## Chapter21Complete a Complete Graph: A Game

###### Project by: Daniel Getz, Alexander Wen, and Solomon Yu.

$\textbf{Summary:}$ Our project is a Unity based recreation of the pencil–and–paper game Sim.

The game Sim takes place on a complete graph $K_6$ and is played as follows: two players alternate colouring an edge of the graph $K_6\text{.}$ The opposing player wins when a player ends up creating a monochromatic triangle in their own colour.

In the generalized form of Sim, the game takes place on a complete graph $K_n$ where Player A loses if they create a monochromatic $K_a$ and Player B looses if they create a monochromatic $K_b\text{.}$

By Ramsey's theorem, if $n\geq R(a,b)$ then this game will always be decisive.

In the case of $K_6\text{,}$ the second player can technically force a win, but the strategy is exceedingly complicated to memorize.

In this project, we recreate this game first introduced by Gustavus Simmons in 1969 in Unity, as well as explore other variations that change the win conditions and size the board.

To play the game go to Solomon Yu's Github site.

Watch the demo video: