Question 117
In every unit square of an $n \times n$ board a positive integer is written. A move consists of choosing a $2 \times 2$ square and adding 1 to exactly three of the four numbers written in the chosen square. We call a positive integer *good* if starting from any initial numbers, there is a sequence of moves that makes all the numbers from the board equal.
a) Prove that $n = 6$ is not good.
b) Prove that $n = 4$, and $n = 1024$ are good.
a) Prove that $n = 6$ is not good.
b) Prove that $n = 4$, and $n = 1024$ are good.