933. Paper Cutting

Starting with one piece of integer-sized rectangle paper, two players make moves in turn.
A valid move consists of choosing one piece of paper and cutting it both horizontally and vertically, so that it becomes four pieces of smaller rectangle papers, all of which are integer-sized.
The player that does not have a valid move loses the game.

Let $C(w,h)$$C(w, h)$ be the number of winning moves for the first player, when the original paper has size $w\times h$$w \times h$. For example, $C(5,3)=4$$C(5,3)=4$, with the four winning moves shown below.

Also write ${\displaystyle D(W,H)=\sum _{w=2}^W\sum _{h=2}^{H}C(w,h)}$$\displaystyle D(W, H) = \sum_{w = 2}^W\sum_{h = 2}^H C(w, h)$. You are given that $D(12,123)=327398$$D(12, 123) = 327398$.

Find $D(123,1234567)$$D(123, 1234567)$.

933. 剪纸游戏

现有一张长、宽都是整数的矩形纸，两名玩家轮流进行剪纸操作。每一次操作中，玩家可以选择一片纸，将其沿水平方向、竖直方向 各剪一刀，并将其剪成四块 长、宽都是整数的 小纸片。无法操作的玩家失败。

记 $C(w,h)$$C(w, h)$ 为：若初始时纸的长为 $w$$w$、宽为 $h$$h$，能让先手获胜的第一次操作的数量。例如 $C(5,3)=4$$C(5, 3) = 4$，如下是 $4$$4$ 个能让先手获胜的第一次操作：

再置 ${\displaystyle D(W,H)=\sum _{w=2}^W\sum _{h=2}^{H}C(w,h)}$$\displaystyle D(W, H) = \sum_{w = 2}^W\sum_{h = 2}^H C(w, h)$，已知 $D(12,123)=327398$$D(12, 123) = 327398$。

求 $D(123,1234567)$$D(123, 1234567)$。

点 这个链接 回到源站。

点 这个链接 回到详细版题目目录。