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Problem 325


Problem 325


Stone Game II

A game is played with two piles of stones and two players. At her turn, a player removes a number of stones from the larger pile. The number of stones she removes must be a positive multiple of the number of stones in the smaller pile.

E.g., let the ordered pair(6,14) describe a configuration with 6 stones in the smaller pile and 14 stones in the larger pile, then the first player can remove 6 or 12 stones from the larger pile.

The player taking all the stones from a pile wins the game.

A winning configuration is one where the first player can force a win. For example, (1,5), (2,6) and (3,12) are winning configurations because the first player can immediately remove all stones in the second pile.

A losing configuration is one where the second player can force a win, no matter what the first player does. For example, (2,3) and (3,4) are losing configurations: any legal move leaves a winning configuration for the second player.

Define S(N) as the sum of (xi+yi) for all losing configurations (xi,yi), 0 < xi < yi ≤ N. We can verify that S(10) = 211 and S(104) = 230312207313.

Find S(1016) mod 710.


取石子游戏II

两名玩家正在用两堆石子玩游戏。每一轮,玩家从较大的堆中取走一定数目的石子,这个数目必须是较小的堆中石子数目的正整数倍。

例如,用有序对(6,14)表示游戏中较小的堆有6枚石子,较大的堆有14枚石子,先手玩家可以从较大的堆中取走6枚或12枚石子。

从某一堆中取走所有石头的玩家赢得游戏。

所谓必胜态指先手玩家必胜的状态。例如(1,5)、(2,6)和(3,12)都是必胜态,因为先手玩家可以立即取走第二个堆中的所有石子。

所谓必败态指无论先手玩家如何操作,后手玩家都必胜的状态。例如,(2,3)和(3,4)都是必败态:先手玩家的任意合法操作都会给后手玩家留下必胜态。

对于所有0 < xi < yi ≤ N的必败态(xi,yi),记S(N)为(xi+yi)的和。我们可以验证S(10) = 211以及S(104) = 230312207313。

求S(1016) mod 710