I've read the article "Two Pile Move Size Dynamic Nim" and I have some problems with proof.
Mainly, we have defined $g(N)$ where for each positive integer $N$, $g(N)$ is the greatest power of $2$ that divides $N$. For example: $g(1) = 1$, $g(24) = 8$ since $8 | 24$ and $16$ does not divide $24$, $g(12)=4$, and so on.
And I have the lemma: For all positive integers $x,N$, if $g(x) < g(N)$ and $x < N$, then $g(x) = g(N−x) = g(N+x)$. If $g(x) < g(N)$ and $x > N$ then $g(x) = g(N +x)$. I think that I have to use the binary representation but I'm not sure how to begin this proof.