Heptiamonds
subtitle goes here.
Introduction
By all rights these should be one of my favourite sets of pieces. Twenty-four of them, giving a lot of divisibility options when constructing shapes with 2-, 3- and 6-fold symmetry, no parity issues to speak of (I think it's the odd number of unit triangles that do this, but don't quote me on that), no holes... On paper these ought to be one of the most versatile sets of pieces out there.
But in reality... well, they are versatile, but actually solving the shapes takes way more elbow grease than similar endeavours with polyominoes or polyhexes. Thus the reason the blog ran for like five years with heptiamonds only getting the occasional cursory mention in passing - I do occasionally make things with the set, but it's a lot of sweat for something that doesn't look particularly interesting next to, say, an octomino rectangle packing. and there's never a lot I can say about the solution process too, no insights other than tipping the set of pieces out onto a tabletop and applying unreasonable amounts of trial and error.
Shapes, shapes and more shapes
There are various simple geometric shapes possible with this set: various trapezia and a family of parallelograms (the 3x28 is at the top of this page, the 6x14 is somewhere here, the rest are an exercise for the reader). But a little more excitingly, there are various nice hexagons, with 2- and 3-fold symmetry, as well as a triangle with a central hole, courtesy of the fact that 7x24=168 is one less than 13² = 169. (By complete coincidence (I think) the octiamonds' total area is one less than a square number too, allowing for a bigger triangle with a single central hole.)
Fig. 1: A couple of example shapes with the heptiamonds. These were all found by hand, and all took way too long to find.
If you have an aversion to central holes in your constructions for whatever reason, then you're in luck too:
Fig. 2: Skinny hexagons.
You know how I mentioned at the start the fact there's 24 pieces makes for nice possibilities? One of those is that the full area the pieces cover is divisible by 6, so shapes with six-fold rotational symmetry are possible. Two of those that I solved by hand are shown below; there are many, many more over on Todor Tchervenkov's page, and even a few where the total possible number of solutions have been enumerated.
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Fig. 3: Simple figures with sixfold rotational symmetry.
Of course, the total area 168 divides by 12, meaning that in theory making two smaller congruent sixfold symmetric shapes, each with half the pieces, should be possible too. But let's not get ahead of myself just yet. Closest I got to that was the following two congruent shapes, which the keen eyed among you will have realised is just a lazy modification of the configuration in the banner image at the top of this page.
Technique for manually solving
Yeah, this is going to be a short section. My general approach is one of try, try, try and eventually they'll all fit... maybe. But there are a few little tips that seem to speed things up a tiny bit. A couple of patented trade secrets I'll let you in on.
The program trisolve by Peter Esser can be used to determine 'frequencies' for each piece in a set of polyiamonds - that is, a small region is tiled with a subset of the pieces, thousands of times, and the pieces which end up being used most often are the most 'useful' ones, ones worth holding onto for the end of a solve.
I was convinced I'd done this in the past with the heptiamonds, but I've looked all over the place and can't find any record of it so I ran it again just now, ten thousand trials packing whichever heptiamonds into the shape below:
The sample shape, which can be packed by 8 heptiamonds. I picked this to include a little bit of all the edge types I'm likely to encounter solving a simplish convex shape - some 60° angles, some 120°, and a few different edge lengths.
However, when I ran this and got the results, they weren't what I was expecting.
I would have included an actual screenshot, but a.) trisolve does a weird thing where the frequencies are partially obscured by the diagram of the polyiamond unless you scale them impractically small, and b.) I did something silly and crashed trisolve before I could take a screenie. Sorry.
Sure, there's some fairly obvious expected things - the teardrop-shaped piece is high-frequency (and therefore quite friendly to tile with), and the straightish pieces are up there too. And the threefold-symmetric propeller piece is in a class of its own for being horrible to utilise. But there are a couple of surprises (to me at least). Things I had just assumed as a sort of conventional wisdom which might have been hurting my ability to pack these shapes all this time.
Notably, the piece with frequency 4456, sixth one down on the left hand column. I had naïvely assumed since it's got a 4-cell triangle inside it it might work like the polyominoes with 2x2 square subsections. But I guess its wiggly perimeter and lack of a really smooth side must cause more harm than the triangle bit does good.
I tried running counts or a few more initial shapes - a few with more wavy edges, or more spikes - and the frequencies fluctuate as you'd expect, but the story is mostly the same in each case. Which means I'm going to need to rethink my solution process for these. I think it may be that the best choice of pieces to hold back until the end will depend massively on the way the perimeter of the outer shape looks in the region that's going to be tiled last. Like if it's a triangle and you're solving down towards a corner, you're maybe going to want to hold onto some shapes with the ability to fit a 60 degree point, and also shapes willing to fit nicely next to that first piece. That kind of thinking.
Triangles as base shapes seem to be too finicky to allow for the kind of 'one size fits all' strategy that does the job with polyominoes.
One more solution I found with them, just for the hell of it. Again, this wasn't intentionally solves as the very restrictive two shapes, more likely I solved a 6x14 or 7x12 parallelogram that just happened to have a seam all the way through in an opportune place.
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Lewis Patterson. Last updated 08/11/2024.