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Cutting planes with polygons (Posted on 2006-09-06) Difficulty: 3 of 5
Into how many regions can you partition the plane with m n-sided regular polygons?

For example, with two squares you can achieve up to 10 regions by choosing the right size and position of your squares.

See The Solution Submitted by JLo    
Rating: 4.0000 (2 votes)

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re(2): I agree. Full explanation. Attempted Proof | Comment 9 of 17 |
(In reply to re: I agree. Full explanation. by JLo)

Proof that there is no more cleverer solution.

[I’m sure there is a more formal proof in existence that someone more learned can offer as a reference, but this is the best I can offer].

<o:p></o:p>

In a 2D plane, 2 straight lines may intersect but do not form a closed space.

Three straight lines will create just one. 

Add to this a fourth. Should it pass through an already formed intersection, then the previous space is divided into 2, however should it cross all other lines at any other point then the regions are 3.

<o:p></o:p>

Here we have:

Lines    Max Regions

2                    0

3                    1

4                    3

5                    6

<o:p></o:p>

Ah! that triangle series again, and it is replayed in every one of Jer’s star points.

 <o:p></o:p>


  Posted by brianjn on 2006-09-07 20:47:09
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