## Theses, Dissertations and Culminating Projects

#### Title

Variations of the Combinatorial Game Tron

5-2013

Thesis

#### Degree Name

Master of Science (MS)

#### College/School

College of Science and Mathematics

#### Department/Program

Mathematical Sciences

Jonathan Cutler

Aihua Li

Philip Yecko

#### Abstract

The combinatorial game Tron is a two player game played on a graph, in which players move to adjacent vertices, but cannot move to any vertex which is currently occupied or which has been occupied by either player earlier in the course of the game. It was introduced by Hans Bodlaender, inspired by the Disney movie of the same name. Bodlaender and, later, Tillmann Miltzow considered the complexity of the question of whether a player has a winning strategy in the game. Miltzow also considered an extremal question regarding the ratio of the vertices taken by the second player over those taken by the first. Others have explored using Monte Carlo Tree Search methods to find a winning strategy.

We have studied variations of Tron that do not appear to have been previously considered. In particular, we have investigated which player has the advantage in two versions of the game in which players may move more than one vertex at a time in a single direction (so long as they do not move through any previously captured vertices in the process), with the object being to capture as many vertices as possible, on grids of varying sizes. For the variation we call Tron^, in which a player may move as far as he or she wants in a single direction on each turn, we have complete or nearly complete results for grids with one dimension less than or equal to 3 and partial results for some larger boards. For the variation we call Tronmax, in which a player must move as far as possible in a single direction on each turn, we have complete results for grids of most sizes and partial results for nearly all of the remaining grids. In most cases, in both of these versions of Tron, it seems that if both players make the best moves they can, it will be the player who moves second who wins.

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