# Matrix Game Theory

I am giving N matrix of size N*M where 1<=N<=4 and 1<=M<=4

Game consist of choosing any rectangle (submatrix) from one of the given N matrix and removing this submatrix.

For Example We are having 1 Matrix of size 4*4 Player can choose submatrix of 4*4 can remove it. or He can choose submatrix of 2*2 or 1*1 or 2*3 or any valid submatrix and remove it from 4*4 matrix and we have remaining matrix left to play.

Player who can't make a move looses ? Which player wins.

Can anyone suggest me wining strategy ? I could not figure out how to solve this.

Both Player Plays optimally.

• Something seems to be missing here. Is the loosing player the one that removes the last element? Because otherwise player one just removes all of the matrix and player two has no moves left, thus player one wins. – Shinja Mar 6 '17 at 9:03
• @Shinja Player can choose only one submatrix at each move he can't remove all the N matrix in one move – Marvel Mar 6 '17 at 9:07
• A related game is Chomp, but there you can only steal/eat upper submatrices, – mlc Mar 6 '17 at 9:38
• 2 questions : 1) is it a really a submatrix (elements are not necessarily neighbours) or a "block" (all removed elements are nighbours) ? 2) I don't understand what you mean by "removing"... Do you mean that once you have removed a submatrix, the elements are filled by zeros or you are left with a staircase pattern ? – Jean Marie Mar 6 '17 at 11:51