# Puzzle: Each entry in a number grid is the average of its neighbors

I'm trying to solve the following puzzle:

Each number should be the average of its four neighbors. For example, $x$ should be equal to $\frac{1}{4}(4+10+y+z)$.

I don't know how to make a formula out of it. What's the trick? Can anyone give me a clue ?

• You can introduce 10 variables and 10 equations. Then use matrices to solve it. – Kamil Jarosz Mar 16 '16 at 17:14
• Do only the numbers in the middle have this property? Because if the ones on the edge do you can puzzle it out by $7=\frac{4+Y+1}{3} \implies Y=16$ – J. Bush Mar 16 '16 at 17:17
• @j-bush , X,y and z , they all have the same property – Ket Mar 16 '16 at 17:21

You start with all unknowns set to zero. Then starting from the boundary, replace each unknown with the average of its four neighbors. Since this is a puzzle, the answer is likely to be integers so you can round the result safely. After two or three iterations you get the answer. Mine is: $$6, 5, 3, 2, 1 \\ 5, 4, 4, 2, 1$$