Congruent Rectangles with Heptominoes

(This post might be Part 1 of many, depending on how much free time I have to look for other sets of rectangles.)

A few weeks ago I found this:

Fig. 1 - Four 10x19 rectangles using one complete set of heptominoes.

There's a number of ways of making sets of same-size identical holey rectangles with the heptominoes. First off, there's three 11x23s which I've done before in an earlier post (and since 11 and 23 are both odd you can get the hole slap bang in the centre of each rectangle.)

Then, there's four 10x19s as above. Then it looks like the next set of rectangles that might be possible is nine 5x17s (with potential for centered holes too!) but whether that's even solvable is another question entirely. I mean, it goes without saying it'll be a frustrating ordeal knocking it out by hand. Actually, Patrick Hamlyn managed twelve 8x8 squares (three quarters of the way down this page) so anything's possible.

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Also, a thing I've found that works really well for heptominoes if you're just after solutions and not the glory of finding them entirely by hand. Solving the first 90% of a shape by hand, saving the nice clumpy blocky bits of course (that goes without saying), then crack open some software like FlatPoly2 to brute-force the last 10 or so pieces. This works with heptominoes especially, since the really wide search tree that is placing the first ninety-odd pieces doesn't seem to play particularly nice with my computer. (More expensive computers may get better results!)

Then there's that other thing. The one that feels partially like cheating, just a teensy little bit. Picture the scene, you're just spend two hours slogging away at a particularly gnarly construction and just can't get the last few pieces in. There's a point at which you start to wonder if what you've got left here is even possible. So occasionally I've succumbed to the temptation to input in the shape of the remaining hole and the last five or six pieces, not to find a solution, but just to check if one exists. Then if it transpires you're looking for a solution in a space where none exist, I tear out one or two pieces, input the new, bigger hole and keep checking until we've got a gap where there is definitely at least one possible solution. Finding that will still be an ordeal though.

So it's computer-assisted solving. I don't know whether it counts as truly properly solving by hand, but when I've just sunk a whole evening into a construction and it's pushing eleven and I've got work tomorrow, I'm usually not too bothered.

Baby steps with Polyhexes

I know, I know, the blog name is 'polyominoes' and this isn't strictly polyominoes but hear me out. A few weeks ago I got my grubby mitts on a set of these:

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 is 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, then.

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 (I think it might be responsible for the triangle with side length 7 not being possible though.)

(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, 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 and 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.

Fig. 6 - The 22 pentahexes squeezed into a little narrow construction that I'm stunned actually works.