Given a set and a family of its subsets, an exact cover selects members of that family so that each element of belongs to exactly one selected member. Thus the selected members are disjoint sets whose set union is . Anchoring a recursive choice at the smallest uncovered element avoids counting a cover in different orders.
Exact cover is a concept from combinatorial mathematics and is particularly well-known in the context of the Donald Knuth's Algorithm X, which is used to solve the Exact Cover Problem. The problem can be described as follows: Given a set \( S \) and a collection of subsets of \( S \), the goal is to find a selection of these subsets such that every element of \( S \) is contained in exactly one of the selected subsets.
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