Beyond the wall: Working with aperiodic tilings using finite-state transducers

* if vertexes have few edges, it's easier to figure out how/if the loop passes through that vertex, but once you do it doesn't imply much about the rest of the puzzle

* if vertexes have many edges, it's harder to figure out how/if the loop passes through that vertex (in particular: just because you know it enters from one face, does not mean it has to exit through any particular other face), but once you do it blocks off many more possibilities

* if faces have few edges, you're more likely to get numbers that constrain the puzzle, but those don't do much for you

* if faces have many edges, you're less likely to get numbers that constrain the puzzle, but the ones you get have major implications

Spectre/Hat (a recent addition, probably not in your distro's version) play a bit funny because unlike the n-gon puzzles there's no instinctive memory of what n-1 is to follow the very-useful S rules (my name):

S-rule, edge version: whenever 2 n-minus-one faces touch at an edge, that edge must be marked as part of the loop, adjacent edges remain unknown (only 2 possibilities remain, an S or reverse S shape), and all others are part of the loop

S-rule, vertex version: whenever 2 n-minus-one faces touch at a vertex (not possible for all tilings), the edges that touch remain unknown but all other edges of the faces can be marked as part of the loop. Additionally, any other edges that touch the vertex (not possible to all tilings) can be marked as not be part of the loop.

Most forms of the S rule are easily shown in the "triangular" tiling by looking for adjacent 2s, though the edge version's "all other edges" set is empty.