Enneiamonds - Two 19x19 Rhombi

Enneiamonds (spell check has a vendetta against that word) have all the frustration of octiamonds and then some, and by far the biggest ball-ache of all is the introduction of holes. You need a hole for the holey enneiamond (the harbour enneiamond?) and then you need more holes to bring the total area up to something that divides nicely. Adding a further three makes the total coverage 1444 triangles which divides up nicely and allows two rhombi with two holes each to be made (Also 4 triangles with one hole each but let's not try to run before we can walk here.)

Click for (slightly) larger.

It looks real pretty. And it looks very human-solved, the pieces getting chunkier and easier towards the top corner of the right hand rhombus. But don't be fooled. the last 15 pieces were done with a little electronic assistance.

I don't like doing this. And in fact when I started keying in the pieces which just wouldn't go, my intention was never to just find a solution - that makes for something of a hollow victory feeling when you put that last piece in and know deep down you didn't properly complete it. I initially just ran a search to check for the existence of a solution, but not the solution itself. If the program came back with '0 solutions found' I'd know to backtrack one more piece and try the search again until I reached a position where one or more solutions existed.

I used to do this back in the early days of heptomino solutions, before I got good enough at them to not need any assistance. And usually I'd need to backtrack 5 or 6 pieces and there'd be a couple of solutions lurking in that piece of search space. Not in this case, though - I had to remove fifteen pieces before it could find even one solution. If that's not irrefutable proof that polyiamonds are just a more obstinate breed than polyominoes I don't know what is. Anyway, I'd found a position where a solution was definitely possible, at least in theory. In reality, I fought on, trying configuration after configuration trying desperately to get the rest of the pieces in. The big triangle belonged in the very corner, that felt like a given; it just fit there so well and didn't seem to sit comfortably anywhere else. But beyond that I was just pushing pieces around pretty much at random, sometimes finding I could get all but one in, other times creating awkward little bays and peninsulas in the edge of the construction that seemed to exclude every other shape.

I spent way too long on this - several days' worth of mornings, lunch breaks and evenings - so I eventually decided I would do the unthinkable. I'd peek at a computer solution for those last 15 pieces. Not the entire thing; just the next few pieces, to point me in the right direction.

Key: Purple pieces were already placed, red bits I looked at the solution to find, yellow bits I did myself after the red bits were placed.

Armed with these handiest of hints, I managed (after another 20 minutes or so) to fit the last of the pieces in and gaze upon the finished construction. It still felt like a bit of a cop-out though. I suppose it's just part of the learning process - I did similar with heptominoes a year and a half ago and it wasn't long after that I was knocking out octomino rectangles in single afternoon sittings. Hopefully the next time I dig out the enneiamonds I'll be able to get a little bit further than this time, and before I know it I'll be able to just solve whatever with them without breaking a sweat.

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Lewis Patterson. Last updated 08/05/21.