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Problem 306

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Paper-strip Game

The following game is a classic example of Combinatorial Game Theory:

Two players start with a strip of n white squares and they take alternate turns.
On each turn, a player picks two contiguous white squares and paints them black.
The first player who cannot make a move loses.

• If n = 1, there are no valid moves, so the first player loses automatically.
• If n = 2, there is only one valid move, after which the second player loses.
• If n = 3, there are two valid moves, but both leave a situation where the second player loses.
• If n = 4, there are three valid moves for the first player; she can win the game by painting the two middle squares.
• If n = 5, there are four valid moves for the first player (shown below in red); but no matter what she does, the second player (blue) wins.

So, for 1 ≤ n ≤ 5, there are 3 values of n for which the first player can force a win.
Similarly, for 1 ≤ n ≤ 50, there are 40 values of n for which the first player can force a win.

For 1 ≤ n ≤ 1 000 000, how many values of n are there for which the first player can force a win?

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纸条游戏

以下游戏是组合博弈论中的一个经典例子：

游戏开始时，有一张划成n个白格的纸条，两名玩家轮流进行操作。
每一轮，一名玩家选择相邻的两个白格，并将它们涂成黑色。
首先无法进行操作的玩家输掉游戏。

• 如果n = 1，不存在合法操作，先手玩家自动输掉游戏。
• 如果n = 2，只有一种合法操作，在此之后后手玩家输掉游戏。
• 如果n = 3，有两种合法操作，但在此之后后手玩家均输掉游戏。
• 如果n = 4，有三种合法操作，先手玩家只需将中间两个白格涂成黑色即可赢得游戏。
• 如果n = 5，有四种合法操作（如下图红色所示）；但无论先手玩家如何操作，后手玩家（蓝色所示）都将赢得游戏。

因此，对于1 ≤ n ≤ 5，共有3种n值使得先手玩家必胜。
同样地，对于1 ≤ n ≤ 50，共有40种n值使得先手玩家必胜。

对于1 ≤ n ≤ 1 000 000，共有多少种n值使得先手玩家必胜？

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