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Paper-strip Game

Problem 306

Published on Sunday, 17th October 2010, 07:00 am; Solved by 636

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?