Dave Harper’s Polyomino Patterns page has some good stuff, looking at patterns of connections between squares in polyominoes, and processes of “integration” and “differentiation” on polyominoes. He enumerates all the possible patterns of connections of the cells in a 2×3 rectangular hexomino that make a connected whole. (There are ten.) These could also be considered as polysticks that touch all six vertices in a 2×3 lattice. The polysticks on a 2×3 lattice are precisely those that can be represented on a 7-segment LED, hence my presentation of them below:

It might be nice to have some puzzle using these. So here is one! Fill in segments on the figure below so that each of the ten patterns above is represented on a 7-segment LED shaped subsection of the figure.

Reflections and rotations of the patterns are considered equivalent. There are 13 7-segment LED shaped subsections of the figure, so three of them either can have other patterns, or can be duplicates.

Are there any other puzzle grids that would make for a puzzle using these patterns that is as good or better than this one?

If this somehow gets distorted, here’s at least where each polystick is (rows alternate sideways and upright):

B

9A

GYx

xEOH

UxS

Funnily enough, the E is upside-down, but none of the characters need to be turned backwards.

Nice. I only found solutions with a repeated pattern. (Hey, a bonus 9.) If I’d known it could be done without duplicates, I’d have tightened up the problem statement to exclude them.

A 1×11 grid works, but it’s less interesting. Also, G and U have upper squares with two segments, which are unique and can’t overlap anything else.

Does any grid composed of fewer than 10 unit squares work? Well, 11 squares work with no wasted space. 10 squares require a loop, 9 squares require two loops, and 8 squares would require three loops, but G and U would have to overlap things. That makes 9 the theoretical minimum.

Depends what you call “repeating.” The extra patterns are a backward S, a backward Y, and a disconnected pattern.