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Prove that it is impossible to color a 2020*2020 square board with two colors such that

1. All unit squares on the boundary have the same color.

2. Every 2*2 square has two colors and no 2*2 square is colored like a mini chess board.

oh fuck me this shouldn't have taken multiple hours. i swear i'm good at math.

https://codeforces.com/gym/102586/problem/I

https://open.kattis.com/problems/distinctivecharacter

You have an arbitrarily large checkers board with a horizontal line across the middle separating the top half from the bottom half. You can place as many checkers as you like anywhere in the bottom half to start. Now you may move the checkers by jumping one over another up down left or right and removing the jumped checker (ie peg solitaire rules). How far above the line can you get a checker?