"There's No Way I'd've Found That By Hand!" Vol. 1

At work I have a notebook. And in it there's surprisingly little pertaining to actual work, but lots of ideas, calculations and sketches for possible shapes that could be filled up with various sets of polyominoes or other polyforms. Most of these end up solved at home, then drawn up all pretty and posted on here, but there are some that I either try and can't do, or that just look so intimidating I don't even try. It's those jagged diagonal edges... I just have a right job doing them.

So, here's a little collection of solutions found with various programs, that I was too weak to suss out by myself.

In hindsight, this one doesn't look that bad, and I've solved similar in the past. I guess at the time I was just not feeling up for the challenge. I think (judging by the colour and scale of the image) this one was found using the solver on Peter Esser's site here.

I'll be honest, I didn't even attempt this one. I did the calculations to make sure that it was permitted by the parity constraints, then just despaired at the thought of having to actually solve it, central holes and all. But FlatPoly2 made short work of it, finding this in about five seconds flat.

Then there's this family of solutions I have no recollection of looking for but are in the folder called 'Polyominoes' so I may as well just post them for the hell of it.

 I think at this point, hexomino solutions just aren't the most impressive thing in the world any more, computer-found or not. But more interesting things like heptominoes take a while to do, and I've got driving lessons and band-related business to contend with on most evenings, so that stuff tends to take a back seat.
Actually, I've got a nice new shiny set of octiamonds that I've yet to do anything of note with... I'll have to have a little play around with those, see if I can create anything worth posting on here with them.

Hexominoes - Scraping the Bottom of the Barrel

In a previous post I made some pretty shapes with the hexominoes and had a good whinge about parity while I was at it. And at the time I thought I'd pretty much exhausted all the possibilities for nice things that could be done with hexominoes. But then a few weeks after that, I found myself on holiday in a little caravan, with the weather being utterly atrocious outside. Luckily, I had decided to bring my little homemade hexomino set along with me, so in the absence of anything else to do I uh... constructed some more constructions, I suppose.

First off, a good (but probably not 100% sure-fire) technique for finding things that are actually constructible (and aren't ruled out by parity constraints) is to start with a rectangle that has odd x odd dimensions (and an area > 210 units) then carve out a central hole until the total remaining area is 210 units. As long as you're fairly careful here, the resulting shape should be solvable.
(Some day I'll do a write up of parity the way I understand it on here... honestly more for my own benefit than anyone else's.)

Fig. 1: Three rectangles with nice symmetrical cross-shaped holes hacked out of them. That third one, I've seen a different solution to before (I think it might be in Solomon Golomb's book).

The third solution, the 15x17 one, took an outrageous amount of time and attempts to solve. It's a shame that processing-power-wise we're a long way off being able to enumerate solutions for a given shape with hexominoes (like we can for pentominoes) - it'd be interesting to see if the ones that take me ages have fewer solutions and are genuinely harder.

I also found a solution to a 13x21 rectangle with a 7x9 removed from the centre. With the central hole being 63 unit squares, that means there's enough space to get all the pentominoes stuffed in there with a little bit of wiggle room to spare.

Fig. 2: This took a while too, as far as I remember. Hexominoes and thin sections (the 3xn bits above and below the pentominoes) don't seem to mix too well.

Then I realised that I didn't need to restrict myself to taking the excess squares out of the middle of the rectangle - I could just as easily round off the edges, and make some stretched-out octagons.
Which I did.

Fig. 3: A way stretched-out octagon.

And here's a slightly more regular, more octagonal-looking one I found a little while later:

Fig. 4: I bet there's a way of doing this with the six internal holes all in a diagonal line, but I'll leave that as an exercise for the reader 'cause I can't be arsed right now.

And just for an extra Brucie bonus, while we're on the subject of octagons, here's one made out of heptominoes that I found a few weeks ago:

Fig. 5: For whatever reason I have no recollection whatsoever of the actual solving process that led to this.

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.

***

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.

A Tale of Colossal Stupidity

After my recent string of (fairly) successful constructions with heptominoes, I had started to get a craving for something even more challenging - and the octominoes were the next logical step. Now, I don't actually own a set of these, but for whatever reason I wasn't going to let something like that get in the way of my fun.
My brilliant idea was as follows - get an image of the 369 octominoes open in one instance of good ol' Microsoft Paint, then get a canvas shaped like the solution I was trying for (in this case, a 29x102 rectangle with six holes) in another. I'd then draw in each piece, deleting it from the piece-list as I went.

Here's a screenshot from part way through my attempt. Solution so far on the left, Pieces remaining on the right.

And for a while this seemed to work swimmingly. Okay, sometimes it was an absolute nightmare finding a piece on the right-hand screen to make sure I hadn't already used it, but aside from that it wasn't much harder than how I'd imagine using actual physical pieces would be. I knew that in the very late stages of the solution, the constant backtracking and retrying would be a ball-ache to say the least, but that was a bridge I was prepared to cross when I came to it.

I got within 11 pieces of completing the thing completely, when I began to feel like something was a wee bit off. And sure enough, on counting the squares left in the remaining area left to fill, I noticed it had 89 squares. At first I thought this could just be something with a perfectly reasonable explanation behind it; a piece with a hole left to place, or something like that. But as I looked at the pieces remaining it became clear what I'd done.

My heart sank.

There was only one plausible explanation for this - at some point during the long late-evening session the previous day, in which I'd done the bulk of this solution, I must have drawn a piece in incorrectly due to tiredness. Somewhere, in that writhing mass of 358 shapes was a lone heptomino, camouflaged almost perfectly. I went back through the pattern, checking each piece to make sure it was made up of eight squares, and colouring in each to mark it off as I checked it. Hopefully the offending piece will be close to the bottom so I don't have to backtrack too far... I repeated to myself as I worked my way through piece by piece, checking and double checking.

