Sunday, August 11, 2019

Hexominoes - Scraping the Bottom of the Barrel

In a previous post I made some pretty shapes with the hexominoes and had a good whinge about parity while I was at it. And at the time I thought I'd pretty much exhausted all the possibilities for nice things that could be done with hexominoes. But then a few weeks after that, I found myself on holiday in a little caravan, with the weather being utterly atrocious outside. Luckily, I had decided to bring my little homemade hexomino set along with me, so in the absence of anything else to do I uh... constructed some more constructions, I suppose.

First off, a good (but probably not 100% sure-fire) technique for finding things that are actually constructible (and aren't ruled out by parity constraints) is to start with a rectangle that has odd x odd dimensions (and an area > 210 units) then carve out a central hole until the total remaining area is 210 units. As long as you're fairly careful here, the resulting shape should be solvable.
(Some day I'll do a write up of parity the way I understand it on here... honestly more for my own benefit than anyone else's.)

Fig. 1: Three rectangles with nice symmetrical cross-shaped holes hacked out of them. That third one, I've seen a different solution to before (I think it might be in Solomon Golomb's book).
The third solution, the 15x17 one, took an outrageous amount of time and attempts to solve. It's a shame that processing-power-wise we're a long way off being able to enumerate solutions for a given shape with hexominoes (like we can for pentominoes) - it'd be interesting to see if the ones that take me ages have fewer solutions and are genuinely harder.

I also found a solution to a 13x21 rectangle with a 7x9 removed from the centre. With the central hole being 63 unit squares, that means there's enough space to get all the pentominoes stuffed in there with a little bit of wiggle room to spare.

Fig. 2: This took a while too, as far as I remember. Hexominoes and thin sections (the 3xn bits above and below the pentominoes) don't seem to mix too well.
Then I realised that I didn't need to restrict myself to taking the excess squares out of the middle of the rectangle - I could just as easily round off the edges, and make some stretched-out octagons.
Which I did.
Fig. 3: A way stretched-out octagon.
And here's a slightly more regular, more octagonal-looking one I found a little while later:
Fig. 4: I bet there's a way of doing this with the six internal holes all in a diagonal line, but I'll leave that as an exercise for the reader 'cause I can't be arsed right now.
And just for an extra Brucie bonus, while we're on the subject of octagons, here's one made out of heptominoes that I found a few weeks ago:

Fig. 5: For whatever reason I have no recollection whatsoever of the actual solving process that led to this.

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