When I had Agincourt made, I purchased a bulk order of 4″ × 4″ × 1″ white cardboard jewelry boxes. They look quite nice, and they fit both Agincourt and L-Topia, but I have enough of them that I’m on the lookout for ideas for polyform puzzles that fit nicely into a few square layers. And now I’ve found one:
I stumbled upon this by noticing that there are 21 pentominoes of this symmetry type, which could make three 5 × 7 layers. I wanted square layers; usefully, squashing the cells into rectangles with a 5 : 7 ratio of width to length simultaneously gave me the square layers and gave the cells the right type of symmetry.
It’s been observed that any of the subgroups of the symmetries of the square can be used as the basis for a type of polyomino puzzle. (See Peter Esser on pentomino variations, and particularly the page on parallel polarized pentominoes, which are equivalent to rectangular pentominoes.) For Agincourt, I physically realized one of these types by laser-cutting symmetrical, arrow-shaped holes in every square cell. Other types have been made by changing the shape of the cells themselves. Rhombic pentomino sets have been produced by Kadon as Rhombiominoes. Sets of rectangular polyominoes, shaped like Meiji chocolate bars, have been produced by Hanayama. (These may not be equivalent to the rectangular polyominoes above, if the top is distinct from the bottom, which isn’t clear from the pictures there.) I’m not aware of anyone who is producing complete sets of rectangular pentominoes, so there’s a gap I’m willing to step into.
If you take out the pentominoes with a diagonal line of symmetry in their non-squashed form, (the green ones above) the remaining 18 pentominoes come in 9 pairs, where each pair contains two different squashed versions of the same pentomino. With these pieces it is interesting to try to tile a pair of shapes with the same orientation such that one piece from each piece pair is in each shape. (Note that if the two shapes had different orientations, you could always make the second shape with corresponding pieces in the same position as the first, but squashed in the other direction.)
Since the set has area 90, the obvious thing to try is two 9×5 rectangles. The next most obvious thing to try is two 7×7 squares with corners removed. Neither of these seem to work, although I have no proof.
One thing that does work is a 7×7 square with a 4×4 square cut out of one corner. But this is again just the case where you can trivially get the solution to the second piece by squashing the pieces differently, because this shape has diagonal “mirror symmetry”.
Another problem is finding three congruent shapes, each of which has the following property: three of its pieces have their twin in one of the other two shapes, and three have their twin in the remaining shape:
I’m looking into having some sets of the rectangular polyominoes made, and if I can do so economically, I’ll sell them through the store. (Sadly, TechShop Portland, the facility where I made Agincourt, has gone away, so I will need to look at other options.)