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Given integer $n \geq 1, \ell \geq 2$, and vector $x = (x_1, \ldots, x_n)$that has an entry which is a multiple of $\ell$ and such that $x_1 \leq \ldots\leq x_n$, the GM-rule is defined as follows: Keep the rightmost minimal entry$x_i$ of $x$, which is a multiple of $\ell$ and reduce the remaining $n-1$entries of $x$ by~1. We will call such $i$ the {\em pivot} and $x_i$ the {\empivotal entry}. The GM-rule respects monotonicity of the entries. It uniquelydetermines a GM-move $x^0 \to x^1$ and an infinite GM-sequence $S$ thatconsists of successive GM-moves $x = x^0 \to x^1 \to \ldots \to x^j \to \ldots$. If $range(x) = x_n - x_1 \leq \ell$ then for all $j \geq 0$:  (i) $range(x^j) \leq \ell$;  (ii) the pivot of $x^{j + \ell}$ is one less than the pivot of $x^j$,assuming that $1 - 1 = 0 = n$.  (iii) $x_i^j - x_i^{j + n \ell} = (n-1) \ell$ for all $i = 1,\ldots,n$.  Due to (iii), we compute $x^j$ in time linear in $n, \ell, \log(j)$, and$\sum^n_{i=1}\log(|x_i|+1)$. For $\ell = 2$ a slighty modified version of theGM-rule was recently introduced by Gurvich, Martynov, Maximchuk, and Vyalyi,"On Remoteness Functions of Exact Slow $k$-NIM with $k+1$ Piles",arXiv:2304.06498 (2023), where applications to impartial games were considered.

GM-rule and its applications to impartial games