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 1 of 6: Work modulo a prime where every degree is invertible
Previous stepNext step
2−1≡3,3−1≡2,4−1≡4(mod5)2^{-1}\equiv3,\quad 3^{-1}\equiv2,\quad 4^{-1}\equiv4\pmod5
Detailed analysis

Every square has 22, 33, or 44 neighbours, and 2,3,42,3,4 are all invertible modulo the prime 55. So the averaging operation can be carried out entirely modulo 55 by multiplying by the corresponding inverse, and if some sequence of turns equalizes the integer board, applying the same sequence to the residues modulo 55 equalizes the board modulo 55 as well.