MathLabs

Problem 4

Consider a 2018×20192018\times2019 board with an integer written in each unit square. Two unit squares are called neighbours if they share a common edge. In each turn, some unit squares are chosen; then, for each chosen square, the average of all its neighbours is computed, and finally, after all these computations are done, the number in each chosen square is replaced by its corresponding average. Is it always possible to make the numbers in all squares equal after finitely many turns?
Step 6 of 6: Conclude the answer is no
Previous stepNext step
board mod 5 is never all-equal ⟹ the integer board can never be made all-equal; answer: No\text{board mod 5 is never all-equal} \ \Longrightarrow \ \text{the integer board can never be made all-equal; answer: No}
Detailed analysis

Take the entries of the tiled board (values in {0,1,3,4}\{0,1,3,4\}) as an actual initial board of integers. By step 5, no finite sequence of turns changes this board modulo 55, and since it is not all equal modulo 55 (step 4), it can never become an all-equal board modulo 55. By step 1, if it could be made all equal over the integers, the same turns would make it all equal modulo 55 too — a contradiction. Hence it is not always possible to make the numbers in all squares equal, so the answer to the problem is no.