# Problem 514: Geoboard Shapes
A geoboard (of order N) is a square board with equally-spaced pins
protruding from the surface, representing an integer point lattice for
coordinates 0 ≤ x,y ≤ N. John begins with a pinless geoboard. Each
position on the board is a hole that can be filled with a pin. John
decides to generate a random integer between 1 and N+1 (inclusive) for
each hole in the geoboard. If the random integer is equal to 1 for a
given hole, then a pin is placed in that hole. After John is finished
generating numbers for all (N+1)2 holes and placing any/all
corresponding pins, he wraps a tight rubberband around the entire group
of pins protruding from the board. Let S represent the shape that is
formed. S can also be defined as the smallest convex shape that contains
all the pins. The above image depicts a sample layout for N = 4. The
green markers indicate positions where pins have been placed, and the
blue lines collectively represent the rubberband. For this particular
arrangement, S has an area of 6. If there are fewer than three pins on
the board (or if all pins are collinear), S can be assumed to have zero
area. Let E(N) be the expected area of S given a geoboard of order N.
For example, E(1) = 0.18750, E(2) = 0.94335, and E(10) = 55.03013 when
rounded to five decimal places each. Calculate E(100) rounded to five
decimal places.