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(2): An idea -- solution? An idea by Hugo)

A night of sleep didn't help.  The 2x triple helix is in fact Charlie's solution and he surely would have found a 7th region.

But I got a better idea to solve this.  I just made a square in Excell from 7x7 units and filled it with the numbers 1...7, such that they all touch each other.  Touching on a donut means also that the top row touches the bottom row and the left column touches the right column.
Just half an hour of trial and error gave me this:

4 1 1 3 3 7 5
4 1 2 2 7 7 5
4 1 2 6 6 5 5
4 1 2 6 3 3 4
1 1 2 6 3 7 7
5 2 2 6 3 7 5
4 4 6 6 3 7 5

I believe there are more solutions.  I tried 8x8, but did not yet found a solution.  I'll wait until Tristan confirms 8 regions are possible.

Posted by Hugo on 2005-04-24 14:50:28