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Problem 899

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DistribuNim I

Two players play a game with two piles of stones. The players alternately take stones from one or both piles, subject to:

1. the total number of stones taken is equal to the size of the smallest pile before the move;
2. the move cannot take all the stones from a pile.

The player that is unable to move loses.

For example, if the piles are of sizes $3$ and $5$ then there are three possible moves.
$$(3,5)\stackrel{(2,1)}{\to }(1,4)(3,5)\stackrel{(1,2)}{\to }(2,3)(3,5)\stackrel{(0,3)}{\to }(3,2)$$

Let $L(n)$ be the number of ordered pairs $(a,b)$ with $1\le a,b\le n$ such that the initial game position with piles of sizes $a$ and $b$ is losing for the first player assuming optimal play.

You are given $L(7)=21$ and $L(7^2)=221$.

Find $L(7^{17})$.

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分布式取石子游戏（一）

两位玩家在玩一个游戏，游戏开始时有两堆石子，玩家轮流从一堆或两堆中取石子，并需遵守以下规则：

1. 取走的石子总数等于行动前较小堆的石子数量；
2. 不能将任何一堆的石子全部取完。

首先无法行动的玩家输掉游戏。

例如，如果两堆石子的大小分别为$3$和$5$，则有三种可能的行动：
$$(3,5)\stackrel{(2,1)}{\to }(1,4)(3,5)\stackrel{(1,2)}{\to }(2,3)(3,5)\stackrel{(0,3)}{\to }(3,2)$$

令$L(n)$为满足以下条件的有序对$(a,b)$的数量：$1\le a,b\le n$，且假设双方都采用最优策略，若游戏开始时两堆石子数目为$a$和$b$，先手玩家必败。

已知$L(7)=21$，$L(7^2)=221$。

求$L(7^{17})$。

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