OMM 2005 Problem 2

Given several matrices of the same size. Given a positive integer $N$ , let's say that a matrix is $N$ -balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$ . (i) Show that every $2N$ -balanced matrix can be written as a sum of two $N$ -balanced matrices. (ii) Show that every $3N$ -balanced matrix can be written as a sum of three $N$ -balanced matrices.