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:

picture showing S_3 + S_4

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

These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK

http://projecteuler.net/problem=228