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4 cents (Posted on 2014-09-12) Difficulty: 3 of 5
Place four pennies on a table such that each touches at least one other.

For what arrangement will the area of the convex hull of this shape be maximized?

No Solution Yet Submitted by Jer    
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Some Thoughts re: About this problem. | Comment 9 of 11 |
(In reply to About this problem. by Jer)

In this problem I made one major assumption - the figure is symmetric.  This reduced a two variable problem down to a one variable problem.  I also have confidence that this is the maximum since there is little reason for the figure to be asymmetric; but that does not guarantee that an asymmetric figure is not maximal.

5 pennies would be a very interesting problem, definitely D4 at a minimum possibly D5.  But I would probably make an initial assumption that not only was the 5 penny figure symmetrical, but also that all the pennies' centers form a cyclic polygon.  In the 4 penny case isosceles trapezoids are always cyclic.

I conjecture that for N pennies that the maximum convex hull occurs when all the centers of the pennies form an N sided cyclic polygon.

  Posted by Brian Smith on 2016-01-31 14:59:56
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