Heptominoes
First Steps
Heptominoes are... intimidating.
I made my original set in 2008, but it wasn't until November 2018 - ten years later - that I finished a complete construction with them. Most of that was because I assumed it would be far more difficult than it actually turned out to be. In my head, constructions with anything larger than hexominoes were just these superhuman feats that you needed to be a serious proper mathematician to solve, and that mere mortals such as me should just stick to pentominoes, which I was having difficulty enough with. So for most of that intervening decade the heptominoes I'd made were just sat around in a box in various cubpoards or the attic. Waiting.
My original (and still main) set of heptominoes. Made on a CNC milling machine at high school when the staff weren't looking, hence the odd beveled edges the pieces have. A drill doesn't quite get the job done like a laser cutter.
What had initially spurred me on to attempt solving something was the earlier successes I'd had with hexomino rectangles. I started easy - area 230 rectangles made up of the hexominoes and tetrominoes (whose respective parity issues cancel each other out). By including the tetrominoes and saving them until last, or at least trying to, I was able to make the endgame of solutions way quicker and more painless. It's more likely a 4-cell hole left will match the 4-cell piece you're holding than it is with a 6 cell piece.
The other thing that used to prevent me from tackling heptomino problems with any degree of seriousness was the decision paralysis I'd get at the beginning of a solution - a fear that starting out with the wrong pieces could have disastrous results. Playing with the hexominoes gave me an understanding of which pieces are best prioritised until the end and these lessons were easily applied to heptominoes. But for that first heptomino rectngle, with every piece I placed there was a dread - what if I shouldn't have used that piece? What if I need it later? What if I accidentally leave a horrible piece until right at the end?
It took a little while to learn that there was a lot of leeway when it came to this. The number of solutions is generally so vast that even piece placements that can only be described as tactical blunders early on in the game don't prune too heavily the 'tree' of solutions still attainable from that point without backtracking.
The above is the first thing I ever completed with the heptominoes. As you can see it's a bit of a cop-out; I didn't place the holes first, instead opting to just use them as monomino pieces as and when needed. I knew I'd need one to stuff inside the harbour heptomino, so that gave me two other 'lifelines' to just drop in as and when. At this point in time just the feeling of completion was more important to me than little trivial things like overall symmetry. So the holes are clustered near each other where the last few pieces went in. This makes it a hell of a lot easier than solving 'properly', instead of waiting for the final space to exactly match the last heptomino, it just has to form an enneomino which the last heptomino is a subset of. It's not exactly pretty, but it works. And it gave me the confidence boost to start tackling slightly more tidy solutions.
Like this one. I'm not sure why I decided on the little diagonal run of holes lowering the overall symmetry of the figure, as opposed to just putting them spaced like a little ellipsis in the centre.
By this point I was confident in my solving ability; heptominoes were no longer to be feared, they just required a slightly different toolbox of sneaky tricks than smaller sets of pieces.
Silly Mistakes
A common theme when looking back over my old blog posts and a good chunk of my newer ones is that I had just as much difficulty with the solving process as I had with actually defining the shape I was solving. So many times I'd build two-thirds of a construction before realising I was building it one unit too long or too short, or the central holes weren't as, well, centred as I'd planned them to be. Have a peep at the following solution from April 2019.
Notice anything unusual?
That's right, I somehow managed to centre the four holes incorrectly. They're all one square up from where they should be, giving the total rectangle only one axis of symmetry instead of two. And I only noticed this right at the very end when it came to drawing up a neat copy of the solution. My excuse at the time was I was looking at the shape from such an angle that it wasn't immediately obvious there was anything wrong - the table's perspective disguised the error. But the real reason was I couldn't be bothered to count to 17 when I was placing those little squares I used as markers for where the holes were going to be.
See?
More Interesting Shapes
Let's face it. Rectangles get boring.
One of the earliest forays into more involved shapes was the below solution. Those wiggly edges were an absolute nightmare to someone who had never tackeld such a shape before. You can see on the left hand side where I started, I use up all the shapes with obvious zig-zag edges and by the time I got to the right-hand side I was just desperately trying any vaguely wiggly piece I could get my hands on. Again, putting the holes in a nice configuration wasn't high on the agenda, I just wanted to prove to myself I could solve the thing, by any definition of 'solve'.
Calling the holes 'monominoes' makes them sound a little more intentional... right?
Next section: More Shapes
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Lewis Patterson. Last updated 15/12/23.