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Dividing a donut (Posted on 2005-04-23) Difficulty: 3 of 5
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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Solution? | Comment 16 of 34 |

I just wake up from the bed and it is 7 a.m. and I see so many debate in regard of this question.

There are two possible solutions:

For instance, if you divide the donut at the ring instead of penetrating it across the hollow part, a normal condition is applied and Hugo, the Genius, has given a full picture of it and there are 8 regions & 4 parts.  His initial answer is absolutely correct.  This is due to the cutting of the donuts was at the corner of it without going through the hollow and a normal condition applies.

For instance, if you divide the donut through the middle of it, it ends up that there are two ends due to the middle of the donut is hollow.  As the two ends of the donut could not touch each other since they are far apart, the answer is 0 region & 0 part.  Remember the question that it asks 'each region must touch each of the other regions'.  Unless cutting the donut into two ends could cause that two ends to link up, there might be other answers.  As the two ends of the donut are far apart & they are not touching with each other, the answer for the cutting in the middle is simply zero.

Hugo is right initially if the cutting of the donut is normal near the corner of the donut without going through the hollow. 

I am wrong in my initial two comments due to I have misread the question.  As the question is clear to me that all the parts have to be touched, I have a full support of Amando's  & Hugo's initial solutions provided that cutting is at the edge without cutting through the hollow.

I not only support Amando's & Huo's initial solution with my hands, I support also with my legs so much so I fall down since I jump up and lift up all my hands and legs to support them at the same time and I do not put it down so much so I fall down and have been injured severely.

No! Just joking! 

Alright men, Amando & Hugo.  Keep it up!  Hold down to your solution if your solution is correct!

Thanks.

Edited on April 23, 2005, 11:55 pm

Edited on April 23, 2005, 11:58 pm

Edited on April 24, 2005, 12:04 am

Edited on April 24, 2005, 12:08 am

Edited on April 24, 2005, 12:11 am

Edited on April 24, 2005, 12:21 am
  Posted by Jonathan Chang on 2005-04-23 23:52:30

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