Sim is a two-player paper-and-pencil game.

Gameplay

Six dots ( vertices) are drawn. Each dot is connected to every other dot by a line ( edge). 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 (only triangles with the dots as all corners count; intersections of lines are not relevant); the player who completes such a triangle loses immediately.

Analysis

Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in R ...

can also be used to show that no game of Sim can end in a tie. Specifically, since the '' Ramsey number'' ''R''(3, 3) is equal to 6, any two-coloring of the

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...

on 6 vertices (''K''₆) must contain a monochromatic triangle, and therefore is not a tied position. This will also apply to any super-graph of ''K''₆. For another proof that there must eventually be a triangle of either color, see the Theorem on friends and strangers. Computer search techniques verified in 1974 that the second player can win Sim with perfect play. A strategy that could be easily implemented by human players was found in 2020. The game of Sim is one example of a Ramsey game. Other Ramsey games are possible. For instance, the players can be allowed to color more than one line during their turns. Another Ramsey game similar to Sim and related to the Ramsey number ''R''(4, 4) = 18 is played on 18 vertices and the 153 edges between them. The two players must avoid to color all six edges connecting four vertices. Because the Ramsey number ''R''(3, 3, 3) is equal to 17, any three-coloring of the complete graph on 17 vertices must contain a

monochromatic A monochrome or monochromatic image, object or palette is composed of one color (or values of one color). Images using only shades of grey are called grayscale (typically digital) or black-and-white (typically analog). In physics, mon ...

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 monochromatic triangle. Finding perfect winning strategies for these variants is most likely out of reach. A technical report by Wolfgang Slany is available online, with many references to literature on Sim, going back to the game's introduction by

Gustavus Simmons Gustavus J. Simmons (born 1930) is a retired cryptographer and former manager of the applied mathematics Department and Senior Fellow at Sandia National Laboratories. He worked primarily with authentication theory, developing cryptographic techn ...

in 1969,Simmons, Gustavus J. "The game of SIM," ''J. Recreational Mathematics'', 2(2), 1969, pp. 66. including proofs and estimates of the difficulty as well as

computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations ...

of Sim and other Ramsey games.

Software

An app including its source code in the visual multi-platform Catrobat programming language is available for playing it against one's smartphone.

References

{{reflist Ramsey theory Mathematical games Combinatorics Combinatorial game theory Paper-and-pencil games Solved games Positional games