Entry tags:

math, pentomino, polyomino

Polyomino cover variations
To recap, since it's been a while since I've done one of these, a polyomino cover (not real terminology, but mine) is a set of cells in a square grid into which any of a set of polyominoes can be placed. A succinct cover is one into which each polyomino in the set can be placed in exactly one position. We[1] are interested in minimal (smallest possible) covers because that's where the fun puzzles are. End of recap.

Looking back at my old graph paper notebooks, I found a variation on the polyomino cover problem that I never bothered to actually explore. A no-slide cover is one in which no piece can be placed where it could slide around. The minimal covers of the pentominoes are not no-slide covers, because the P pentomino could slide around. Here's an 11 cell no-slide cover of the pentominoes, which I'm guessing is probably minimal:

Another problem I came up with this morning: A somewhere succinct polyomino cover is one where for each polyomino in the set being covered, there is a piece in the cover in which it can be placed in exactly one spot. Here's my best somewhere succinct cover of the pentominoes:



The Z occurs once in both pieces, the Y occurs twice in the left one, and the L twice in the right one, but each pentomino occurs exactly once in at least one of them.

[1] We meaning me.