Enneominoes!

It had to happen sooner or later.

Ever since not long after I got my octominoes made, I'd wondered occasionally whether the next set up would even be feasible. For the longest time I dismissed the idea outright; octominoes was as high as I'd be able to go, for three reasons:-

Firstly, the total area of 1285 pieces as 10mm per unit edge was just far too much to handle. The octominoes already pushed the limit of what the table in my room could hold, and even if I resorted to building enneomino constructions on the floor I would struggle with particularly long and skinny rectangles. (That and the fact it's all carpet so not really a suitable surface.)

Secondly, the amount of pieces would just be too much to work with - I knew well the pain of searching through the box of 300+ octominoes for that one specific piece (usually only to find that I'd already used it in the construction...) and even making the enneominoes in 10 colours, that'd still be 128 or so pieces of each colour, so even if I knew the colour of the piece I was after it'd still be a tedious hunt every time. And more often than not I wouldn't even know the colour of the piece I was searching for.

And finally, making this set wasn't going to be cheap. And I'm averse to spending money at the best of times, so was very reluctant to look too hard into this if I was laser cutting something that was going to spend most of its life in a box in my room for the above two reasons.

But even then these reasons felt more like excuses than actual set-in-stone barriers; to paraphrase Lilly Satou, "I knew the possibility of it happening was there."

Making the Set

Gradually I realised a few things that sort of tipped the scales. Firstly, I let go of the idea that the pieces had to be 10mm/edge. This was the case for the smaller sets because it allowed me to line pieces up to the grid printed on the cutting mats, a handy help given my track record for misaligning holes and building things to the slight wrong dimensions. I realised that by making each unit edge 6 and two-thirds millimetres I'd still be able to line some of the pieces up with the grid, and the tradeoff for this slightly awkward compromise was that I'd be able to actually fit more pieces on the table. Potentially even all of them if I'd done my maths right.

I then found a construction by Patrick Hamlyn which had the entire set of unholey enneominoes in 48 congruent rectangles, which I could use as a basis for the CAD files when I got the pieces cut. The number of rectangles meant I had a bit of flexibility with how many batches I got them cut in, and as a result the number of different colours. Eventually I settled on eight (plus a ninth for the holey pieces), which was more influenced by the number of colours of perspex the laser cutting place offered than anything else. It gives 156 pieces of each colour, which is not ideal but it's marginally better than 1248 of each colour I suppose.

I drew up the .SVG files of the pieces while on holiday in Arran, on the evenings where the midges were out in full force and I couldn't do a great deal else. This wasn't the most fun task, just manually copying out the pieces from the reference image. Almost as tedious as digitising octomino solutions from a photo after finding them. And then when I'd finally uploaded all the files to the laser cutting people and parted with an eye-watering amount of money, I set about devising something to actually solve with the pieces once they got here. Something easy to start with, or at least as easy as enneominoes can be. A 79x147 rectangle with the holes all in the centre in a 4-fold symmetrical configuration, for instance.

The Solve

By this point I naively assumed that the enneominoes wouldn't be that much harder to solve than the octominoes - they were wigglier, sure, but there was more of them and therefore more possible pieces to fill a particular niche whenever one should arise in the boundary of the solved area. Right? And lots of pieces with inner 2x2 squares too. 436 of them in fact - a whole 34% of the pieces, up from 23% of the heptominoes and 30.6% of the octominoes.

I misunderestimated these bastards.

Fig. 1: The start of the peeling process. To call it gruelling would be an insult to gruel.

I started the solving and the peeling of the protective sticky plastic pretty much simultaneously.  It was less that the solve was too exciting to wait for, and more that the peeling process was too boring to do on its own. And at this point I made my first mistake.

In planning the boundaries and hole placements of the rectangle, I worked in units and units squared, and then had to convert back to mm in order to place markers on the table. But somewhere here clearly I cocked up, because I built the ring of holey pieces, filled it with 100+ pieces and built a good chunk of solution connecting the centre to the bottom edge before realising it was two thirds of a millimetre higher than it needed to be. So I had to rip out a strip of enneominoes, move a huge patch down carefully without disrupting everything, and then re-solve the little strip I'd taken out with a completely different set of pieces and some new ridiculous constraints on the pieces. Fun stuff. An afternoon well spent.

