Btw, what is the combination for the impossible #35?
That combination is Touching outside - Touching outside - Contained. The problem with this is that one of these interactions between two circles constrains others involving the same circle.
For example, if we start by drawing two circles touching, then we would need to draw the third circle so that it touches one on the outside, but is contained within (or "contains", this is not a symmetrical property) the other.
There seems to be only one problem like this with three circles - but if we think about four or even more circles, these problems will surely pop up more often because we will be more and more constrained by having to draw all of them in the 2D plane. It would be great if we had the means to rule out some of the potential 4-circle candidates automatically, for example if they contain this impossible 3-circle configuration as a subset.
So there would be 4 potential results with only two different types of interactions.
2 types of interactions, 4 potential results
3 types of interactions, ?? potential results
4 types of interactions, ??? potential results
5 types of interactions, 35 potential results
This is very interesting. I calculated the number of potential results for 3 and 4, and the whole list seems to be 4, 10, 20, 35. Just pasting these numbers into a Google search, I found that these are the first few tetrahedral numbers:
Now, what we actually want to do is not to increase the number of interaction types (that is a constant 5), but the number of circles involved. Still, the observation that having three circles (or, more likely, three two-circle interactions between them) and a variable number of interaction types leads to tetrahedral numbers (which themselves are partial sums of triangle numbers) might be a clue about how to deal with four circles.
Four circles have six two-circle interactions between them, so maybe the number of possible configurations of those will have something to do with partial sums of hexagonal numbers?
More research necessary! :D