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Partitioning Space (Posted on 2004-09-13) Difficulty: 5 of 5
From Pizza Cut, we know the formula for maximum partitioning (pieces) of the circle, given n straight lines (cuts).
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  1. Determine the maximum number of regions of the plane produced by n intersecting circles.

  2. Determine the maximum number of regions of the plane produced by n intersecting ellipses.

  3. Determine the maximum number of regions of space produced by n intersecting spheres.

No Solution Yet Submitted by SilverKnight    
Rating: 4.5000 (4 votes)

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Solution Part 3?? | Comment 5 of 13 |

The spheres are tricky cuz they are hard to visualize. I could visualize up to 4, but not 5. So I donít know if my generalization will hold even to 5.

I donít think that the answer is the same as for circles, though, which you can see after the fourth sphere is added. If you place your first three spheres such that the top view looks like a Venn Diagram (just like the circles), you can place the fourth sphere above the other three (such that it looks like a squashed pyramid). The fourth sphere can cut through all of the previous regions (the circle couldnít do this Ė it would always miss one region, or miss 2 regions but add another in space) plus make a new region in space. I donít know how to explain that better.

Spheres:
n      regions added      regions     
0                                    0
1               1                   1
2               2                   3
3               4                   7
4               8                 15

Based on this, it seems that you are adding 2^(n-1) each time.  So the formula is [sum of 2^(n-1)] = [(2^n Ė 1)/(2-1)] = 2^n Ė 1.

Like I said, I donít know if itís right.


  Posted by nikki on 2004-09-14 10:51:42
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