1646: 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.

For each test case output whether the specified square is homogeneous or not. Adhere to the format shown in the sample output.

2
1 2
3 4
3
1 3 4
8 6 -2
-3 4 0
0

homogeneous
not homogeneous