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: A seond step?
The way I understand what you are saying is per the following diagram:
| | | |
|A______ 1 | |A_________ 2 |
| \___________________ | | \________________ |
| 5 \----- B| | 6 \----- B|
_______________ ___________________________________ __________________
| | C | | C
3 | | \ 4 | | \3
| | \ | | \
\ | | 7 \ | | 8
_8_D___________| |_____________________D_____________| |__________________
where the ends wrap into each other as well as the top and the bottom. Numbers 1-4 refer to armando's original regions, which correspond to the full rectangle of each, undivided by athe A-B and C-D lines. The number 3 appears at both left and right due to the wrapping.
However, region 3 does not touch region 1. Nor does region 4 touch region 2.
If the top were to receive C-D type cuts instead of A-B type, so as not to cut off regions 1 and 2 from the bottom, then 5 would be cut off from 4 and 6 from 3.
Posted by Charlie
on 2005-04-23 19:59:50