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 5 of 6: An equilibrium board is a fixed point of every turn
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every cell already equals its neighbours’ average ⟹ any turn leaves the board unchanged\text{every cell already equals its neighbours' average} \ \Longrightarrow\ \text{any turn leaves the board unchanged}
Detailed analysis

In a turn, each chosen square is replaced by the modulo-55 average of its neighbours' current values; but by construction every cell of the tiled board already equals that average. So whichever subset of squares is chosen, every chosen square is replaced by its own current value, and unchosen squares are untouched by definition. Hence the board modulo 55 is unchanged by any single turn, and therefore by any finite sequence of turns.