Way back in late Spring/early Summer of 2019, I had just about got the hang of solving things with heptominoes, provided the outer shape wasn't too restricting. After a string of these (fairly) successful constructions, I had started to get a craving for something even more challenging - and the octominoes were the next logical step. At the time I didn'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 a second window. I'd then draw in each piece, deleting it from the piece-list as I went. Foolproof, right?
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 specific 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 imagined 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.
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.
Of course, I wasn't just going to give up that easily. After a few weeks I found myself drawn back to the challenge of an octomino solution, and pretty quickly found myself cracking open two instances of Paint again, this time tackling a slightly differently shaped rectangle (20x148 this time, with eight holes, six of which are due to the holey octominoes.) And this time I ran into a different set of problems. I got about 90% of the way through the construction and did the usual sanity check, counting the amount of free area left and praying it divides by eight and I haven't drawn a heptomino (or a nonomino for that matter) in there anywhere by mistake.
I counted 168 unit squares remaining. So far so good. But then I counted the remaining octominoes. And got 21. And 23x8 is not 168. Something was afoot, but it being fairly late on Friday night (because I lead such an exciting life...) I was far too tired to work out exactly what was up. so I hit the hay and resolved to see what was up tomorrow.
Oddly enough, as I was falling asleep I had some kind of major tetris effect going on, seeing endless visions of octominoes fitting together in various ways all wiggly like; a veritable kama sutra for tetris blocks. And, as I'd been reading a fair bit on organic chemistry recently, in my bizarre sleep-deprived state I was also fruitlessly trying to assign the 'systematic name' to each octomino based on the way it branched and twisted, not quite awake enough to notice that the octomino was not in fact a molecule.
Well, the next morning and with a fresh pair of eyes, I took another look at the almost-complete rectangle and couldn't immediately spot any foul play just by eyeballing it. My guess was that I'd somehow used a piece twice, forgot to cross it off the 'used bits' list the first time round. And so began the laborious task of verifying this - getting a fresh image of all the octominoes up, then crossing each one off as I highlighted it in my construction. And if I found one that had been previously used... well, I'd cross that bridge when I came to it.
And so it transpired I had inadvertently duplicated two pieces. Thankfully, they were both right near the bottom of the construction; I only had to backtrack about 15 pieces to be back in a state where the rest was solvable. I guess I had started to get a bit careless just as I was becoming too tired to think properly and in hindsight it was lucky I called it a night when I did on the Friday. So I continued (being just a little more careful this time) and managed to get the rest of the pieces in without incident.
Of course I used FlatPoly2 as a further sanity check when I had about 12 pieces left, just to make sure I hadn't solved myself into an impossible endgame. And in running that check I might have accidentally glimpsed the position of two or three pieces that allowed a solution. But I did the rest of it. All by hand! And it only took me, what, six hours or so? (Actually, putting it that way, it feels like a colossal waste of time I could have spent doing something useful, but...)
Anyway, here it is, in all its glory:
All 369 octominoes in a 20x148 rectangle with symmetrically placed holes. Not pictured: enough blood, sweat and tears to fill an Olympic swimming pool.
This success spurred me on to attempt more octomino constructions in this way, all of which went wrong somewhere in the solution process. The above rectangle remains the only octomino solution I actually managed to complete until I got my hands on an acrylic set of pieces in January 2020.
[ Home > Polyominoes > Octominoes (the hard way) ]
Lewis Patterson. Last updated 08/05/21.