Mathematico | Mathematico

The table below is called a magic square because, if you add together the numbers in any column, row or diagonal you get the same number.

\begin{array}{ccc} 2 & 7 & 6 \\ 9 & 5 & 1 \\ 4 & 3 & 8 \end{array}

In this case, the number you get is $15$ , and so we say that the magic number for this magic square is $15$ .

The below table is also a magic square:

\begin{array}{ccc} 3 n - 2 & 2 n + 1 \\ n^{2} - 3 n \\ 2 (2 n - 3) \end{array}

There are two possible values for $n$ , giving two different possible magic squares. Work out the two possible magic squares.

Give the sum of all the numbers in the two possible magic squares.

Hint

The sum of the top row must equal the sum of the middle column.

Solution

The sum of the top row must equal the sum of the middle column, so:

\begin{aligned} (3 n - 2) + (2 n + 1) & = (n^{2} - 3 n) + (4 n - 6) \\ 5 n - 1 & = n^{2} + n - 6 \\ n^{2} - 4 n - 5 & = 0 \\ (n + 1) (n - 5) & = 0 \\ n & \in {- 1, 5} \end{aligned}

When $n = - 1$ we have

\begin{array}{ccc} - 5 & 18 & - 1 \\ 8 & 4 & 0 \\ 9 & - 10 & 13 \end{array}

When $n = 5$ , we have

\begin{array}{ccc} 13 & 6 & 11 \\ 8 & 10 & 12 \\ 9 & 14 & 7 \end{array}