NIM with Cash

Original · EN

Let A be a finite subset of. Then NIM(A;n) is the following 2-player game: initially there are n stones on the board and the players alternate removing a∈ A stones. The first player who cannot move loses. This game has been well studied. We investigate an extension of the game where Player I starts out with d dollars, Player II starts out with e dollars, and when a player removes a∈ A he loses a dollars. The first player who cannot move loses; however, note this can happen for two different reasons: (1) the number of stones is less than min(A), (2) the player has less than (A) dollars. This game leads to more complex win conditions then standard NIM. We prove some general theorems from which we can obtain win conditions for a large variety of finite sets A. We then apply them to the sets A=1,L, and A=1,L,L+1.