# Sim (game)

Game board of Sim

Sim is a game for two people. The game board consists of six points, each of which is connected to each other by a line. Each player is assigned a color and each player alternately colors a line with his or her color. If you produce a triangle in your own color, you lose immediately.

The Ramsey theory shows that no sim game can end in a draw. This is especially true because the Ramsey number R (3,3) = 6. Every 2-coloration of the complete graph with 6 nodes ( ) must contain a single-colored triangle. This also applies to every supergraph of the . The reason for this is very easy to understand: you pick any point. Let's call this P1. This is connected to the five other points. Of these five lines, at least three must be in the same color, let's call this color F1. We now look at the three points reached by these three lines of the same color. Either the lines between these three points are all in one color, then these three points form a triangle of the same color or at least two of the three points are connected with a line in color F1, then these two points form a triangle of the same color with P1. ${\ displaystyle K_ {6}}$${\ displaystyle K_ {6}}$

By means of complete enumeration with the computer, it has been found that the second player always wins if the game is error-free. Finding a perfect game strategy that people can also remember has not yet succeeded.

Sim is an example of a Ramsey game. Other Ramsey games are possible. For example, according to the Ramsey theory, every 3-coloration of a complete graph with 17 nodes must contain a monochrome triangle. In the accompanying Ramsey game, the two players use any of three colors. It is still unknown who will win.