The problem is here.

The solution I've learned is:

Color all the integer points in black and white, a point (x, y) is colored black when 2|(x+y).

If there are both black and white points in the n points, divide them by color.

Otherwise, assuming all the points are black, we rotate and zoom all n points, transforming (x,y) into ((x+y)/2, (x-y)/2).

Then, we solve the sutask, there could be O(logV) subtasks in total before the points don't change.

My problem is:

I know the subtask is equivalent to the original problem, but why is the solution always found in this way when there is a solution?

Thank you for helping!