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Minkowski Sums

Problem 228

Published on Saturday, 17th January 2009, 05:00 am; Solved by 920; Difficulty rating: 70%

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:

picture showing S_3 + S_4

How many sides does S1864 + S1865 + … + S1909 have?