Problem 4

Consider a

2018\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 4 of 6: Tile the whole board by repeated reflection

Previous step Next step

2018=2\cdot1009,\quad 2019=3\cdot673\ \Longrightarrow\ \exists\ 2018\times2019\ \text{board in equilibrium mod 5, not all equal}

Detailed analysis

Starting from the $2\times3$ equilibrium block of step 2, repeated reflection (step 3) doubles the dimensions while staying in equilibrium. Since $2018=2\cdot1009$ and $2019=3\cdot673$ , enough reflections build a full $2018\times2019$ board, every cell of which is a value from $\{0,1,3,4\}$ copied from the original block, still satisfying the equilibrium condition at every cell; since the block itself has unequal entries ( $0\ne1$ ), the resulting large board is not all equal.