Bram Cohen (bramcohen) wrote,
Bram Cohen
bramcohen

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.
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