# Number of paths through an incomplete graph (with restrictions)

Here's a question I came upon while fiddling around with yarn on spindles.

I joined three spindles so that they were orthogonal.. then, beginning at the base of a particular spindle (A), wound it around another spindle, creating an arc connecting the two. After making some interesting figures I started to wonder about the combinatorial question of how many possible patterns could be created in this manner.

Assuming that each half spindle can hold some number ( arbitrary, but the same for all spindles) of winds (visits, essentially), and not excluding redundant patterns due to symmetry (once the basic question is answered, I will consider excluding these degenerate patterns).

A spindle can have arcs going back to itself (loops).

I've tried figuring it out using markov graphs, basic linear algebra and combinatoric techniques, but am very curious as to how other people would go about it.

-There will be times when the yarn 'paints itself into a corner', e.g. when 4/6 spindles have filled their holding capacity, and the thread must travel within the remaining nodes, ultimately cornering itself to eventually loop back its own node until completion (possibly even excluding some nodes entirely, leaving them untouched and unreachable). How could this be figured in to the problem?

I'm very excited to see how more mathematically mature problem solvers attack this :)