Arnold's Puzzle ?

Rules

1. Several lines are drawn on the plane, intersecting at one point per pair.
2. The lines divide the plane into chess-like painted regions.
3. You can toggle a triangular region by clicking on it.
4. The goal is to get as many dark regions as possible.

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

Your score is the total number of dark regions. The theoretical upper estimate for the selected number of lines is displayed alongside the score.

Mathematical basis

This puzzle is named after Russian mathematician Vladimir Arnold and 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 have dedicated considerable time to solving Arnold's problem. The result of our research is available in the report (PDF, in Russian).

The source code for this puzzle is published on Github.