  A. Mitschke showed that a variety with an $m$-ary near-unanimity term has
J\'onsson terms $t_0, \dots, t _{2m-4} $ witnessing congruence distributivity.
We show that Mitschke's result is sharp. We also evaluate the best possible
number of Day terms witnessing congruence modularity. More generally, we
characterize exactly the best bounds for many congruence identities satisfied
by varieties with an $m$-ary near-unanimity term.
