
4. Consider the following $3 \times 2$ array formed by using the numbers $1,2,3,4,5,6$ :

$$
\left(\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{array}\right)=\left(\begin{array}{ll}
1 & 6 \\
2 & 5 \\
3 & 4
\end{array}\right)
$$

Observe that all row sums are equal, but the sum of the squares is not the same for each row. Extend the above array to a $3 \times k$ array $\left(a_{i j}\right)_{3 \times k}$ for a suitable $k$, adding more columns, using the numbers $7,8,9, \ldots, 3 k$ such that

$$
\sum_{j=1}^{k} a_{1 j}=\sum_{j=1}^{k} a_{2 j}=\sum_{j=1}^{k} a_{3 j} \quad \text { and } \quad \sum_{j=1}^{k}\left(a_{1 j}\right)^{2}=\sum_{j=1}^{k}\left(a_{2 j}\right)^{2}=\sum_{j=1}^{k}\left(a_{3 j}\right)^{2}
$$


Solution:Consider the following extension:

$$
\left(\begin{array}{llllll}
1 & 6 & 3+6 & 4+6 & 2+(2 \cdot 6) & 5+(2 \cdot 6) \\
2 & 5 & 1+6 & 6+6 & 3+(2 \cdot 6) & 4+(2 \cdot 6) \\
3 & 4 & 2+6 & 5+6 & 1+(2 \cdot 6) & 6+(2 \cdot 6)
\end{array}\right)
$$

of

$$
\left(\begin{array}{ll}
1 & 6 \\
2 & 5 \\
3 & 4
\end{array}\right)
$$

This reduces to

$$
\left(\begin{array}{llllll}
1 & 6 & 9 & 10 & 14 & 17 \\
2 & 5 & 7 & 12 & 15 & 16 \\
3 & 4 & 8 & 11 & 13 & 18
\end{array}\right)
$$

Observe

$$
\begin{array}{ll}
1+6+9+10+14+17=57 ; & 1^{2}+6^{2}+9^{2}+10^{2}+14^{2}+17^{2}=703 \\
2+5+7+12+15+16=57 ; & 2^{2}+5^{2}+7^{2}+12^{2}+15^{2}+16^{2}=703 \\
3+4+8+11+13+18=57 ; & 3^{2}+4^{2}+8^{2}+11^{2}+13^{2}+18^{2}=703
\end{array}
$$

Thus, in the new array, all row sums are equal and the sum of the squares of entries in each row are the same. Here $k=6$ and we have added numbers from 7 to 18 .
