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?
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