Most prolific hexomino cover piece:
Most efficient hexomino cover piece:
My program for finding minimal succinct covers first must find all of the polyominoes that could be legal pieces in a succinct cover, that is, polyforms with at most one placement for every polyform in the original set. These pieces are mathematically interesting in their own right.
We can look for pieces with interesting characteristics, for example, the largest piece, and the piece within which the largest number of polyforms can be placed. (I call the latter prolific in analogy with my use of the term succinct. A prolific piece is one with a lot to say.)
Of interest for the minimal succinct cover problem is the ratio of number of polyforms in the set of interest that can be placed in the piece to the size of the piece. We would expect a minimal succinct cover to have pieces with high ratios. And in fact, the piece with the highest ratio for the hexominoes is part of a minimal succinct cover, as we can see below. A good greedy algorithm for finding a small cover would be to repeatedly take the piece with the best ratio among the pieces that only contain polyforms with no placements in the pieces we've already taken.
A minimal succinct cover of the hexominoes