Magic Square Generator
Build a magic square of any size from 3 to 12, or a fill-in-the-blanks puzzle with exactly one answer. Every square is verified line by line.
Magic square
How these are built
- Every row, every column and both long diagonals add up to the same number, using each whole number from 1 to n² exactly once.
- That total is not a choice — it is forced. The numbers 1 to n² add up to n²(n²+1)/2, and that has to be shared equally between n rows.
- Which construction works depends only on n mod 4. Odd sizes, sizes divisible by four, and the awkward middle cases like 6 and 10 each need a different method — which is why many generators quietly stop at 5.
- There is no 2 × 2 magic square, and not because nobody has found one. Four cells with every pair summing alike forces all four numbers to be equal.
- A magic square stays magic when rotated or mirrored, so each size has eight versions of the same square. For 3 × 3 those eight are all the magic squares that exist.
- Puzzles only ever have one right answer — every arrangement of the missing numbers is checked before the puzzle is offered.
How to use
- Pick a size and press Make one.
- Switch to puzzle mode for a grid with blanks to fill in.
- Press Print for a clean copy.
Frequently asked questions
Why is there no 2 × 2 magic square?
Not because nobody has found one — because it cannot exist. Four cells where every row, column and diagonal must share a total forces all four numbers to be equal, and a magic square has to use four different numbers.
Where does the magic constant come from?
It is forced, not chosen. The numbers 1 to n² add up to n²(n²+1)/2, and that total has to be shared equally between n rows — so the constant is n(n²+1)/2. For a 3 × 3 that is 15, and for a 4 × 4 it is 34.
How is a magic square actually built?
It depends only on the size modulo four. Odd sizes use the Siamese method — step up and to the right, drop down when blocked. Sizes divisible by four use a complement pattern. Sizes like 6 and 10 need the LUX method, which is fiddlier and is why many generators quietly stop at 5.
How many 3 × 3 magic squares are there?
Exactly eight, and they are all rotations and mirrors of the same one. This is not folklore — checking all 362,880 arrangements of 1 to 9 finds precisely those eight.
Do the puzzles always have one answer?
Yes. Every arrangement of the missing numbers is checked before a puzzle is offered, and a grid with more than one valid completion is thrown away and redrawn. If it cannot find a unique one at the number of blanks you asked for, it tells you rather than handing over an ambiguous grid.
Why does pressing Make one give a different-looking square?
Rotating or mirroring a magic square leaves it magic, so each size has eight versions of the same underlying square. It picks one at random so the output does not look identical every time.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.