This question was part of the problem set for PROMYS Europe 2020, a maths (or math, just in case I made anyone unhappy) camp held at Oxford every year. It was the one which I couldn't do, and I am still puzzled by it to this day:
The Mathematical Forest is grown in a two-dimensional plane, where trees can only grow on points with integer coordinates. To start with, there are no trees at all. The foresters plant the first tree at (0, 0). Each year, they carry out tree planting according to the following rule. If there is a tree on the point (m, n) but there are no trees on the points (m + 1, n) and (m, n + 1), then they can choose to remove the tree on (m, n) and plant new trees on the points (m, n + 1) and (m + 1, n).
For an integer k ≥ 1, the kth diagonal consists of all points (m, n) with m + n = k − 1. Is it possible for the foresters to arrange their planting so that eventually there are no trees on the first 2 diagonals? What about the first 3 diagonals? 4 diagonals? Can you generalize?
P.S. The deadline has already passed, so please do not think that I am cheating! The camp was actually cancelled due to obvious reasons, which sucks, because I really looked forward to going.