# Fill each box of the magic square, with a number such that the number adds up to 18 horizontally, vertically and diagonally? [(, ,5), (,8 ,), (, ,9)]

Apr 18, 2016

The constraints are incompatible, resulting in no solution.

#### Explanation:

Attempting to find a solution, fill in one digit at a time to satisfy the constraints:

[[?, ?, 5], [?, 8, ?], [?, ?, 9]] ->[[1, ?, 5], [?, 8, ?], [?, ?, 9]] ->[[1, 12, 5], [?, 8, ?], [?, ?, 9]]
->[[1, 12, 5], [?, 8, 4], [?, ?, 9]]->[[1, 12, 5], [6, 8, 4], [?, ?, 9]] ->[[1, 12, 5], [6, 8, 4], [11, ?, 9]]
$\to \left[\begin{matrix}1 & 12 & \textcolor{red}{5} \\ 6 & \textcolor{red}{8} & 4 \\ \textcolor{red}{11} & - 2 & 9\end{matrix}\right]$

The rows, columns and one of the diagonals sum correctly, but the back diagonal does not.