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?