MathDB

The cells of an

n \times n

table are filled with the numbers

1,2,\dots,n

for the first row,

n+1,n+2,\dots,2n

for the second, and so on until

n^2-n,n^2-n+1,\dots,n^2

for the

n

-th row. Peter picks

n

numbers from this table such that no two of them lie on the same row or column. Peter then calculates the sum

S

of the numbers he has chosen. Prove that Peter always gets the same number for

S

, no matter how he chooses his

n

numbers.

3.