Puzzle ring question answer
Times up on my puzzle ring pattern challenge. Nobody gets the cookie.

To refresh, the question was what the pattern/generalization of this design is:

___   ___   ___   ___   ___   ___   ______
   \ /   \ /   \ /   \ /   \ /   \ /
    \     \     \     /     /     /
___/ \   / \   / \   / \   / \   / \   ___
      \ /   \ /   \ /   \ /   \ /   \ /
       \     \     \     /     /     /
___   / \   / \   / \   / \   / \   / \___
   \ /   \ /   \ /   \ /   \ /   \ /
    /     /     /     \     \     \
___/ \   / \   / \   / \   / \   / \   ___
      \ /   \ /   \ /   \ /   \ /   \ /
       /     /     /     \     \     \
___   / \   / \   / \   / \   / \   / \___
   \ /   \ /   \ /   \ /   \ /   \ /
    /     /     /     \     \     \
___/ \___/ \___/ \___/ \___/ \___/ \______

And the answer is that if you chop the pattern up into four quadrants, and the crossings all go the same way within each quadrant, like so:

___   ___   ___   ___|  ___   ___   ______
   \ /   \ /   \ /   \ /   \ /   \ /
    \     \     \    |/     /     /
___/ \   / \   / \   / \   / \   / \   ___
      \ /   \ /   \ /|  \ /   \ /   \ /
       \     \     \ |   /     /     /
___   / \   / \   / \|  / \   / \   / \___
---\-/---\-/---\-/---\-/---\-/---\-/------
    /     /     /    |\     \     \
___/ \   / \   / \   / \   / \   / \   ___
      \ /   \ /   \ /|  \ /   \ /   \ /
       /     /     / |   \     \     \
___   / \   / \   / \|  / \   / \   / \___
   \ /   \ /   \ /   \ /   \ /   \ /
    /     /     /    |\     \     \
___/ \___/ \___/ \___/ \___/ \___/ \______

This can be scaled up or down to any number of bands, for example the classic design is four bands

___   ___   ___   ___   ______
   \ /   \ /   \ /   \ /
    \     \     /     /
___/ \   / \   / \   / \   ___
      \ /   \ /   \ /   \ /
       /     /     \     \
___   / \   / \   / \   / \___
   \ /   \ /   \ /   \ /
    /     /     \     \
___/ \___/ \___/ \___/ \______

And it can be chopped into quadrants as well

___   ___   ___|  ___   ______
   \ /   \ /   \ /   \ /
    \     \    |/     /
___/ \   / \   / \   / \   ___
------\-/---\-/+--\-/---\-/---
       /     / |   \     \
___   / \   / \|  / \   / \___
   \ /   \ /   \ /   \ /
    /     /    |\     \
___/ \___/ \___/ \___/ \______

I also gave the three banded version in a previous post. The five-banded design works as well, but I don't like that as much because it doesn't have a strong canonical position which the bands want to go into like they do with four or six.