How to Solve a Magic Square
Find the magic sum and the center first. The rest fills in by subtraction.
Play Magic Square (free)Solve it in 3 steps
- Magic sum: add all the numbers and divide by the number of rows (15 for 1 to 9)
- Center of a 3×3: the magic sum ÷ 3 (always 5 for 1 to 9)
- Lines with one empty square (rows, columns, diagonals): fill in by subtraction
A 3×3 with some numbers: fill lines with one gap first
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Puzzle
- Use each of 1 to 9 once so that every row, column and diagonal has the same total.
- Magic sum: 1 to 9 add up to 45. Split over 3 rows: 45 ÷ 3 = 15
- Center: the magic sum ÷ 3, so 15 ÷ 3 = 5
- Top row: 2 + 7 + □ = 15, so □ = 6
- Middle row: □ + 5 + 1 = 15, so □ = 9
- Diagonal: 2 + 5 + □ = 15, so □ = 8
- Diagonal: 6 + 5 + □ = 15, so □ = 4
- Bottom row: 4 + □ + 8 = 15, so □ = 3. Solved!
An empty 3×3 with 1 to 9: the center is always 5
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Puzzle
- Place each of 1 to 9 once in an empty 3×3. The magic sum is 15.
- Center: the 4 lines through it add up to 60 = 45 + 3 × center, so it is 5
- 9 goes on an edge: a corner 9 uses 1 and 5 on its diagonal, so only 2 + 4 is left to make 6
- Opposite the 9: 9 + 5 + □ = 15, so □ = 1
- Corners next to the 9: they make 6, so 4 and 2 (either way round)
- Diagonals: 15 − 4 − 5 = 6 and 15 − 2 − 5 = 8
- Last 2 squares: 3 on the left, 7 on the right. Solved!
Even numbers always go in the corners, odd numbers in the middle of the edges.
A 4×4: the magic sum is 34, then fill lines with one gap
16321351011896712415141
Puzzle
- The square in Albrecht Dürer’s 1514 engraving Melencolia I. Use 1 to 16 once each.
- Magic sum: 1 to 16 add up to 136. Split over 4 rows: 136 ÷ 4 = 34
- Top row: 16 + 3 + 2 + □ = 34, so □ = 13
- Left column: 16 + 5 + □ + 4 = 34, so □ = 9
- Diagonal: 13 + □ + 6 + 4 = 34, so □ = 11
- Second row: 5 + □ + 11 + 8 = 34, so □ = 10
- Diagonal: 16 + 10 + 7 + □ = 34, so □ = 1
- Bottom row: 4 + □ + 14 + 1 = 34, so □ = 15. Solved!
Negative numbers: the center is the average of opposite squares
−32140−4−1−23
Puzzle
- Only −3, 1 and 3 are given. Make every row, column and diagonal add up the same.
- Center: the average of the opposite −3 and 3 is 0, so the magic sum is 0 × 3 = 0
- Top row: −3 + □ + 1 = 0, so □ = 2
- Diagonal: 1 + 0 + □ = 0, so □ = −1
- Left column: −3 + □ + (−1) = 0, so □ = 4
- Right column: 1 + □ + 3 = 0, so □ = −4
- Bottom row: −1 + □ + 3 = 0, so □ = −2. Solved!
Tips and tricks for solving magic squares
- Opposite squares add up to twice the center (10 for 1 to 9: 1 and 9, 2 and 8…)
- In a 4×4, the corners and center four also add up to 34 (all 880 with 1 to 16)
- For a line with two gaps, find two unused numbers that make up the rest
- Check every line at the end: 8 lines in a 3×3, 10 in a 4×4
Magic square practice puzzles (with answers)
Puzzles 1 and 2 use whole numbers, and puzzle 3 uses negative numbers.
| 8 | ||
| 5 | ||
| 9 |
| 3 | 8 | |
| 2 | ||
| −1 | ||
| 2 | ||
| 3 |
Show the answers
The colored numbers are the ones given in the puzzle.
| 8 | 1 | 6 |
| 3 | 5 | 7 |
| 4 | 9 | 2 |
| 3 | 8 | 7 |
| 10 | 6 | 2 |
| 5 | 4 | 9 |
| −1 | 4 | −3 |
| −2 | 0 | 2 |
| 3 | −4 | 1 |
FAQ
- What is the magic sum (magic constant)?
- The total of each line: n(n² + 1) ÷ 2 for 1 to n². That is 15 for 3×3, 34 for 4×4 and 65 for 5×5
- What number goes in the middle of a 3×3 magic square?
- 5 with 1 to 9. With other numbers, the center is always the magic sum ÷ 3
- How many 3×3 magic squares are there?
- With 1 to 9, just 1 if rotations and reflections count as the same. Otherwise 8
- How do I find the magic sum without knowing all the numbers?
- Add up a full line. In a 3×3 you can also multiply the center by 3
- What is the rule for a 4×4 magic square?
- With 1 to 16, every line, the four corners and the four center squares add up to 34
- How do I make my own magic square?
- 4×4 and odd sizes like 5×5 have simple methods: see how to make a magic square
- Where can I practice?
- Here, for free: 3×3 to 5×5 with line totals shown and a Hint button
