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Circles Abound (2) (Posted on 2004-10-17) Difficulty: 4 of 5
I've a black circle of radius 5.

I wish to create 5 identical white circles of some lesser radius, which I will place on top of the black circle and completely obscure (cover) the black circle.

What is the smallest radius which I can make the smaller circles and still meet my requirement?

No Solution Yet Submitted by SilverKnight    
Rating: 3.3333 (3 votes)

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Some Thoughts No Subject | Comment 10 of 11 |

As Richard pointed out, the 3.09 answer is not the smallest. Unfortunately, I haven't been able to compute the smallest radius either.

Heres what I have tried:

I start with five circles, spaced in the pentagram as described. Then, shrink the radius, with four of the circles spreading out around the rim (which will leave the center uncovered) and the fifth moving in to cover the triangle between the two points on the perimiter and the closer intersection of the two opposite circles. If you take this to the maximum point where the fifth circle still just covers this triangle, you can set up a set of simultaneous equations to try to solve for the radius and enough angles to determine the system. Unfortunately, I come up with five equations, five unknowns but the equations are too messy (lots of trig functions). I hate messy math, so this is where I stop.

Anyway, that's why I rate this a 4 (I'm new here so I'm not sure about the scale). Good luck to all solvers.

  Posted by SteveH on 2004-10-20 01:41:55
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