The same total in every direction
A magic square is a square grid whose rows, columns and two main diagonals all add to the same total. In the familiar 3 × 3 version using each number from 1 to 9 once, that total is 15.
Here is one arrangement. Each number appears once, and you can check every line yourself.
| 8 | 1 | 6 |
| 3 | 5 | 7 |
| 4 | 9 | 2 |
Three rows, three columns, two diagonals
The top row gives 8 + 1 + 6 = 15. The first column gives 8 + 3 + 4 = 15. The diagonal from the top left gives 8 + 5 + 2 = 15, and the other diagonal gives 6 + 5 + 4 = 15.
There are eight lines to check in this grid. Matching the three rows alone is not enough. NRICH’s introduction explains why all these directions matter.
Why does this version need 15?
The numbers 1 to 9 add to 45. The three rows divide those numbers into three equal totals, so each row must total 45 ÷ 3 = 15.
This argument gives the target; it does not prove that any arrangement works. You still have to check the columns and diagonals. Other sizes or choices of numbers can have different totals.
Find one missing entry
Hide the 9 in the bottom row. You can recover it from 4 + ? + 2 = 15: the missing number is 15 − 4 − 2 = 9. Then check its column too: 1 + 5 + 9 = 15.
That habit of testing a rule beyond the first example also helps with number patterns. For a different arrangement of dots and numbers, try triangular numbers.