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    • At this point, it occurs to me that not all configurations are actually possible. For example, if A and B are in the "separate" configuration, it isn't possible to add a circle C so that A includes C and B includes C. I currently see no way to solve this other than to check each possible configuration manually. I wonder if this could be solved by some clever form of notation, but I don't really see it.

      Regarding a "clever notation", it occurs to me that my previous attempt at generating all potential configurations of n circles by taking all possible configurations of n-1 circles and adding another one might also lead to duplicates.

      For example starting with

      "two circles intersecting" and then adding a third one that "is separate from both"

      leads to the same configuration as starting with

      "two separate circles" and then adding a third one that "intersects one but is separate from the other".

      To avoid this, we'd need to generate potential solutions in a way that skips duplicates but does not skip those where the same set of individual configurations leads to a different overall solution.

      Can this be done by first finding all different sets of SUM(1..(n-1)) circle-circle configurations ignoring their order, and then find all unique permutations of that set?