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Problem 331


Problem 331


Cross flips

N×N disks are placed on a square game board. Each disk has a black side and white side.

At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus 2×N-1 disks are flipped. The game ends when all disks show their white side. The following example shows a game on a 5×5 board.

It can be proven that 3 is the minimal number of turns to finish this game.

The bottom left disk on the N×N board has coordinates (0,0);
the bottom right disk has coordinates (N-1,0) and the top left disk has coordinates (0,N-1).

Let CN be the following configuration of a board with N×N disks:
A disk at (x,y) satisfying $N-1 \le \sqrt{x^2+y^2} \lt N$, shows its black side; otherwise, it shows its white side. C5 is shown above.

Let T(N) be the minimal number of turns to finish a game starting from configuration CN or 0 if configuration CN is unsolvable.
We have shown that T(5)=3. You are also given that T(10)=29 and T(1 000)=395253.

Find $\sum_{i=3}^{31}T(2^i-1)$.


交叉翻转

N×N个碟子被放置在方形的游戏板上。每个碟子一面是黑色,另一面是白色。

每一轮,你可以选择一个碟子,并将与其同一行或同一列的碟子都翻转过来:这样一共要翻转2×N-1个碟子。当所有碟子都是白色面朝上时,游戏结束。下图所示是一个5×5游戏版上的例子。

可以证明,最少需要3次翻转才能结束游戏。

记N×N游戏板左下角碟子的坐标为(0,0);
记右下角碟子的坐标为(N-1,0),左上角碟子的坐标为(0,N-1)。

将N×N个碟子按如下方式摆在游戏板上,这样的设置记为CN
位于(x,y)的碟子,若$N-1 \le \sqrt{x^2+y^2} \lt N$,则黑色面朝上;否则白色面朝上。上述样例即为C5

若从CN开始的游戏能够结束,记T(N)为最少需要翻转的次数;若不能结束,则记T(N)为0。
我们已经得出T(5)=3。此外,还已知T(10)=29以及T(1 000)=395253。

求$\sum_{i=3}^{31}T(2^i-1)$。