NIM with Cash: A Concrete Approach

Let A be a finite subset of the naturals and let n be a natural. Let NIM(A;n) be the two player game in which players alternate removing $a\in A$ stones from a pile with $n$ stones; the first player who cannot move loses. This game has been researched thoroughly. We discuss a variant of NIM in which Player 1 and Player 2 start with d and e dollars, respectively. When a player removes a stones from the pile, he loses a dollars. The first player who cannot move loses, but this can now happen for two reasons: (1) The number of stones remaining is less than min(A), (2) The player has less than min(A) dollars. This game leads to much more interesting win conditions than regular NIM. We investigate general properties of this game. We then obtain and prove win conditions for the sets A={1,L} and $A={1,L,L+1}.