In my previous post I offhandedly tossed off a taxonomy of polyform positioning problems:
| No Overlap | Overlap | |
| No holes | Tiling | Piling |
| Holes | Packing | Tacking |
The vast majority of the problems you will find in the wild are tiling problems, with an occasional sprinkling of packings. The other side of the matrix is rare enough that it didn’t already have established terms. Piling in particular is a topic that I haven’t focused on since before the blog, so it was overdue for another look.
The earliest appearance of pentomino piling that I’m aware of is a set of problems that appeared in Puzzle Fun #7. Ariel Arbiser filled a 6×9 rectangle with 6 pairs of pentominoes that overlap in one cell, and asked if the positions of overlaps could be made symmetric. Pieter Torbijn’s solution was printed in Puzzle Fun #9:
If pentominoes overlap two different other pentominoes, you get a chain. I explored this type of problem before the blog, and wrote up some results. One problem that I set at that time was making such a chain in a 7×7 square such that every overlap cell was a knight’s move from the next. (The X and I pentominoes are the only ones that do not contain two cells a knight’s move apart, so they must be at the ends of the chain.) Recently, Bryce Herdt solved it:
Remarkably, Herdt reports that the only other solution is the one made by flipping the F so that it overlaps with the I in a different cell.
There are 12 different tetrominoes with a single marked cell. It was these that inspired me to look at overlap problems again, since, (with three T’s) they have a parity problem that makes it impossible for them to tile a rectangle. It seemed natural to ask if they could pile a rectangle with the overlaps occurring at the marked cells. Herdt found a symmetrical solution:
Herdt noted that this is the only type of symmetry that a solution can have. If a piling had vertical reflection symmetry or rotation symmetry, it would have unworkable parity.
There are 20 tetrominoes with two marked cells. These are the tetrominoes in Kate Jones’s Fill-Agree puzzle. They could make chains, and should not have any parity issues. (Edit: There is, in fact a parity issue. There is an odd number of pieces where the marks have opposite parity. Since the chain must end on the same checkerboard color it begins on, we can’t close the loop. Thanks to Bryce for pointing this out.)
Problem 56: Pile a 6×10 rectangle 5×12 torus with the tetrominoes with 2 marked cells, such that overlaps only occur on the marked cells, and the overlaps form a single circuit containing all of the pieces.
Is there more that we can do with polyomino piling? Will tacking be useful for future puzzles? Will I stop trying to make “piling” and “tacking” happen? Stay tuned for the answers to these questions and more! (Well, more silly questions, at any rate.)