## Cross flips

### Problem 331

`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 C_{N} 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. C_{5} is shown above.

Let T(`N`) be the minimal number of turns to finish a game starting from configuration C_{N} or 0 if configuration C_{N} is unsolvable.

We have shown that T(5)=3. You are also given that T(10)=29 and T(1 000)=395253.

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