Whiskey. Tango. Foxtrot.

Fab called today. He's in Pennsylvania somewhere. He said something to the effect of, "Didn't I mention that I was travelling?" No. You didn't. But that's OK, Fab. That's why we love you. He also gave me a series of connected problems that I solved while at work today. I'm now trying to figure out the appropriate generalization to higher dimensions. The problems have to do with a collection of line segments intersecting at a single point and integer values associated with each end of each line subject to the following constraints:
(1) the two numbers on each line sum to the same value, call it S.
(2) adjacent numbers going around the circle have a common difference, call it D.

That is, if we call the values going around the circle a₀, a₁, ... , a_2n, then we can rewrite the constraints as:
(1*) a_i + a_i+n = S for all i
(2*) |a_i - a_i+1| = |a_2n - a₀| = D for all i

A simple case:


My problem is that now that I've solved the problems, I'd like to generalize. If we think of a line segment as a simplicial complex of dimension 2, then numbering the endpoints is just labelling simplices of dim 1. The obvious thing to do is look at simplices of dim 3 (triangles) and label the simplices of dim 2 (edges). But how to generalize the arrangement of "in a circle?"