I noted a while back that triangular polysticks hadn’t received the study they deserved. Perhaps we can say more generally that a fruitful way to find understudied polyforms is to take the polysticks that correspond to connections between cells in some other polyform set. Since I was recently looking at polykings, it makes sense to ask what nice sets we can make out of polykingsticks, and what we can do with them. There are only six dikingsticks. Here are the 41 trikingsticks: [Edit: numbers fixed per communication from GS]
That’s a lot! We can distinguish some categories. First, “proper” polykingsticks (in red) exclude those that have only one type of connection. “Snakes” (bolder lines) are polysticks for which there is a single path that visits each vertex once. Most of the non-snake trikingsticks are branched. There are a couple of oddballs, a right triangle and one self crossing snake, which you might need to omit if you have a problem where you want to avoid crossings.
What kinds of problems can we explore with these? They’re a bit awkward for tiling. Compatibility problems should work, as long as the pieces have the same balance between diagonal and orthogonal segments. Common superform problems should also be doable.
We can also try Hamiltonian circuit problems like the one that I looked at with regular polystick snakes. There is a parity issue here. If we checker the vertices of the grid, we require an even number of snakes where the two ends are opposite colors in order to make a circuit. This means we can’t use the set of proper trikingstick snakes with the self-crossing snake omitted. Perhaps with it included, we could do a Hamiltonian circuit of a 6×11 grid. If we just make a path, rather than a circuit, and exclude the self-crossing snake, visiting all of the points on an 8×8 grid might be possible.
With so many trikingsticks, tetrakingsticks might seem too large of a set to contemplate, but I have a couple of ideas for interesting reasonably sized subsets. These will have to wait for another post, however.
