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 Partitioning Space (Posted on 2004-09-13)
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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 re(3): Part 3?? what was I thinking? | Comment 8 of 13 |
(In reply to re(2): Part 3?? by nikki)

Ummmm, yeah, I don’t know what I was thinking...

I still think my beginning was right… that [n^3 + 3n^2 + 2n]/3 puts me in the right ball park. I got way messed up, and wasn’t looking at the right then when I tried to find a pattern for what I was off by.

I THINK I have it right this time =) I think I’m off by the sum of 4*(n-1). The sum of 4*(n-1) 4*[(n-1)(n)/2) = 2n(n-1) = 2n^2 – 2n.

So the final answer should be
[n^3 + 3n^2 + 2n]/3 – [2n^2 – 2n]
= [n^3 + 3n^2 + 2n]/3 – [6n^2 – 6n]/3
= [n^3 - 3n^2 + 8n]/3

Wait, I’m still off though, but only by one. I’m not sure what I missed there, but my really really final answer is:

[n^3 - 3n^2 + 8n - 3]/3

 Posted by nikki on 2004-09-14 15:20:23

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