But it wasn't.

(Click on the image to feel the disappointment in all its full-size glory.)

In blue are the pieces I checked, and in red is the heptomino that managed to sneak in undetected. And at this point, after considering the hours I'd sunk into this to get this far and the countless more I'd have to spend to complete it, I just packed it in on the spot, gave it up as a bad job.

Octominoes are just gonna have to wait for another time, I think.

Tetracubes - A Very Short Introduction

Polycubes. Like polyomines but pure chunky like.
There's one monocube. That's the cube. And there's one dicube, I guess it would be called, a 2x1x1 rectangle that you can't do a lot with. There are two tricubes, and these are just the 'chunky' versions of the two triominoes. You can't really do too much with these either.

Fig. 1 - the monocube (a.k.a. the cube), dicube and two tricubes

Sure, there's the 3D versions of the little trivial things you can make with the triominoes: the stairstep shape from the two tricubes, or the 3x3x1 from the 1, 2 and 3-cubes all together. But these aren't exactly challenging or interesting, let's face it.

Fig. 2 - Truly riveting stuff

I never thought the tetracubes would be particularly interesting. I assumed that if they could make any cuboids or other shapes, I'd be able to buy a set from somewhere. And I assumed that the existence of the Soma Cube was because the mathematically-complete set of tetracubes wasn't much fun and this was the next best thing.

I thought wrong.

There are 8 tetracubes, if you count the top-left and top-right in the image below as two distinct shapes since you can't rotate one and get the other. Five are just the tetrominoes with an extra sprinkling of depth (the planar tetracubes) and the other three are brand spanking new. Together these have a total of 8x4=32 unit cubes. And since these aren't limited by parity issues the way tetrominoes are, there are a few nice rectangles you can knock together with these.

Fig. 3 - The tetracubes.

Making a set of these yourself is strongly encouraged. I did. Or I attempted to anyway. I grabbed a bunch of wooden cubes off eBay and glued them up one morning, only to realise afterwards that the cubes weren't as cubic as I'd have liked, so the shapes were a tad uneven. Still, they do the job.

The first thing I was delighted to find out was that these eight pieces can form a 2x2x4 cuboid. There are 1,390 ways of doing this but it's bloody hard. Or at least I had a right job with it.

Then, just after I'd calmed down from the sheer excitement of finding out a 2x2x4 was possible, I found out that a 2x2x8 was possible too!
In fact, the pieces can be made into two little 2x2x4 cuboids as in the image below, then these two cuboids can be arranged into either a 2x2x8 or the 2x4x4:

Fig. 4 - Cuboid solutions, colour-coded because I have no idea how else to represent 3D solutions as images.

Exciting stuff!
...Actually, this is about as far as I got playing with these. Obviously, 1xnxn rectangles are ruled out because three of the pieces stick out 2 squares in every direction. And I haven't found many other nice constructions (so far) while just playing around. (But that doesn't mean they're not out there, just that I'm far too lazy.)

Really, deep down, part of me just wants to buy some more wooden cubes and make a set of wonky-but-usable pentacubes. There are 29 of these bad lads (12 planar pentacubes and 17 that really utilise that third dimension), which is a bit of a shame because that gives 29x5 = 145 unit cubes which can't be split up into 3 factors, which means that solid cuboids are out. But some computer searching revealed that a 7x7x3 cuboid with a 2-cell deep central 'well' is totally possible. So if I was to make a set, I'd maybe include the 2x2x1 dicube and sacrifice a little bit of mathematical completeness for prettiness. And if my woodworking skills weren't utterly atrocious I'd consider making a nice little box that holds all 30 pieces in the 3x7x7 shape. And give it a hinged lid or something too while we're at it.

Having said that, I hate puzzles that have to be solved in order to be put away properly. I have this set of wooden pentominoes (and one tetromino) that fit into an 8x8 box but to put the lid on and have it all nice and tidy, you've got to solve the thing first, and it drives me up the wall.

Even more Heptominoes!

As per the title really. The set is just sat there in a little box on my desk, tempting me, so when an evening comes along and I've got nothing else to do, it just kind of... happens.

Adding 8 more holes to the heptomino set (plus one hole for the harbour heptomino) gives 765 unit squares, which can form (among other things) a 17x45 rectangle. Here's one with the holes arranged in a grid, like a little keypad for someone who doesn't mind not being able to type zeroes:

Fig. 2: 17x45 with 9 holes.

The question now is how big a grille like this can you put in the middle of a heptomino rectangle before everything gets all out of hand? Actually, that can be a nice challenge for the reader, can't it?


Taking the full set of heptominoes then removing the one with the hole leaves 107 pieces, which (since both 7 and 107) are prime can only make a 7x107 rectangle. Which has been done before, in loads of different ways (there's an example about half way down this page.) Below is a way I found, done as two 7x53's with the I-heptomino separating them.

Fig. 1: The heptominoes (excluding the holey one) in a 7x107 rectangle, with the I-heptomino right in the middle. Click for a full-size image, it's a little bit too long to fit on the page.

I also found this symmetric pinwheel shape, in response to the challenge about half way down Kadon Enterprises' page showing various heptomino solutions.

Fig. 3: A rotationally-symmetric shape with a central hole.

And just for funsies, here's a whopping cross made from all the pentominoes, hexominoes, and heptominoes combined:

Fig. 4: I need to just decide on one scale to draw every image in. And maybe pick a colour scheme and stick to that too, while I'm at it.