Nemotron-IMO-Bench 2026 C10

Prove that for any $n\geq 1$ there is a subset $S$ of $\mathbf{Z}^n$ such that no two points in $S$ are neighbors, and any point in $\mathbf{Z}^n\setminus S$ has exactly one neighbor in $S$. Here two points $p=(p_1,\ldots, p_n)$ and $q=(q_1,\ldots, q_n)$ in $\mathbf{Z}^n$ are called neighbors if $$\sum_{i=1}^n |p_i-q_i|=1.$$