Magic squares
A 3 by 3 grid holds the numbers 1 to 9. A diagonal goes from one corner to the opposite corner. Every row, column and diagonal adds to the same sum. What sum, and what goes in the middle?
Loading lab…
Step 1 of 5
- The numbers 1 to 9 add to 45.
- Three equal rows share 45. Each adds to 15.
- Four lines pass through the centre: 4 × 15 = 60.
- They count every number once, and the centre 3 more times.
- So 60 = 45 + 3 × centre. The centre is 5.
The rule
The sum is 15, and the middle number is 5.
The equal sum is called the magic sum.
A magic square has equal sums in every row, column and diagonal. Use nine numbers that follow each other, like 1 to 9. The centre cell holds the middle one. The magic sum is 3 times it.
Call the centre number m. The other cells are m plus or minus a small number.
| Row | Left | Middle | Right |
|---|---|---|---|
| Top | m − 1 | m + 4 | m − 3 |
| Middle | m − 2 | m | m + 2 |
| Bottom | m + 3 | m − 4 | m + 1 |
Every row, column and diagonal adds to 3m. For 1 to 9, m = 5. The magic sum is 3 × 5 = 15. For m = 20, the nine numbers are 16 to 24. The magic sum is 3 × 20 = 60.