yaworsw/euler-manager

---
:id: 228
:name: Minkowski Sums
:url: https://projecteuler.net/problem=228
:content: |+
  Let <var>S</var><sub>n</sub> be the regular <var>n</var>-sided polygon – or _shape_ – whose vertices <var>v</var><sub><var>k</var></sub> (<var>k</var> = 1,2,…,<var>n</var>) have coordinates:

  | | <var>x</var><sub><var>k</var></sub>&nbsp; &nbsp;=   cos( <sup>2<var>k</var>-1</sup>/<sub><var>n</var></sub> ×180° ) |
  | | <var>y</var><sub><var>k</var></sub>&nbsp; &nbsp;= &nbsp; sin( <sup>2<var>k</var>-1</sup>/<sub><var>n</var></sub> ×180° ) |

  Each <var>S</var><sub><var>n</var></sub> is to be interpreted as a filled shape consisting of all points on the perimeter and in the interior.

  The _Minkowski sum_, <var>S</var>+<var>T</var>, of two shapes <var>S</var> and <var>T</var> is the result of adding every point in <var>S</var> to every point in <var>T</var>, where point addition is performed coordinate-wise: (<var>u</var>, <var>v</var>) + (<var>x</var>, <var>y</var>) = (<var>u</var>+<var>x</var>, <var>v</var>+<var>y</var>).

  For example, the sum of <var>S</var><sub>3</sub> and <var>S</var><sub>4</sub> is the six-sided shape shown in pink below:

   ![picture showing S_3 + S_4]({{ images_dir }}/p228.png)

  How many sides does <var>S</var><sub>1864</sub> + <var>S</var><sub>1865</sub> + … + <var>S</var><sub>1909</sub> have?