2941：Homogeneous Squares

Assume you have a square of size n that is divided into n × n positions just as a checkerboard. Two positions (x₁, y₁) and (x₂, y₂), where 1 ≤ x₁, y₁, x₂, y₂ ≤ n, are called “independent” if they occupy different rows and different columns, that is, x₁ ≠ x₂ and y₁ ≠ y₂. More generally, n positions are called independent if they are pairwise independent. It follows that there are n! different ways to choose n independent positions.

Assume further that a number is written in each position of such an n × n square. This square is called “homogeneous” if the sum of the numbers written in n independent positions is the same, no matter how the positions are chosen. Write a program to determine if a given square is homogeneous!

The input contains several test cases.

The first line of each test case contains an integer n (1 ≤ n ≤ 1000). Each of the next n lines contains n numbers, separated by exactly one space character. Each number is an integer from the interval [−1000000, 1000000].

The last test case is followed by a zero.