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.