Pillowminoes (From the Pillowbook VI)

The admission of an abundance of pillows with straight edges to the zoo raises the question whether these new citizens are any good. We have employed the ones with two straight edges to form arrows and combine to Hamiltonian circuits. Today we will look at those eight pillows with a single straight edge (let’s call them the singles)

Eight 01

Can we use them to tile a curvy 7×7 square, say? The answer is clearly no, because the singles have to hide their straight edges by combining in pairs to pillowminoes. This means we can only tile curvy shapes with an even number of singles. Here, for instance, is a simple solution that shows how to tile a 7×7 curvy square with a gap at the center. It also shows how to tile this shape with a single pillowmino.

7x7 01

This looks too easy? Can we also tile the same curvy 7×7 square so that the missing square at the center has four straight edges?

7x7no 01

A little trial and error shows that this is not possible, but we would like to have a reason for this. We need for singles to neighbor the missing square at the center with their straight edges. I have indicated their position by a slightly darker shade of green. Thus the remaining lighter green squares will be entirely curvy, so needs to be tiled with singles that have combined into pillowminoes. That, however, is impossible: Color the squares alternatingly yellow and pink, as in the solution above. Each pillowmino will cover a pink and a yellow square, but the light green shape that needs to be tiled consists 24 pink and 20 yellow squares. This argument also shows that the missing square needs to have all edges curved.

What else can we do with the pillowminoes? There are 36 of them, and not all of them tile by themselves. If we want to tile the curvy 7×7 square with a circular gap in the middle, we will also need to balance the convex and concave edges, as explained earlier.

Here are the ten balanced pillowminoes:

Balanced 01

Surprisingly, only the top left will tile the 7×7 (or any larger) curvy square with a central gap.
Below is an example that tiles with four individually unbalanced but centrally symmetric pillowminoes.

7x7b 01

More to follow!

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