I want to divide the surface of a donut into as many different regions as possible. The regions can be any shape, as long as they are each in one piece. Each region must touch each of the other regions (touching on a corner doesn't count). How many regions can I make?

(In reply to re(3): A seond step? by Hugo)

You are right.  The bottom of 3 touches the top of 1.  It is region 4 that does not touch region 1.  And it is region 3 that does not touch region 2.

The very top is divided into only 2 segments--1 and 2.  The bottom into 4 segments--3, 7, 4 and 8.  Wherever boundaries are drawn at the bottom, there have to be four of them, which means that one section at the top must contain more than one, which then isolates the area(s) bounded by them from the other of the top regions.  Similar considerations apply to the middle boundary, where 7 doesn't touch 6 and 8 doesn't touch 5.

Posted by Charlie on 2005-04-23 20:32:40