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There are 18 hexominoes that can be traversed with orthogonal moves without revisiting cells. This tiling has a closed tour, where all of the cells in each hexomino are visited in an uninterrupted sequence. (I have a blog post in the works about this stuff, but it's not quite done, and Tuesday very nearly is.)
@two_star very satisfying loop!
@two_star Every time you post something like this I spend a minute leaning back in my chair, looking at it and smiling.
@two_star I noticed that some of these, such as the 3×2 rectangle, have multiple traversals. When looking for a closed-tour tiling did you have to backtrack and try different versions of the tile set?
@mjd PolySolver conveniently lets you input multiple shapes as aspects of the same piece, so you don't have to do that manually. But in this case, there is a bit of a wrinkle. I used connector pieces (the pairs of diamonds in the screenshot) to enforce having piece traversals that connect up. A couple of the pieces have a forced connector position, so I fused a connector to the piece to help the solver out.
Note that the solver knows nothing about global topology; I have to manually sift through solutions that may have multiple loops in order to find a good one. Now you might notice that there is a problem with the 2×3 rectangle specifically. It admits a loop within itself, and I don't want the solver to waste time with those placements. So for the 2×3, I just included one of the two possible aspects. I could have combined the fused connector technique with multiple aspects to use the traversal that starts and ends on adjacent positions while excluding the internal loop, but I was lazy.
It turned out that the solver was just within the bounds of being fast enough to find a solution without me giving up, and I might have been lucky to get a single loop solution in the first few. The 8/8 doesn't mean that there were 8 solutions total, just that that was when I aborted the search after about 6 hours.
@two_star I found two pentomino solutions, both symmetrical. The first looks like a hip bone, or a ram's head. (I missed the second the first time around. Well, there's only 23 to sort through, so....)
ETA: Did the split octagon thing. Yep, just two.
@SevenNinths Cool! I don't know why I didn't bother to look at the pentominoes alone.