Octiamonds

Adapted from a nasty blog post I wrote back in 2020


Polyiamonds are tricky. Even the smaller sets seem harder than their similarly-sized polyomino counterparts. Hexiamonds, for example. There's 12 of them, same as the pentominoes, and in a geometric sense they don't seem any more jagged or otherwise unruly, but for whatever reason solving anything with them seems a lot harder.
I mean, it could just be that I'm more used to polyominoes and that a sort of intuition for other polyforms would build naturally with time. And with polyominoes I know all the handy tips and tricks, which pieces (or types of pieces) are most useful in which situations, whereas with polyiamonds I don't have that (yet).
With larger sets of polyominoes, (i.e. hexominoes and above) a technique emerges of saving the more cooperative shapes for the end game, and as the sets get larger this pool of 'nice' pieces increases rapidly in size. But with (say) octiamonds it's not so obvious which shapes are the most useful. They're all pretty hideous, actually, at least to the untrained eye. The little hexagon made of six triangles is the closest analogue I can think of to the 2x2 square block that makes for nicely-behaved polyominoes. But there are only 4 out of 66 octiamonds which contain it and even these 4 pieces don't play especially nicely together with each other.

The solution below was the result of about an hour and a half of stumbling about cluelessly followed by a flash of pure luck.

Another fun fact: Drawing these out neatly is really hard too. Pixel art and triangles don't mix too well.

And the solution below, found in May 2020, took longer still. From the old site, "I solved the bottom edge first, and the rest followed, four hours later...".

Fig. 2: Equilateral triangle with edge length 23.



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Lewis Patterson. Last updated 18/06/2022.