  For every prime number p and integer $n>1$, a simple, involutive,
non-degenerate set-theoretic solution $(X,r$) of the Yang-Baxter equation of
cardinality $|X| = p^n$ is constructed. Furthermore, for every
non-(square-free) positive integer m which is not the square of a prime number,
a non-simple, indecomposable, irretractable, involutive, non-degenerate
set-theoretic solution $(X,r)$ of the Yang-Baxter equation of cardinality $|X|
= m$ is constructed. A recent question of Castelli on the existence of singular
solutions of certain type is also answered affirmatively.
