Open Problems

There are many open problems based on the above links as well as those enumerated below.

22 holes in a one-sided pentomino construction (or a proof that this cannot be done)
Maximum number of holes in a hexomino construction

Multiple replications of a one-sided hexomino based on 2-2-2-2-2-2-2-4-4 or 2-2-2-2-2-2-3-3-3-3.

Other constructions with the 363 octominoes without a hole.
The remaining hexomino pairs in rectangles.
 Further 1-3-3 constructions with the one-sided hexiamonds
 More multiple replications with the heptiamonds based on 2-2-4.
 More than 25 internal holes with the heptiamonds or a proof that 25 is the maximum.
 More multiple replications with the one-sided heptiamonds.
 62 congruent shapes with the 620 one-sided heptahexes

Find a 3 x m x n box with the tetracube at the right or prove one cannot be made.

Other Polyforms
 Other figures with the sliced heptiamonds
 Other rectangles with the thirty nine perimeter 9 polyares