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{\bf Arthur Holshouser and Harold Reiter}
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{\bf Dynamic Single-Pile Nim Using Multiple Bases}
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In the game $G_{0}$ two players alternate removing positive numbers of
counters from a single pile and the winner is the player who removes
the last counter. On the first move of the game, the player moving
first can remove a maximum of $k$ counters, $k$ being specified in
advance. On each subsequent move, a player can remove a maximum of
$f(n,t) $ counters where $t$ was the number of counters removed by his
opponent on the preceding move and $n$ is the preceding pile size,
where $f:N\times N\rightarrow N$ is an arbitrary function satisfying
the condition (1): $\exists t\in N$ such that for all $n,x\in N$,
$f(n,x) =f(n+t,x) $.  This note extends our earlier paper [E-JC, Vol
10, 2003, N7]. We first solve the game for functions $f:N\times
N\rightarrow N$ that also satisfy the condition (2): $\forall n,x\in
N$, $f(n,x+1) -f(n,x) \geq -1$. Then we state the solution when
$f:N\times N\rightarrow N$ is restricted only by condition (1) and
point out that the more general proof is almost the same as the
simpler proof. The solutions when $t\geq 2$ use \emph{multiple bases}.



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