In the 7-by-7 grid above, one can draw a simple1 closed curve using nothing but quarter-circle segments. Two examples are shown above: one enclosing a region of 4-π (in blue), and one enclosing a region of area 6 (in red).
It can be shown that there are 36 ways to enclose a region of area exactly 4-π. How many ways can one draw a curve enclosing a region of area exactly 32?
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Note that a simple curve is not allowed to self-intersect. ↩