Predicting the Winner in the Game of Nim: توقُّع الفائز في لُعبة نِيم

Game Theory is defined as a means of mathematical analysis when interests collide with each other to reach the best possible decision-making options taking into consideration the given circumstances to get the desired results. Even though Game Theory is related to well-known games such as checkers, XO, and poker. In fact, it is associated with more serious problems pertaining to sociology, economics, politics, military sciences. Game Theory includes several sorts of games like Combinatorial Games. This Paper gives the reader a detailed outlook of a special Combinatorial game called Nim Game. Before the start of the game we will develop a strategy to determine the winner in advance with the help of some basic mathematical concepts like Minimum Excluded (MEX), Grundy Numbers and The XOR Number.

Circular Nim Games

A circular Nim game is a two player impartial combinatorial game consisting of $n$ stacks of tokens placed in a circle. A move consists of choosing $k$ consecutive stacks, and taking at least one token from one or more of the $k$ stacks. The last player able to make a move wins. We prove results on the structure of the losing positions for small $n$ and $k$ and pose some open questions for further investigations.