# Problem 228: Minkowski Sums
Let Sn be the regular n-sided polygon – or shape – whose vertices vk
(k = 1,2,…,n) have coordinates: xk   =   cos( 2k-1/n ×180° ) yk   =  
sin( 2k-1/n ×180° ) Each Sn is to be interpreted as a filled shape
consisting of all points on the perimeter and in the interior. The
Minkowski sum, S+T, of two shapes S and T is the result of adding every
point in S to every point in T, where point addition is performed
coordinate-wise: (u, v) + (x, y) = (u+x, v+y). For example, the sum of
S3 and S4 is the six-sided shape shown in pink below: How many sides
does S1864 + S1865 + … + S1909 have?