MathDB

Consider the following

3\times 2

array formed by using the numbers

1,2,3,4,5,6

\begin{pmatrix} a_{11}& a_{12}\\a_{21}& a_{22}\\ a_{31}& a_{32}\end{pmatrix}=\begin{pmatrix}1& 6\\2& 5\\ 3& 4\end{pmatrix}.

Observe that all row sums are equal, but the sum of the square of the squares is not the same for each row. Extend the above array to a

3\times k

array

(a_{ij})_{3\times k}

for a suitable

k

, adding more columns, using the numbers

7,8,9,\dots ,3k

such that

\sum_{j=1}^k a_{1j}=\sum_{j=1}^k a_{2j}=\sum_{j=1}^k a_{3j}~~\text{and}~~\sum_{j=1}^k (a_{1j})^2=\sum_{j=1}^k (a_{2j})^2=\sum_{j=1}^k (a_{3j})^2

Problem Statement