A minimal succinct cover of the pentahexes:
A minimal succinct cover of the tetrasticks:
A minimal succinct cover of the checkered pentominoes:
A 31-cell contiguous irreducible cover:
The area of polyomino covers is still relatively unexplored, and plenty of new problems are waiting to be discovered. If you discover new problems of interest or better solutions to the problems I have discussed, please send them to me, and I will credit you on here.
All of the puzzles discussed in these pages can be adapted to larger polyominoes. My python program for solving minimal succinct cover problems is, unfortunately, very slow when handling relatively large polyomino sets. Even the heptominoes are currently out of the range of what it can solve on my computer.
Likewise, the principles used here can be extended to other polyforms. Ed Pegg has a page with a section about the minimal cover of tridominoes on mathpuzzle.com. There was a contest based on a cover of Chaos Tile combinations there, but it appears that nobody entered. My minimal succinct cover program is able to find solutions for polysticks, polyhexes and polyplets, a.k.a. polykings (polyominoes with cells that may be connected by corners instead of just edges.)
A cover can be said to be reducible if it is possible for a cell to be removed from it with the result still being a cover. A minimal cover is of course irreducible. How many cells does the largest contiguous irreducible cover of the pentominoes have? My best solution has 31. The coloring scheme I used shows the pentomino that would be lost if any of the cells within were removed. The lone cell is part of two T pentomino placements that would both be lost if the cell were removed.
Different coloring schemes than the one used for the panchromatic covers could be considered. For example, we could have 3 colors and require that they occur in a 2:2:1 ratio in each pentomino.
With 3 colors there are exactly 12 combinations for filling 5 cells with 2 of the 3 colors. Can you find a minimal cover for which each pentomino occurs with a unique combination? (Pentominoes with all 3, or only 1 color would not be counted.) There are two ways to interpret this probem. Either the same pentomino may occur more than once, if it contains the same combination of colors in each occurance, or it may occur only once in the entire cover. I've been able to make a cover with the latter rule with 25 cells.