  A quandle is an algebraic structure with three axioms: idempotency,
right-invertibility, and self-distributivity. Right-invertibility allows us to
regard a finite quandle as a finite sequence of permutations. For a finite
connected quandle and any two of its permutations, there is a 1-1
correspondence between the lengths of their disjoint cycles. This list
(possibly with repeats) of lengths is the profile of the connected quandle. In
this article we prove that if this list does not have repeats then the finite
connected quandle it refers to is latin. This is the criterion mentioned in the
title. Furthermore, we define super quandle to be a quandle whose opposite
structure is again that of a quandle. Since latin Alexander quandles are super
quandles, then if a latin quandle is not a super quandle then it is not an
Alexander quandle. This is a criterion for a latin quandle not to be an
Alexander quandle.
