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$\sum_{i=3}^{31}T(2^i-1)$。