This might, at first glance, appear to just be a random tiling of a bunch of dominoes and trominoes.
But what would be the point of such a thing? In fact, it’s a complete* set of dominoes and trominoes where edges may be marked, and the marked edges are exactly the ones that can occur on the border of this tiling. That thick rectangle around the tiling is part of the pieces!
*There is, in fact, one more tromino with markings that could fit in that rectangle. But if we included it, there would be a single cell in a corner that would be unfillable, since we’ve specified that the tiling has only dominoes and trominoes. Therefore, by the rule of exclusion of things that would be awkward to keep around, it has to go.
That this works at all, even with this minor fudge, feels like a pretty bit of luck. Not only do we have a perimeter and area that are compatible, we have exactly the usual number of corners for a rectangle.
With polyiamonds, I thought my luck ran out. No combination of sizes gave me compatible perimeter, area, and corners. But when I abandoned corners entirely, and focused on pieces with only one marked side, I found that a parallelogram with opposite sides marked could be made using the 2-, 3-, and 4-iamonds. Since it wouldn’t do to have any unmarked edges lying bare to the outside world, I wrapped that parallelogram into a cylinder:
And here are the pieces individually:
Sometimes when I have a novel polyform puzzle idea, I feel like I’m tapping into a rich vein of possibilities. Here, I’m not so sure. The problem is that when you move up to sets with larger sizes of polyforms, the area and border segment length are unlikely to scale in a way that gives you tilings with completely marked borders. But I would love to be surprised!
