#### Event Title

#### Location

Parker-Reed, SSWAC

#### Start Date

1-5-2014 1:00 PM

#### End Date

1-5-2014 2:00 PM

#### Project Type

Poster

#### Description

This poster presentation covers our research on a mathematical game that is somewhat similar to the popular game Nim. In this game, Players play a variation of Nim where the number of piles must be a perfect square and where the piles are actually situated within a matrix. In addition, players are only allowed to subtract one from an entry on their turn, where as in Nim there are typically more options. The game is over either when a player does not have a move to make (because all entries have been reduced to zero) or the when the game matrix becomes singular. Hence, the game is similar to Nim except that it will end sooner due to a zero determinant. In our research, we formulated a description of when two different games are considered to be equivalent and proved a theorem which characterizes when exactly two games are equivalent. In addition, we studied the 2x2 case in depth and developed methods for deterring which games are winnable, with perfect play, by the first player vs second player. One interesting result was that whenever the sum of the elements of the matrix is one more than a prime, the player who is about to go can never win if the other player plays with perfect play. The result did not extend to games of higher dimensions.

#### Sponsoring Department

Colby College. Mathematics and Statistics Dept.

#### CLAS Field of Study

Natural Sciences

#### Event Website

http://www.colby.edu/clas

#### ID

211

Mathematical Analysis of a Nim-like Matrix Game

Parker-Reed, SSWAC

This poster presentation covers our research on a mathematical game that is somewhat similar to the popular game Nim. In this game, Players play a variation of Nim where the number of piles must be a perfect square and where the piles are actually situated within a matrix. In addition, players are only allowed to subtract one from an entry on their turn, where as in Nim there are typically more options. The game is over either when a player does not have a move to make (because all entries have been reduced to zero) or the when the game matrix becomes singular. Hence, the game is similar to Nim except that it will end sooner due to a zero determinant. In our research, we formulated a description of when two different games are considered to be equivalent and proved a theorem which characterizes when exactly two games are equivalent. In addition, we studied the 2x2 case in depth and developed methods for deterring which games are winnable, with perfect play, by the first player vs second player. One interesting result was that whenever the sum of the elements of the matrix is one more than a prime, the player who is about to go can never win if the other player plays with perfect play. The result did not extend to games of higher dimensions.

http://digitalcommons.colby.edu/clas/2014/program/138