Sunday, December 1, 2019

More heptomino things

Recently I've been way busier than usual with non-polyform-related things. There haven't been as many evenings where I could just whip out the ol' heptominoes set, so constructions have been a little thin on the ground. Additionally, I can't help but feel I'm beginning to exhaust all the possibilities with just heptominoes (well, all the interesting possibilities anyway, I'm sure there's plenty more rectangles with holes in them possible but after the first five or so they don't make for particularly gripping blog posts.)
Anyway, to kick things off here's a squarish shape with minimal holes and maximal possible symmetry, that I'm surprised I hadn't thought of sooner:

Here's two parallelograms, using the full set of 108 split between them. Getting the holes centered here was hard, it's impossible to eyeball it and it's a bit of a pain to work out where they should be by counting. And it never looks quite right either, just due to the sloping nature of the diagonal sides. Still, far as I can tell this is correct:

For reference, the first one took about 10-15 minutes, and the second one took a bloody age because I'd foolishly forgot to use the [ shaped piece up earlier. And it doesn't play nicely with other pieces.
Things like three 11x23 or 4 10x19 parallelograms are probably possible too. Next time I've got a free evening I'll have a crack at one of them maybe.

A while ago I found a nice hexomino solution (about half-way down that page) that had a 7x9 rectangular hole in it, which could fit a set of pentominoes inside it with 3 cells to spare. Well, I went one better and found a heptomino frame that could accommodate that inside it, creating the following three-layer pattern. With the holes spaced nicely around the edge too. Forget 45° zig-zag edges - five-cell-high tubes with heptominoes are my new least favourite thing to solve.


To me right now, this just screams 'find a fourth layer with the octominoes!' I haven't checked yet if it's mathematically possible to, or at least if it's possible without introducing more holes than the 6 required for the six holey octominoes. I'll have to have a check.

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