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?
Initially, cut one end of the donut and leave another end joining to this donut. After that, follows the picture below to cut the inner part of the donut from one end to another with the identical shape.
The drawing as shown below is one end of the donut after the cutting (Note: another end I need not to elaborate since it will be the same shape across from one end of the donut to another. Not only that, the cutting is parallel from one end to another so that both ends can be perfectly joined with each other): a) Follow the instruction as follows to draw a line across in accordance to the alphabetical order as follows: Ia, Ib, Ic, Id...up to 1n & after that, join back to In (this forms the external part of the one end of donut). We then join 2a to 2k & that forms the second cutting (Note: the external of it we name it as section A and its internal as section B as shown in the picture). We follow the same pattern of cutting as 2a to 2k to do the cutting for 4a..up to 4g and to link up to 3g & that form section D internally.
We join 3a and 3b. We then join 3a & 3h. We then join 3c..3d..3e..3f..3g..3h by a straight line. We leave 3c and 3b to be open end.
The rest of the joining would be: a) from Im to 5a and even until 5v b) from Ia to 6a even until 6s; c) from 3a to 7a and even until 7q; d) from 3b to 8a even until 8u; e) from Ic to 9a even until 9m
I do not draw another end of donut since the cutting of donut from one end to another must be parallel in such a way that an end of any parts of the donuts will perfectly join with the same parts of donut of another end.
5h..............................................................5i
5g 6c............................................6d 5j
5f 6b 7b...........................7c 6e 5k
5e 6c 7c 8c...............8d 7d 6f 5l
5d 6b 7b 8b 9b..9c 8e 7e 6g 5m
5c 6a 7a 8a 9a 9d 8f 7f 6h 5l
5b Ia......Ib3a.......3bIa.....Ib 9e 8h 7g 6i 5m
5a In 2k......3h 3c2a....2b Ic 9d 8l 7h 6j 5n
Im A 2j B 4g...3g C 3d4a.4b 2c Id 9e 8m 7l 6k 6o
Il 2i 4f D 3f........3e 4c 2d Ie 9f 8n 7j 6l 5p
Ik 2h 4e................4d 2e If 9g 8o 7k 6m 5q
Ij 2g.......................2f Ig 9h 8p 7l 6n 5r
Ii...............................Ih 9i 8q 7m 6o 5s
9j 8r 7n 6p 5s
9k 8s 7o 6q 5t
9l 8t 7p 6r 5u
9m 8u 7q 6s 5v
Take note that Region A touches Region B due to it is placed side by side. Region A touches Region C due to the open end between 3b and 3c. As 3b and 3c is open end, sections C & D have a direct contact with section A. Thus, section A touches B, C & D.
Region B touches Region A & D since Region B is placed in the middle of Region A & D. Region B touch Region C since Region A & Region C join with each other in open end at 3b and 3c. Thus, Region B touches A, B, C & D.
Region C is surrounded by Regions A, B & C.
One could perform many cuts from any point between 3b and 3c along the curve lines to another end of Section C between 3a and 3h. However, one must not cross over any lines during the cutting. Through this way, one could perform numerous cuttings to form numerous regions.
The above is just one of the way that one could have numerous cuttings. One might change strange lines to different kind of curve lines and follow the above pattern to have numerous ways of cutting too.
Thanks
Edited on April 26, 2005, 1:49 pm
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