Thanks, for the good problem. I have an intuition regarding the distinct planes between three points in the 3D case, and I'm not fully successful chasing it. In particular, my n(n-1)(n-2) conjecture is clearly wrong (too small). The basic idea itself is simple. For every such plane, with infinitesimal movements, you can generate six distinct orders. The difficulty here is figuring out when identical orders can be generated by distinct planes. In the 2D case, those pair up nicely by linear rotation of the disk. In the 3D case, it's not as nice. I guess this is the intersection of three of your great circles. I guess the next step is to figure out the pattern on the surface you want those great circles to form, and then prove it is maximal. On Fri, Oct 17, 2014 at 6:57 PM, David Wilson <davidwwilson@comcast.net> wrote:
Just for the record, I authored this problem.
Anyway, here is as far as I got:
For the 2D case without reflection. N = 0 and 1 each have 1 possible order, they are what they are.
Now consider two bubbles A and B in the disk. Draw line L through these bubbles, then draw diameter line D through the center of the disk perpendicular to L. Let X and Y be the endpoints of the diameter on the disk (antipodal points). As you roll the disk, the order of A and B changes precisely as X or Y touches the floor. Call X and Y the "exchange points" for A and B. The exchange points slice the disk boundary into two open semicircular arcs. When one of these arcs touches the floor, A and B are in order AB, when the other touches the floor, they are in order BA.
Now consider N >= 2 bubbles in general position on the disk. There are choose(N, 2) = N(N-1)/2 distinct pairs of bubbles. We can perturb the bubble locations so that no two lines are parallel (I think this is implied by general location). Since no two lines are parallel, each line determines a unique pair of antipodal exchange points on the disk, giving are N(N-1) exchange points on the disk. These exchange points divide the disk into N(N-1) arcs, let's call these "order arcs".
Now suppose the disk touches the floor at point A on the disk, and roll it so it touches point B on the disk. I leave it to you to show - If A and B are on the same order arc, the bubbles are in the same order. - If A and B are in different order arcs, the bubbles are in different order.
I believe this shows that in the 2D case, - For N = 0 or 1, there is 1 bubble order - For N >= 2, there are at most N(N-1) bubble orders.
The 3D problem is harder. In this case, for 2 bubbles A and B, you draw a line through the bubbles, then take the plane through the center of the marble perpendicular to this line. This plane intersects the marble surface in an "exchange great circle". If the marble touches the floor in one hemisphere of this great circle, the bubble order is AB, if in the other, BA.
Again, for N >= 2 bubbles, there are N(N-1)/2 lines, hence N(N-1)/2 exchange great circles. However, these great circles cannot be placed in general position on the marble surface, and the maximal number of "order regions" on the marble surface dissected by the great circles is not apparent.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of David Wilson Sent: Friday, October 17, 2014 5:55 PM To: math-fun Subject: [math-fun] Marble problem
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?
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