Question 41
On a $2023 \times 2023$ board, some of the squares contain a beetle, with at most one beetle per square. After one minute, every beetle moves one square to the left or right, or one square up or down. After each following minute, every beetle again moves one square further, but they always make a 90-degree turn. So if a beetle moved left or right in one minute, it moves up or down in the next minute, and vice versa. What is the minimal number of beetles that must be on the board so that you know for sure that at some point in time two beetles end up in the same square at the same moment, no matter where the beetles start and how they move (while obeying the rules)?