This is a group research project done in collaboration with Yacob Ozdalkiran and Elmehdi Haress during our second year in the classe preparatoire Lycee Pierre de Fermat from September 2016 to July 2017. This project was supervised by Guillaume Brevet and presented at the entrance competitive exams of several scientific schools such as Mines Paristech and ENS.
Our aim was to be able to provide a winning strategy for a game we imagined: the sum of a Wythoff game and N Nim games, N being any integer. This game being an accessibility game, our winning strategy consisted in finishing each round on a position whose Grundy number is 0. The Grundy function of a Nim game is obviously the number of allumettes left. All that was left was to compute efficiently the Grundy function of a wythoff game and to be able to infer from the grundy functions of all the sub games the grundy number of a position in the global game. Regarding this last matter, the Sprague-Grundy theorem stipulates that the grundy function of a sum of games is the Nim sum of the grundy functions of all the games. The Nim sum is composition law for positive integers that can be defined simply using the binary decompositions of integers, however in order to gain time it is more efficient to compute beforehand the table of all values of this law (until an acceptable upper bound) by using a much more implicit algebraic construction described in \ref_quadrature. This work allows us to compute efficiently and recursively the table of this law for all integers inferior to 2^(2^n). Having found a scalable method to compute the grundy number of any position in the global game, we provide an algorithm that can indicate for each position whose grundy number is not 0, which accessible position has a zero gundy number.