# Polyhexes

I know, I know, the site name is 'polyominoes' and this isn't strictly polyominoes, but hear me out...

## Introduction

Back in summer 2019 I got my grubby mitts on a set of these bad lads:

Polyhexes! The 1- through 5- hexes to be more specific, fresh from Kadon Enterprises. And after playing with nothing but polyominoes for years, switching to these was weeeird. The whole '120-degree angles' thing.

With months of solving polyomino constructions I had developed a kind of sixth sense for instinctively knowing whether a piece would fit in a given place, and I had a pretty good idea of which pieces I needed to hold onto for late in the solution. With this bunch, no such luck however. It didn't help that I had no familiarity with the pieces as a set either - with hexominoes (and even heptominoes to a degree) you start to individually know each piece in the set, and can generally rely on memory to get a vague idea of which pieces have been used so far. And the pieces end up with little nicknames based on their shape, so that when I'm frantically scrabbling around looking for a piece I can better remember exactly which one I'm after. With polyhexes it was like starting from scratch again. So I started with just the easy pieces and worked my way up...

## Tetrahexes

There are seven of these, and they can do a surprising amount. Their total area is 28 units, meaning that a 4x7 parallelogram should be possible... and it is. While there is a sort of restriction similar to the parity issue with polyominoes that can occur in polyhexes, it doesn't impact constructions like this the way it does tetrominoes (Although I think it might be responsible for the triangle with side length 7 not being possible.)

(Also surprisingly challenging: drawing hexagonal things in MS Paint.)

There's also this 3-cell-high pattern too. There are two possible solutions for this; finding the second one is an exercise for the reader.

And here's two patterns based on the 5x6 parallelogram with symmetrical holes.

Difficulty-wise, I'd put these somewhere between tetrominoes and pentominoes. Which sort of makes sense, as there are 7 of these, right between the 5 tetrominoes and 12 pentominoes. And that propeller-looking piece is a royal pain in the arse.

There's bound to be more fun stuff to be done with these pieces, but this was all I managed to find before the allure of the pentahexes became much too strong to resist...

## Pentahexes

Pentahexes, for those not in the know, are the shapes made by sticking five hexagons together edge-to-edge. And there are 22 of them, giving a total coverage of 110 units. Which is promising, since 110 can be divided up in various nice ways - we ought to be able to get a nice selection of parallelograms out of them. Sadly, I've been a tad lazy and only attempted the 10x11 so far; my solution is shown below.

If you look at the top-left you'll see that I've tried to carry over my usual technique for polyominoes, which is holding onto the clumpy, blocky bits. But this technique... needs work. This was still a right hassle to solve, I think it took about an hour by hand (and just to rub it in, search software finds solutions to this in like 3 seconds.)

Oh yeah, and there's one other fun thing I noticed with the pentahexes. None of them extend for more than 3 cells in more than one direction. They all could fit in a three-cell-high construction, if someone was masochistic enough to go look for it...

I remembered how deceptively tricky getting the tetrahexes into that 3-cell hexagon thing was. And at this point I could have done the right thing and put down the pentahexes and, I don't know, gone outside or talked to girls or something. But~! Once a challenge like this presents itself, you can't just back down, so I began knocking together little segments of three-cell-high, to be hopefully worked into one big long construction. Remember the infuriating propeller-shaped piece in the tetrahexes? (Maybe you own a set, and know the frustration first-hand!) Well, the pentahexes have a good selection of pieces related to the propeller but with an extra hexagon tacked on, and these have all the infuriating properties of their 'parent' tetrahex, and then some!

So after quite a while (I lost track of time, as tends to happen once you get right into a good polyform construction) I eventually stumbled upon the following solution. And vowed never to tackle something like this again - not for next few hours anyway.

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Lewis Patterson. Last updated 18/06/22.