Magic Square Puzzles
Every row, column and diagonal of a magic square sums to the same total. Find that constant once — in a 3×3 it is always three times the centre — and every missing number falls out of one subtraction. Sixty patterns across eight hiding arrangements, each with the arithmetic written out.
Also known as the magic square puzzle and missing number square. The classic 3×3 board here is the Lo Shu square and its shifted, turned and reflected versions.
The Constant Decides Everything
Every row, column and diagonal of a magic square sums to the same total — the magic constant. In a 3×3 square that constant is always three times the centre cell. Find it once and every missing number falls out of one subtraction along a line through the gap.
How To Solve Magic Square Puzzles
- Multiply the centre cell by three: that is the constant.
- Pick any complete line through the missing cell.
- Add the visible cells of that line.
- Subtract from the constant; the difference is the answer.
- Check a second line through the same cell.
The centre sits on four lines — middle row, middle column, both diagonals — which is why the constant is exactly three times the centre. Every board here obeys it.
| Family | What it covers |
|---|---|
| A corner missing | One corner cell hidden — its row and its column both give the answer. |
| An edge missing | The middle of an edge hidden: the row gives the answer and the column checks it. |
| The centre missing | The centre is the magic constant divided by three — the fastest cell of all. |
| Two in one row | Two cells of the same row hidden, so one line is not enough any more. |
| Two in one column | The same idea down a column. |
| Two on a diagonal | A diagonal with two gaps, plus the rows and columns to cross-check. |
| Two anywhere | Two hidden cells in different lines — you choose which line to read first. |
| Constant given | The magic constant is stated up front, which removes one step and adds speed. |
Every Board Is A Lo Shu Square In Disguise
Up to turns and reflections there is exactly one 3×3 magic square on 1 to 9 — the Lo Shu square. Add the same number to every cell and the square stays magic while every entry changes, which is how these boards vary honestly.
Magic Square Puzzles Frequently Asked Questions
What is a magic square?
A square grid of distinct numbers in which every row, every column and both main diagonals sum to the same total, called the magic constant. In a 3×3 magic square the constant is always three times the centre cell.
How do you solve a magic square with missing numbers?
Find the magic constant first — in a 3×3 it is 3 × the centre. Then take any row, column or diagonal through the missing cell whose other cells are visible, add them, and subtract from the constant. The missing number is the difference.
Why is the magic constant three times the centre?
In a 3×3 square the centre cell sits on four lines: the middle row, the middle column and both diagonals. Adding those four lines together counts the centre four times and every other cell once — three times the constant in total — so the centre is the constant divided by three.
What numbers can a 3×3 magic square use?
The classic Lo Shu square uses 1 to 9 exactly once, but any shift of it works too: add the same number to every cell and the square stays magic while every number changes. The boards here include those shifted versions.
How many puzzles are there?
There is no fixed number. Each puzzle is rebuilt from the short seed in the page address, so the supply is effectively unlimited. Copying the puzzle link saves that exact puzzle permanently.
Can I practise one pattern at a time?
Yes. Pick any pattern from the catalogue above and the board switches to that pattern only, with its own address you can bookmark or hand out.
Are magic square puzzles good for exams?
Yes. Missing-number magic squares appear in school olympiads, SSC non-verbal and quantitative papers, and puzzle books generally. The method is always the same: constant first, then one line.
What should I do when two cells are hidden?
Do not start with the hidden pair. Find a line through one of them whose other cells are all visible, place that number, and the second usually falls out of the line the first completed.