## Flipping game

### Problem 459

Published on Sunday, 16th February 2014, 10:00 am; Solved by 72The flipping game is a two player game played on a N by N square board.

Each square contains a disk with one side white and one side black.

The game starts with all disks showing their white side.

A turn consists of flipping all disks in a rectangle with the following properties:

- the upper right corner of the rectangle contains a white disk
- the rectangle width is a perfect square (1, 4, 9, 16, ...)
- the rectangle height is a triangular number (1, 3, 6, 10, ...)

Players alternate turns. A player wins by turning the grid all black.

Let W(N) be the number of winning moves for the first player on a N by N board with all disks white, assuming perfect play.

W(1) = 1, W(2) = 0, W(5) = 8 and W(10^{2}) = 31395.

For N=5, the first player's eight winning first moves are:

Find W(10^{6}).