# 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 , 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 .