Wednesday, May 27, 2020

A first foray into Multiple Congruent Rectangles with Octominoes

This one had been on my to-solve list for ages now.
369 is 3x3x41, which means that the octominoes can be partitioned into equal-sized groups of 3, 9 and 41 pieces. And also 123 groups if you're so inclined, but in this case each group would contain only 3 octominoes so you can't really do a lot with them. In fact it sounds like there's an impossibility proof lurking there - prove there's no shape that allows 123 congruent copies to be constructed with the set of octominoes. It would have to have a hole, and be able to accommodate the 1x8 piece, so if there is a shape it's likely going to be a real odd-looking beast. Anyway...

Splitting the set into 3 yields the most manageable challenge (nine looks possible but a little intimidating*) so of course it was the one I attempted first:


This was a rare case of one of those blissfully hassle-free solves where there's pretty much no back-tracking and the pieces just fit first time, followed by momentarily sitting back stunned because I now had a free afternoon I hadn't previously banked on. The whole thing came together in three hours, tops.
I've still got gripes with octominoes though. Mainly, that there are too many of the damn things. One thing I kept encountering was I'd have a space that could be nicely filled by one specific octomino that I knew I hadn't used yet (keeping track of which pieces have already been used is something you just begin to get a feel for after a while without really trying to), but then not being able to locate that piece. Or not without several minutes of scrabbling around the little box I keep them in looking for it. They're transparent acrylic too, which makes spotting the one you want even harder since the edges of any piece and the pieces under it all kind of blur together. But between these frantic moments of digging around for a piece which may or may not be there, the solving process gets strangely therapeutic in a way.

Next time, nine 15x22 rectangles, two holes each?

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* 41 is pushing it, massively, but it is doable - I've seen a solution for 41 copies of a rotationally symmetrical near round-looking shape with a central hole, but a.) that was (as far as I'm aware) found with computer search, and b.) I seem to be unable to re-find the image right now.

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