We say that an algebra A has sparse relational clone (or equivalently, has few subpowers) if there is a positive constant k so that |Sub(An)| ≤ 2nk for all integers n > 1. It is known that a finite algebra A has sparse relational clone iff A has one (or equivalently, all) of the following: a cube term, an edge term, a parallelogram term. Recently, Aichinger, Mayr and McKenzie have shown that every finite algebra F with sparse relational clone is finitely related; that is, there is a finitary relation ρ over F so that the finitary operations that respect ρ constitute precisely the clone of all term operations of F. Let F be any finite set. We work with subclones of the clone I of all idempotent operations over F. In this talk, we describe finitely many clones E1,... , Ek included in I which turn out to be precisely the maximal subclones of I that fail to contain a cube operation. They are also precisely the maximal not finitely related subclones of I. For each of these clones, there is a trivial polynomial-time algorithm to determine if an operation belongs to it. This yields an algorithm to determine, given finitely many operations f1,... ,fℓ over a finite set F, whether or not the clone generated by these operations contains a cube operation. With F fixed, the algorithm is polynomial-time as a function of its input; but with variable F it becomes exponential-time. There is a problem here, to determine if this computational problem with input F,f1,... ,fℓ has a polynomial-time algorithm. Applying the theorem of Aichinger-Mayr-McKenzie, we obtain the corollary that a finite idempotent algebra F has sparse relational clone if and only if F is inherently finitely related; i.e., every algebra obtained by enriching the set of operations of F is finitely related. Using a recent result of Libor Barto, and an application of commutator theory, we find interesting equivalent conditions for the truth of Valeriote’s conjecture that a finite algebra in a congruence modular variety is finitely related if and only if it has a cube term.