Problem 306

Paper-strip Game

The following game is a classic example of Combinatorial Game Theory:

Two players start with a strip of `n` white squares and they take alternate turns.

On each turn, a player picks two contiguous white squares and paints them black.

The first player who cannot make a move loses.

- If
`n`= 1, there are no valid moves, so the first player loses automatically. - If
`n`= 2, there is only one valid move, after which the second player loses. - If
`n`= 3, there are two valid moves, but both leave a situation where the second player loses. - If
`n`= 4, there are three valid moves for the first player; she can win the game by painting the two middle squares. - If
`n`= 5, there are four valid moves for the first player (shown below in red); but no matter what she does, the second player (blue) wins.

So, for 1 `n` 5, there are 3 values of `n` for which the first player can force a win.

Similarly, for 1 `n` 50, there are 40 values of `n` for which the first player can force a win.

For 1 `n` 1 000 000, how many values of `n` are there for which the first player can force a win?

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