By this point it had dawned on me these were an entirely different beast to the octominoes - the difficulty plateau I'd envisioned after comparing the octominoes to the heptominoes just wasn't there, and this wasn't going to be as fast and easy a 'warm-up solution' as I'd initially hoped. Additionally, it was taking absolutely ages; by this point I must have sunk 6 or 7 hours into the solution and only placed about 350 pieces or so. And a lot of this time was spent digging around looking for specific pieces, just like I'd feared. Additionally, this being the first time with the pieces meant I wasn't familiar enough with them to know which colour I was looking for half the time either. So I began trying to organise things a little bit. A separate dish where I'd put all the 'really nice' bits - those containing a 2x3 rectangle as sub-polyomino, a dish for the pieces with 2x2 subsections, and a dish for pieces that would work to fill 1xn cavities at the edge of the board - which I internally refer to as edge pieces despite the fact you can use them anywhere really. And this made things a tiny bit easier.

Fig. 2: How the solution looked at about this point. The various trays were my method of sorting the pieces - pieces with 2x3 sections in the smallest tub, ones with 2x2 sections on that big tray on the floor, etc. This was also the last time I could use my desk for normal desk things like writing and eating for a while.

The going got tougher still when it came to filling those two narrow parts between the centre and the long edges. They're not even particularly narrow by most standards - like 20 squares or something, would have been a walk in the cake with heptominoes - but it was enough to make me sneakily reach into the pile of pieces marked 'save for later' multiple times. By the time I had extended the top edge along to the top-left corner I had pretty much no 'normal' pieces left, just 2x2-box ones. Plain sailing from here onwards, I thought.

A Cock-up Most Spectacular

The remainder of the solve was, just as I'd hoped, fairly simple - no worse than putting the last third of the octominoes in. Until I got down to about 19 or so pieces and slowly realised with a creeping sense of dread that the space I had left was concerningly smaller than it should be. I had checked and double-checked all the measurements as I went to prevent exactly this type of thing; I didn't want a repeat of the off-centre centre a few days back. But obviously I hadn't checked hard enough, and sure enough after another tedious bout of counting rows I discovered I had in fact built a 79x146 rectangle - one row shorter than it should be.

I'm still to this day unsure of how I managed to screw this up, I had left little post-it notes tucked under the top edge of the solution as I went denoting the number of columns to each edge in each direction. I must have just been exceptionally tired that day or something.

And so I ripped out a huge wedge of solved pieces right back up to the top left corner, probably 150 or so in total, and began again to the correct dimensions this time. Which was a lot harder than the first time around, given that I was working in a long, narrow space.

The Endgame

The final bit of the solution wasn't too bad all things considered. Once I've whittled the set down to a dozen or so squarish blocky shapes, and the space they need to go into doesn't have any tricky bits, it's just a matter of trial and error. And in this case it maybe took only an hour or so to find something that worked.

Fig. 3: Complete!

The total time to solve was probably 12-14 hours, spread out over several days. Many little solve sessions crowbarred in between work, sleep, gym and whatever else I had to do. And some of that was peeling the plastic backing off pieces too. In fact there's a few of the pink pieces where I don't know quite what was going on, but the stuff just wouldn't come off (or would come off in tiny bits) so I just left it.

Here's a nice clear image of the solution where you can actually see the individual pieces. Which took another couple of hours to draw up, that wasn't a fun task... Normally I find the digitising of solutions kind of therapeutic, but this one just took the piss.

Fig. 4: The completed rectangle in all its glory.

So I suppose the question now is, what next? Dekominoes are definitely out of the question, but I reckon it's only a matter of time before I try something else with the enneominoes, something a little more interesting. Sets of five congruent shapes are definitely possible. For next time I don't feel like using my desk for a week or so.

And there's another fun side effect of solving these I noticed. When you go back to hexominoes (or even heptominoes) afterwards, they seem like baby toys in comparison. There's barely any of them, and they're all such simple shapes... This holds for a little while, until you're about half way into a solution, and then they kindly remind you that they can still be fiendishly tricky.