Algebraic structures associated to flexagons?

Solution 1:

Flexagons by C. O. Oakley and R. J. Wisner. The American Mathematical Monthly Vol. 64, No. 3 (Mar., 1957) (pp. 143-154) looks like the first hardest look at them. Somewhere it says they are analyzed into sets of "recursively defined permutations." If you look at that, and then trace forward other things that reference it, I think you'll get somewhere.

I saw a good bit of combinatorics in these articles so far, and there is a definition of an "abstract flexagon" on page 146. The article is labeled as free access at JSTOR.

As for question B, I see 11 hits in mathscinet, lots of which are in the Monthly. They seem like decent articles, but I didn't notice any bigger journals. Two of the hits are from 2012.