2 problems

Let $d$ be a positive integer and let ${\cal D}\subset\mathbb{Z}^{d}$ be a finite difference set satisfying … for every $(a_1,\dots,a_d)\in{\cal D}$. The positions are vectors…

Let ${\sc Nim}^1_{n,=k}$ be the impartial game on nonnegative integer vectors in which a move decreases exactly $k$ coordinates by one. A position from which the previous player ca…