NIM with Cash: A Concrete Approach

TL;DR

This paper studies a novel variant of NIM involving players with dollars, analyzing how monetary constraints affect game outcomes and establishing win conditions for specific sets of moves.

Contribution

It introduces a new NIM variant with monetary constraints and derives win conditions for particular move sets, expanding understanding of combinatorial game variations.

Findings

Win conditions for A={1,L} are established.

Win conditions for A={1,L,L+1} are derived.

The game exhibits more complex outcomes due to monetary constraints.

Abstract

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}.