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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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Solution I agree. Full explanation. | Comment 6 of 16 |

I agree with Dej Mar and his formula nm(m-1)+2

I see that Charlie just edited his recursive definition anyway.

If we make the n-gons congruent and concentric, we can also arrange the m of them with equal rotations to get a nice symmetrical star shape.

The star has n*m points, one for each corner of each polygon.  The tip of each point is a kite.  Spiraling inward from these kite are more kites (depending of m) with isosceles triangles bordering an (n*m)-gon as the central region.

If you consider each of the n*m spiral arms you can see that the number of kites increases by one if m increases by one and that the number of kites + triangles in an arm is m-1

So the total number of kites and triangles is (n*m)(m-1)
Add the inner polygon and outer region to get the complete formula.

  Posted by Jer on 2006-09-07 11:46:34
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