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?

Your solution and the solutions from Hugo and the rest of the people are all correct since the question does not mention that it needs maximum number of ways of touching.

Observing carefully the 1st solution and the 2nd solution that I have given whereby the phrase, solution is on process, is mentioned.  I have provided numerous regions with numerous cuttings.  Especially the 2nd solution, the complete one whole slices through straight line cuttings and each part of cutting touches all parts of donut.  All the slices of donuts attach each other.  If not, do a real cutting in the donut and join it in accordig to the picture as shown and you will realize it.

The 3rd set of my solution might not be acceptable by some of the people.  However, if one interprets the question of the word, touch as it is considered to be touched if the side of the donut is interlinked and they are not cut, the 3rd set of solution is still to be considered as part of the solution.

As the question does not mention that maximum of regions is to be met,  all the solutions and that includes yours must be accepted as the solutions of this question.

Thanks Tristan of your contribution of this puzzle so that we could have hours of fun of solving.

Not to worry!  Hugo and Amando.  Thanks for your comments.  All solutions are acceptable to this question since it does not mention the maximum of regions is needed.

Thus, nobody is to criticize any one of the solutions that is provided for the question since the question does not mention the maximum is required.

Thanks.

Edited on April 28, 2005, 2:50 am

Posted by Jonathan Chang on 2005-04-28 02:22:51