MathDB

Positive integers 1 to 9 are written in each square of a

3 \times 3

table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers

a, b, c

with either non-negative numbers

a-x, b-x, c+x

a+x, b-x, c-x

, where

x

is a positive number and can vary in each operation. (1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)?

a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )

b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )

(2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

Problem Statement