Question 172
Let $S=\{s_{0}, \ldots, s_{n}\}$ be a finite set of integers, and define $S+k=\{s_{0}+k, \ldots, s_{n}+k\}$. We say that $S$ and $T$ are equivalent, written $S \sim T$, if $T=S+k$ for some $k$. Given a (possibly infinite) set of integers $A$, we say that $S$ tiles $A$ if $A$ can be partitioned into subsets equivalent to $S$. Such a partition is called a tiling of $A$ by $S$.
Exhibit a set $S$ which tiles the integers $\mathbf{Z}$ but not the natural numbers $\mathbf{N}$.
Exhibit a set $S$ which tiles the integers $\mathbf{Z}$ but not the natural numbers $\mathbf{N}$.