach can have three players compete, while another would allow two players to alternately select any of the three colors to paint an edge of the graph, until a player loses by completing a mono-chromatic triangle. It is unknown whether this latter game is a first or a second player win.
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