MathDB

Let

n

be a positive integer. Lavi Dopes has two boards

n \times n

. On the first board, he writes an integer in each of his

n^2

squares (the written numbers are not necessarily distinct). On the second board, he writes, on each square, the sum of the numbers corresponding, on the first board, to that square and to all its adjacent squares (that is, those that share a common vertex). For example, if

n = 3

and if Lavi Dopes writes the numbers on the first board, as shown below, the second board will look like this.
Next, Davi Lopes receives only the second board, and from it, he tries to discover the numbers written by Lavi Dopes on the first board.
(a) If

n = 4

, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?
(b) If

n = 5

, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?

Problem Statement