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 2 of 6: An explicit invariant 2×3 block
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[010434](mod5)\begin{bmatrix}0&1&0\\4&3&4\end{bmatrix}\pmod5
Detailed analysis

Label a 2×32\times3 block of cells a,b,ca,b,c (top row) and d,e,fd,e,f (bottom row). The values a=0,b=1,c=0,d=4,e=3,f=4a=0,b=1,c=0,d=4,e=3,f=4 satisfy, for every cell, that the cell's value equals the modulo-55 average of its neighbours within the block: 2a≡b+d2a\equiv b+d, 3b≡a+c+e3b\equiv a+c+e, 2c≡b+f2c\equiv b+f, 2d≡a+e2d\equiv a+e, 3e≡d+f+b3e\equiv d+f+b, 2f≡c+e(mod5)2f\equiv c+e\pmod5, which a direct check confirms (0≡1+40\equiv1+4, 3≡0+0+33\equiv0+0+3, and so on, all mod 55). Since the block is already in equilibrium and b≠ab\ne a, choosing any subset of its cells to update leaves every value unchanged.