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 Dividing a donut (Posted on 2005-04-23)
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?

 See The Solution Submitted by Tristan Rating: 4.0000 (5 votes)

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 An idea -- solution? (don't know if maximum is achieved) | Comment 9 of 34 |

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.

 Posted by Charlie on 2005-04-23 20:22:59

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