Question 7
Decide whether the integers $1, 2, \ldots, 100$ can be arranged in the cells $C(i,j)$ of a $10 \times 10$ matrix (where $1 \le i, j \le 10$), such that the following conditions are satisfied:
(i) In every row, the entries add up to the same sum $S$.
(ii) In every column, the entries also add up to this sum $S$.
(iii) For every $k = 1, \dots, 10$ the ten entries $C(i,j)$ with $i-j \equiv k \pmod{10}$ add up to $S$.
(i) In every row, the entries add up to the same sum $S$.
(ii) In every column, the entries also add up to this sum $S$.
(iii) For every $k = 1, \dots, 10$ the ten entries $C(i,j)$ with $i-j \equiv k \pmod{10}$ add up to $S$.