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NIM with Cash | | William Gasarch
; John Purtilo
; Douglas Ulrich
; | | Date: |
12 Nov 2015 | | Abstract: | Let A be a finite subset of $
at$. Then NIM(A;n) is the following 2-player
game: initially there are $n$ stones on the board and the players alternate
removing $ain A$ 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 ain 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 $min(A)$ 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}. | | Source: | arXiv, 1511.4035 | | Services: | Forum | Review | PDF | Favorites | |
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