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?

The two solutions that I have provided are the complete cuttings.

The third solution is not to make a complete cutting since the question does not mention that a region must be formed with a complete cutting.  Thus, it could be as follows:

 

                    a            b

              c..............               d

              e.............                f

                   g               h

Again a, b, d, f, h, e, c & a form one end of donut after breaking one side of the donut.  Follow the dotted lines to cut from c & e horzontally.  Open up c and you will find two regions are formed.  Open up e and you will find another two regions are formed.  Draw as many lines horizontally as possible but not to break the donut.  Many regions are formed.

Thanks.

Edited on April 26, 2005, 3:48 pm

Posted by Jonathan Chang on 2005-04-26 15:46:10