How to split average back into its original numbers? (When you have surrounding averages as well).

Lets say you have a matrix of numbers a 4x4 group.

[20 23 37 40]
[29 30 45 55]
[19 35 55 67]
[15 12 40 45]

The columns will be a, b, c, d and the rows 1, 2, 3, 4

I want to concentrate on the central 4 numbers b2, c2, b3, d3. Lets average that and you get

41.25, lets also average a block of four to the right ((c2+d2+c3+d3)/4) = 55.5, down ((b3+c3+b4+c4)/4) = 35.5, to the left ((a2+b2+a3+b3)/4) = 28.25, and up ((b1+c1+b2+c2)/4) = 33.75.

Using these 5 averaged from surrounding groups and the central group average, can you recalculate the original 4 numbers from the central group? If not, can you do it by including diagonal group averages (1 to left and up etc).

I am sure it can be done I just cant figure out how.