--description--

The 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 $W({10}^2) = 31\,395$.

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

Find $W({10}^6)$.

--hints--

flippingGame() should return 3996390106631.

assert.strictEqual(flippingGame(), 3996390106631);

--seed--

--seed-contents--

function flippingGame() {

  return true;
}

flippingGame();

--solutions--

// solution required