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 4 cents (Posted on 2014-09-12)
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?

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 Overlooked problem | Comment 1 of 11
This puzzle just came up randomly.  Hard to believe that nobody has commented in 16 months, so I will kick it off.

Instead of pennies, let's use 4 disks of unit radius and 0 thickness.

If laid in a straight line, the convex hull is 12 + Pi

If arranged in a square, the convex hull is also 12 + pi.

If arranged in a diamond shape, the convex hull is 8+ Pi + 2sqrt(3), which is approx .6 less than the other two.

It is hard to imagine a better arrangement, so I am going with a diamond.

Unless, of course, this is a trick question.  If the four are stacked one on top of another, the convex hull is only Pi

 Posted by Steve Herman on 2016-01-30 21:15:26

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