Subsets of the Hexominoes

Like hexominoes, but are overwhelmed by the sheer number (35) (!) of them? Well, have I got just the puzzle for you. It's these:

The set consisting of the eight hexominoes that contain a 2x2 block is... less than fantastic. Sure, at first glance it looks interesting, a manageable set of cooperative pieces, but scratch the surface and you hit the snag that seems to ruin everything in the world of tetrominoes and hexominoes (how the octominoes avoid it is a mystery to me) - parity. The set has an imbalance of ±2, so kiss goodbye the hopes of being able to make any rectangle with an even edge length, and many other symmetrical constructions. To alleviate this problem somewhat, I've added to the set a single monomino which allows a 7x7 square to be filled. It's natural to want to solve it with the monomino in an aesthetically pleasing place - the centre or a corner - but due to the aforementioned parity constraints none of these is possible. In fact, the only places the monomino can go are the black squares on the diagram below, which means there are effectively four different positions for it excluding rotation and reflection.

All of these are solvable - holes at positions 1 through 4 have 11, 22, 11 and 5 solutions respectively. But the ones with the notch at the centre of an edge are undeniably the best.

If you take the pieces off-road, away from the confines of the tray, there are other shapes you can wring out of them too with or without the supplemental monomino. A 5x10 with two corners on the short side removed is possible with the eight hexominoes, and there are (if FlatPoly2 is to be believed) 47 ways of doing this.

Fig. 1: Here's one of them

There are also a few configurations of the 5x10 with two holes down its long line of symmetry which have solutions, and I'm beginning to think that maybe I should have made the wooden puzzle a 10x5 frame with two monominoes and it would have been more interesting.

Again, numbers are excluding rotations and reflections. And these numbers are of unique solutions, it doesn't guarantee interesting solutions. Several may be related to each other in that reflecting a symmetrical subpart of one solution made of two or three hexominoes could lead to another. Still counts. I always felt a little bit cheated by this as a kid, looking at the 2,339 distinct solutions of the pentominoes in a 6x10 rectangle. But I grudgingly came to accept that cases like these were in fact solutions in their own right.

Fig. 3: Like this. This would have made nine-year-old me's blood boil. (Not really.)

In fact, that example solution before is even more egregious an example than I'd first realised. The two pieces at the top (the fish and the one that looks like a backwards δ) can also be flipped over, giving four possible solutions (two for each of the configurations of the orange part). And then the entire non-orange section of the image can be reflected and rotated, giving a total of 2x2x4 = 16 solutions for the price of one.

The Remaining 27 Hexominoes

Perversely, taking a parity-unbalanced subset away from the hexominoes doesn't leave a nice balanced set remaining. Of the 27 hexominoes that don't include a 2x2 sub-square, seven of them have 4-2 balancing when checkerboard-coloured, which means we've still got all the same problems as with the full set. In addition, we've took out eight pieces which are some of the easiest to work with, meaning that the resulting set is going to be a real challenge to work with, at least if we're solving by hand.

Fig. 4: Here's the set crammed into the first shape I could think of, solved by computer because I got lazy. It's like 28°C here right now, it's lucky I can be arsed to even write a blog post.

The other subgroup of the hexominoes that suggests itself is the set of 15 symmetrical hexominoes (including both rotational and mirror symmetry). However, these pieces are just so uncooperative I've found it pretty much impossible to solve anything pretty or interesting with them, and I've tried on and off for the past couple of months.