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?

These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK