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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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 Another simple way-please don't open-on process | Comment 29 of 34 |

Break one end of the donut.  After breaking, you will have two end of the donuts as shown below (Note that: draw a line from the small letters from a to b & from b to c & so on & so forth up to the final p and a circle is formed and this is the shape of one end of donut):

First, cut a straight line vertically so as to follow the sequence of letters & numbers as shown here, ie. o, 4, 3, 2, 1 & j.  Not only that, the cutting should follow the same pattern across the ring to another end of the donut so that another end of donut would have a vertical line cutting as the same as this end as shown in the picture below.

Secondly, cut another straight line vertically so as to follow the sequence of letter & numbers as shown here, ie. a, 5, 6, 7, 8 & f.  Not only that, the cutting should follow the same pattern across the ring to another end of donut so as another end of donut would have a vertical cutting as the same as this end as shown in the picture below.

Thirdly, follow the dotted lines to cut from o to a; from 4 to 5; from 3 to 6; from 2 to 7; from 1 to 8; and from j to f.  The above cuttings must be horizontally as shown below and to cut across the ring to another end of the donut parallelly.

p

o.................................a

n     4.................................5        b

m           3.................................6           c

l             2.................................7           d

k       1.................................8       e

j..................................f

i

Turn another end of donut 90 degree clockwise and leave one end of the donut as shown above.  Another end of donut would shown like this:

l               m

k                                     n

j      1          2               3         4    o

i          l           l                l          l        p

f      8           7               6        5    a

e                                      b

d                c

Note that the horizontal lines turn up to be vertical after turning it 90 degree clockwise.  Now join one end of the donut that has horizontal lines with another end vertical lines.  It ends up the vertical & horizontal lines forms numerous squares after the joining.  Not only that, every part of the cutting touches with each other.  Thus, numerous cutting with numerous regions are formed after joining one end of the donut as vertical lines and another end as horizontal lines.

For instance, if you cut as many straight lines as you could, many regions are formed with touching after turning another end 90 degree clockwise in joining.

The above straight lines can be cut in different kinds of curve, again this causes numerous ways of cutting.

Simple!

Thanks.

Edited on April 26, 2005, 3:28 pm

Edited on April 26, 2005, 3:34 pm
 Posted by Jonathan Chang on 2005-04-26 14:33:07

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