# 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 (n1,n2,n3) indicates a Nim position consisting
of heaps of size n1, n2 and n3 then there is a simple function
X(n1,n2,n3) — 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 ≤ 230 does X(n,2n,3n) = 0 ?