This challenge is spans all the way from graph theory, geometry, and calculus—to algebra and combinatorics.
Consider a square with diagonal 1. In it, N points are randomly arranged. A graph is created from these points, such that the probability any 2 points touch is equal to 1 minus their distance. Choose a point at random on this graph. The challenge is to calculate a probability regarding this point: what is the probability that a randomly selected node connected to this one is connected to more nodes? Each node has a loop, so if the node was connected to no other nodes the probability of this is 0. For N=1 or 2, this probability is 0. Try to calculate at least one of the following: chance for a specific N > 2, limit of chance as N approaches infinity, and most difficult, generalized probability in terms of N.
I love that you are still doing these problems and publishing then and the solutions. You are so industrious. Gramps