Optimal Strategy for Fibonacci Nim

Theorem

Consider a game of Fibonacci nim with $n$ counters.

Let it be the turn of player $\text A$.

Let the maximum number of counters that can be taken by $\text A$ be $q$.

Let $n$ be expressed in Zeckendorf representation as:

$n = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$

Then $\text A$ can force a win if and only if:

$F_{k_r} \le q$

and by taking those $F_{k_r}$ counters.

Proof

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