Very nice solution, J.P.! This reminds me a bit of an old problem I proposed a long time ago, whose generalization to one more like David's marble problem is as follows: QUESTION: Suppose given N bi-infinite cylinders of radius 1 in 3-space, whose axes all contain the origin. What is the maximum number of (curved) faces that their common intersection can have? --Dan On Oct 17, 2014, at 7:05 PM, J.P. Grossman <jpg@alum.mit.edu> wrote:
The solution for the 3D case is similar. Each pair of points defines a great circle across which the order of the points changes, so we need to count the number of disjoint regions on the surface of the sphere defined by all n(n-1)/2 great circles. . . . . . . F = 2 + E - V = 2 + n(n-1)(n-2)(3n-1)/12 J.P.
On Oct 17, 2014, at 2:54 PM, David Wilson <davidwwilson@comcast.net> wrote:
A clear marble has N tiny bubbles in it, numbered 1 through N.
Roll the marble on the floor, then list the bubbles in order of their distance from the floor (ignore situations where two or more bubbles are at the same height).
Given an optimal distribution of bubbles, what is the largest possible number of bubble orders you could record?
What if you measure distance from the contact point of the marble and floor?