You win!


  1. There are several lines on the plane and each pair of lines intersects at one point.
  2. The lines divide the plane into regions painted chess-like.
  3. You can flip over a triangular region by clicking on it.
  4. The goal is to get as many dark regions as possible.

When a triangle is flipped over, the number of dark regions in its neighbourhood changes from 3 to 4 and vice versa:

1 2 3 4

Your score is the total number of dark regions. For the selected number of lines, a theoretical upper estimate is shown next to the score.

Mathematical basis

This puzzle is named after Russian mathematician Vladimir Arnold. It is inspired by one of the open mathematical problems published in his book “Arnold's problems”:

Let N lines be given in the real plane, and their complement be chess-like painted black and white. What is the greatest difference between the number of black and white regions?

This problem is equivalent to the goal of the puzzle: maximizing the difference between the number of black and white regions is almost the same as maximizing the number of black regions.

There are another closely related open mathematical problem–Kobon triangles problem.

About author

I am a graduate of MIPT. My colleagues and I spent a lot of time trying to solve Arnold's problem. The result of our research is in the report (pdf, in Russian).

The source code of this puzzle is published on Github.

© 2020 Roman Parpalak
Click triangles to flip. Get the maximum number of dark regions.