# 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?