Let $x$ is the largest natural number in the grid. But we get that $x=\frac{y+z}{2}$ for $y,z$ in adjacent cell. So $x=y=z$

Well it would really be x=(a+b+c+d)/4 or x=(a+b+c)/3 or x=(a+b)/2 (depending on how many cells border the cell with the largest number), but yeah, that's the gist of it.

an 5 year old solution fill it with 1.