Question 45
A $3 \times 3$ square is filled with the numbers $a, b, c, d, e, f, g, h$ and $i$ as follows:
| $c$ | $f$ | $i$ |
| :--- | :--- | :--- |
| $b$ | $e$ | $h$ |
| $a$ | $d$ | $g$ |
We know that it is a magic square, that is, there is a value $S$ that equals the sums of the numbers in each row, each column and each of the two diagonals. Verify that:
a) $2(a+c+g+i)=b+d+f+h+4 e$.
b) $S=3 e$.
c) $a c+c i+a g+g i=e(b+d+f+h)$.
d) $2\left(a^{2}+c^{2}+g^{2}+i^{2}\right)=b^{2}+d^{2}+f^{2}+h^{2}+4 e^{2}$.
| $c$ | $f$ | $i$ |
| :--- | :--- | :--- |
| $b$ | $e$ | $h$ |
| $a$ | $d$ | $g$ |
We know that it is a magic square, that is, there is a value $S$ that equals the sums of the numbers in each row, each column and each of the two diagonals. Verify that:
a) $2(a+c+g+i)=b+d+f+h+4 e$.
b) $S=3 e$.
c) $a c+c i+a g+g i=e(b+d+f+h)$.
d) $2\left(a^{2}+c^{2}+g^{2}+i^{2}\right)=b^{2}+d^{2}+f^{2}+h^{2}+4 e^{2}$.