Saturday, November 2, 2019

'Measure twice, construct once'

Building something only to discover that I've somehow put the central holes in the wrong place seems to be a common theme for me. I need to implement some 'measure twice, construct once' philosophy maybe.
A short while after I wrote that big long post about combining pentominoes and hexominoes into one set, I thought about how just possible it would be to knock out a 4xn rectangle with those pieces. I mean, each piece would physically fit into a 4 cell high rectangle, and if it's possible to squeeze all 1- to 7-ominoes into a 5x211* then this could well be possible too.

The whole 4-cell-high thing made it tricky but not as big a challenge as I had initially suspected it was going to be. After 45 minutes with my hexomino set (and a shoddy set of pentominoes cut out of graph paper to supplement them) I found the pattern below. With the two holes off-centre by one...

Fig. 1a: Side note, I'm not mad on this arrangement of holes but it's the most symmetrical one I can think of for two holes in an even x even rectangle.
Thankfully, as I was sketching down this solution, I notices it wasn't as bad as I'd feared - there was a central section (marked in light gray above) which could be flipped over and had the effect of transposing the two dots over by one cell - into the middle!

Fig. 1b: The finished construction.

Of course, there's rarely a nice quick fix like that, as I discovered with a heptomino construction from a few weeks ago. The harbour heptomino and its little central hole must have been accidentally knocked during the solve process and I hadn't realised. Only upon drawing the solution down onto graph paper did I notice it was slightly off. There was no quick little rearrangement of pieces that would fix it this time. I just had to solve the entire thing again in a few days' time.

Fig. 2: The heptominoes in an octagon with central hole. Coloured by how far each heptomino is from the edge of the construction, just to see how it would look really.
Notice the way the diagonally-symmetric pieces are all grouped nicely along a diagonal running from near the south-west edge up through the centre. Basically, I noticed in one of David Bird's nonomino constructions (typically, the one I can't find on the internet anywhere to link to; it's in S.W. Golomb's 'Polyominoes', page 116) that all (or at least a good chunk) of the rotationally symmetric pieces were grouped together in the central section surrounded by holes**. And somehow I'd never noticed that before. I just like the idea of little things like that, 'hidden' in plain sight within the construction that reveal themselves on closer inspection. (Another one I like is in a heptomino construction by Nick Maeder, third one from the bottom on this page, which has the crucifix-shaped heptomino in the centre of a triplicated version of itself, all positioned centrally within the pattern.)

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* I've seen this done but can't find it anywhere on the internet, but can't find in anywhere or I'd link it. Maybe it was in the middle of an Internet Archive binge of polyomino-related sites that are no longer up.

** Actually, there's something similar going on in this nonomino construction, also by David Bird. A lot of symmetrical pieces are grouped around the line of holes running upwards from the middle.

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