Hexahexes

I always sort of overlooked polyhexes in the past. Polyominoes were the main event, so to speak, and the first polyforms I really got properly into, and on the rare days I wanted a really infuriating challenge I would usually turn to the polyiamonds. But polyhexes for whatever reason just weren't really on my radar. Sure, I had a physical set of the 1- through 5-hexes from Kadon, and I solved a couple of things with them. And I even made a half-arsed blog post a while back. But that was about it really. Until now.

A few weeks ago on a whim I got a set of hexahexes cut out of the cheapest MDF money could buy. I didn't even shell out the extra two quid for the laser cutting people to cover the wood with protective masking tape, instead opting to let the bits get gently toasted around the edges by the laser. And then I took them on holiday, to a chilly weekend in a caravan in Northumberland where I knew I'd be a captive audience in the evenings. And while there I slowly began to realise that I'd missed out... Polyhexes were fun. In fact they weren't just fun, but were in fact... very fun.

This photo doesn't really give any sense of scale, but each hexagon is 8mm to an edge, and the full solution has a diameter of about 40cm or so on average. I think. Nice and chunky. I actually checked the scale this time before cutting unlike my positively tiny enneiamonds.

The slight browning of the edges turned out to be something of a blessing in disguise - it makes the borders between adjacent 'hexes stand out a bit in photos which is handy. Sometimes I'm too lazy to draw up a pen-and-paper record of a solution, so just being able to take an aerial photo that I can work from to create a digital image is a nice time-saver. And talking of digital images:

Here are solutions to two different hexagons, the more compact one is the shape of the solution Kadon uses for Hexnut II; the larger thinner hexagon I haven't seen anywhere before. I haven't ran the numbers for hexagons larger than this; it could be that there is an even bigger thinner (and therefore harder to solve) hexagon ring out there waiting to be found.

Speaking of, solve difficulty is the best thing about the hexahexes. It's somewhere between that of hexominoes and heptominoes, I'd say. A good, meaty challenge but one that I don't need to set aside a whole evening for. There are a couple of kinks to be ironed out with my solving technique, though, mostly the fact I'm not used to hexagons so it's often not immediately obvious whether a piece will fit in a certain place without actually trying it a few different ways.

Unholey Hexahexes

If you discard the holey hexahex (as we sort of unintentionally did for the rings above) you get 81 pieces and a total area of 486 hexagons, which divides up very nicely indeed. So far all I've done with this set is the really easy stuff - a couple of approximations of parallelograms, of which one is shown below for your perusal.

The 81 unholey hexahexes squeezed into an 18x21 parallelogram. Solve time approx. 45 minutes manually.

But there's a lot more out there than just parallelograms. It's fairly easy to work out formulae tying the edge lengths to the area for various hexagons, triangles and other such shapes that hexagons lend themselves well to. And from there just a little bit of searching for edge lengths that give the magic number, 486.

Which will all be a nice excuse to post a bunch more blog posts. I need to pick up the pace - this year my posting rate on here has gone right down. That's partly because I've been putting some things directly to polyominoes.co.uk (and discovering the joys of trying to display characters like '°' in html), but it's also partly because my interest in polyforms seems to come and go in phases. And summer this year I've just been preoccupied with other things (recording an album, teaching myself to read Japanese, and dusting off the Rubik's cubes and getting back into speedsolving). But now with winter drawing in, and with its long cold rainy evenings with nowhere else to go and not much else to do, there's a non-zero chance I'll dedicate a bit more time to the sacred art of polyform-ing. And to the subsequent rambling about it on here.