Saturday, December 28, 2019

Octominoes: The Hall of Shame

Hoo boy, where do I start with this one?

Fig. 1:  ;~;
This is what happens when you solve without a physical set of pieces. Pieces that don't belong in the set (in this case, the two blue nonominoes) creep in undetected, and only when you're down to the last little corner do you realise that the remaining space is the wrong size. I do a quick manual check to see if the number of free squares are divisible by 8 at about this point, and in this case it wasn't so I engaged panic mode (i.e. looked for the offending pieces, found that they were nowhere near the edge of the solution, and gave up.)

This was my second crack at the same solution, a month or so later:

Fig. 2: Another 4 hours I'll never get back.
This time, the area of the final space was divisible by 8, but it wasn't the multiple of 8 I was expecting. I had 15 octominoes left to place, and an area of 14*8 = 112 unit squares. So I reckon I've used a piece twice somehow. I've looked over the solution but can't see it, so if anyone actually reads this and has a better eye than me, see if you can spot what I did wrong here.

A while after all this, I read somewhere that octominoes have parity constraints of sorts. Or at least the 363 unholey ones do. (See here, click through to 'Other octomino constructions' and it's about 2/3 the way down.) I'm still trying to get my head around this; I'm still not sure if and how it will impact constructions like this. Maybe this shape wasn't even solvable to begin with, since I put down the holey octominoes first, reducing the construction to a solution with the set of 363 unholey ones?

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