Let $G$ be the group of permutations on $\mathbf{Z}$ that preserve distance:
$$G = \{\sigma:\mathbf{Z}\to\mathbf{Z} : \, |\sigma(i)-\sigma(j)| = |i - j| \mathrm{\,for \, all \,} i,j\in\mathbf{Z} \}.$$
Show (by elementary methods) that $G$ is generated by two elements of order two.
I can do this for the dihedral group but that used geometric intuition that I don't have for the group of distance-preserving permutations on $\mathbf{Z}.$
Two elements that might interest you are $\phi, \psi$ defined by $\phi(x) = -x$ and $\psi(x) = 1 - x$. After confirming that $\phi, \psi \in G$ let $H = \langle\phi, \psi\rangle$, we want to show that $H = G$.
Start by figuring out what the product $\phi\psi$ is. Use this to show that for $g \in G$ there is an element of the coset $Hg$ that fixes $0$. Then show there's an element that fixes $0$ and $1$. Then show that any element that fixes $0$ and $1$ is the identity. Thus for all $g$, $1 \in Hg$ so $g \in H$.