One-Sided Hexominoes

It's been a long time since I last posted anything on here. Too long. But sadly there just hasn't been that much in the way of polyomino-related goodness to post. Partly because I've just been too busy doing other things, and partly because there's only so much you can do with polyominoes - the easy stuff isn't interesting enough to write blog posts about, the hard stuff is too hard to do without the assistance of a beefy computer, and the stuff in the sweet spot is hard to come by.

I've never written up anything about the one-sided hexominoes before on here. Basically, earlier this year I got a second set of heptominoes laser cut, which opened me up to the possibility of solving shapes using the set of one-sided heptominoes by combining the sets and then trying not to flip the pieces over. Then I realised the same thing could be done using my two sets of hexominoes too. And the set of one sided hexominoes is a much more versatile set than the plain ol' regular hexominoes.

There are 60 one-sided hexominoes in comparison to the standard set's 35. But this new set doesn't have those pesky parity constraints which means that a lot more shapes are possible to tile - rectangles without unsightly internal holes, for example.

The total area is 60x6 = 360, which means that rectangles of size 4x90, 5x72, 6x60, 8x45, 9x40, 10x36, 12x30, 15x24 and 18x20 should all be possible. Sadly, due to the length of the perimeter compared to the amount of perimeter squares the pieces can provide, 4x90 isn't possible.

Solving manually is a little trickier than the normal hexominoes - each piece having only one accepted 'right side up' means that any given piece is slightly less practical than its two-sided equivalent, and you do get those cases where you're down to one piece left and the hole is the mirror-image of the piece you're holding.

Here's two 9x20's which can be combined to make either a 9x40 or an 18x20, in a two-rectangles-for-one type deal. As 360 divides up really nicely, this gives a lot of possibilities for tiling groups of congruent shapes, but that'll be a blog post for the future. Others have already solved congruent sets of ten or twelve shapes, so go look at those. Scroll about three-quarters of the way down the page for them. In fact just read the entire page, it's all good.

Something else nice you can do with the one-sided hexominoes is square rings, i.e. squares with a centred square hole. Here are three possibilities (these might be the only three actually), with the rings getting progressively thinner, and as a result a little harder to solve manually:

And here's one more shape, a diamond with a central hole and those tricky diagonal edges.

And I'm purposely leaving this post a bit less thorough than usual so I've got an excuse to post 'One sided Hexominoes - Part Deux' in a few weeks. Or a few months, if my recent posting schedule is anything to go by.