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Project Euler 301: Finding Nim positions consisting of heap sizes n, 2n and 3n for n ≤ 230 that result in a losing game

Problem Description

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 ?

Analysis

Classic use of the Fibonacci sequence. Just find the Fibonacci Number FL+2, where L is the 2’s exponent.

Project Euler 301 Solution

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Afterthoughts

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