Putnam 1972 A2

Let $S$ be a set with a binary operation $\ast$ such that

1. $a \ast(a\ast b)=b$ for all $a,b\in S$ .
2. $(a\ast b)\ast b=a$ for all $a,b\in S$ . Show that $\ast$ is commutative and give an example where $\ast$ is not associative.