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 An idea -- solution? (don't know if maximum is achieved) by Charlie)

I see that in my original diagram 6 touches 5 along two boundaries, as do 6 and 2, and 6 and 3, so introducing a new area that cut off one of each of these accesses is ok.  The following diagram introduces a seventh mutually touching region:

 ___________           ____________          _____________  
           |          |           |                       
           |  4       |     1     |     5    
           |          |           |______________________ 
|    5     |          |________   |_____                 |
|          |__________|        |__|     |                |
|_____________________|           |     |                |
                      |     2     |     |   6
                      |           |     |
                      |___________|     |________
         6            |           |              |
                      |           |     7        |
                      |     3     |__________    | 
___________ __________|           |          |   |_______
     7     |          |           |     5    |
___________|     4    |___________|          |___________

Posted by Charlie on 2005-04-24 15:22:29