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?
Using the diagram where the left side wraps to touch the right side and the bottom to the top:
___________ ____________ _____________
| | |
| 4 | 1 | 5
| | |______________________
| 5 | |___________| |
| |__________| | |
|_____________________| | |
| 2 | 6
| |
|___________|
6 | |
| |
| 3 |__________
__________| | |
| | | 5 |
___________| 4 |___________| |___________
The lines around the edges are missing if the wrap-around is to a different portion of the same region. The line is shown on both sides if the region is different on the other side of the wrap-around.
Note that 1 touches 2, 4, 5 and 6 in the obvious fashion and 3 via the wrap-around. Region 2 touches all the other regions in the obvious fashion. Region 3 touches 4, 5 and 6 in the obvious fashion, as do regions 5 and 6 with each other.
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Posted by Charlie
on 2005-04-23 20:22:59 |
An idea -- solution? (don't know if maximum is achieved)
