Problem: Show that all horses are of the same color. “Solution”: We will show, by induction, that for any set of $ n$ horses, every horse in that set has the same color. Suppose $ n=1$, this is obviously true. Now suppose for all sets of $ n$ horses, every horse in the set has the same color. Consider any set $ H$ of $ n+1$ horses. We may pick a horse at random, $ h_1 \in H$, and remove it from the set, getting a set of $ n$ horses.