Place the numbers 1 through 9 in a $3\times 3$ grid, using each number only once, so that the sum of each row, each column, and each diagonal is 15.
Give a valid arrangement of the magic square.
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The Lo Shu magic square (China)
The It Shu is one of the oldest and most elegant magic squares. With only the numbers 1 to 9, force rows, columns and diagonals to obey the same sum: 15.
Hints
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- The total sum of the numbers 1 through 9 is 45, so each of the three rows must add up to 15.
- The central number participates in four lines; the corners and sides participate less.
- In It Shu, 5 occupies the center and the opposite numbers around it add up to 10.
Solution
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Answer: a valid provision is $$
\begin{matrix}
4 & 9 & 2\\
3 & 5 & 7\\
8 & 1 & 6
\end{matrix}
$$
Let's check the sums. Rows: $$4+9+2=15,\qquad 3+5+7=15,\qquad 8+1+6=15.$$ Columns: $$4+3+8=15,\qquad 9+5+1=15,\qquad 2+7+6=15.$$ Diagonals: $$4+5+6=15,\qquad 2+5+8=15.$$ Therefore, the square meets the condition. Additionally, in any magic square with the numbers 1 through 9, the center must be 5. The numbers opposite the center must add up to 10: $1+9$, $2+8$, $3+7$ and $4+6$. By placing those pairs around the 5 in a way compatible with rows, columns and diagonals, all the solutions obtained are the same structure seen through rotations or reflections. So this solution is unique except for symmetries.
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