Problem 301: Nim

Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any heap until no stones remain. We'll consider the three-heap normal-play version of Nim, which works as follows: - At the start of the game there are three heaps of stones. - On his turn the player removes any positive number of stones from any single heap. - The first player unable to move (because no stones remain) loses. If ($n_1$, $n_2$, $n_3$) indicates a Nim position consisting of heaps of size $n_1$, $n_2$ and $n_3$ then there is a simple function $X(n_1,n_2,n_3)$ — that you may look up or attempt to deduce for yourself — that returns: - zero if, with perfect strategy, the player about to move will eventually lose; or - non-zero if, with perfect strategy, the player about to move will eventually win. For example $X(1, 2, 3) = 0$ because, no matter what the current player does, his opponent can respond with a move that leaves two heaps of equal size, at which point every move by the current player can be mirrored by his opponent until no stones remain; so the current player loses. To illustrate: - current player moves to (1,2,1) - opponent moves to (1,0,1) - current player moves to (0,0,1) - opponent moves to (0,0,0), and so wins. For how many positive integers $n ≤ 2^{30}$ does $X(n, 2n, 3n) = 0$?