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# Problem 228

Minkowski Sums

Let Sn be the regular n-sided polygon – or shape – whose vertices vk (k = 1,2,…,n) have coordinates:

$x_k= \text{cos(}\frac{2k-1}{n} \times 180^\circ \text{)}$ $y_k= \text{sin(}\frac{2k-1}{n} \times 180^\circ \text{)}$

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?

$x_k= \text{cos(}\frac{2k-1}{n} \times 180^\circ \text{)}$ $y_k= \text{sin(}\frac{2k-1}{n} \times 180^\circ \text{)}$