Magic Square Solver and Generator

How to solve a magic square with missing numbers

Start with the magic sum, the total every line has to reach. With the numbers 1 to 9 in a 3 by 3 it is 15, and in general it is the total of all the numbers divided by the number of rows, so 1 to 16 in a 4 by 4 gives 34. In a 3 by 3 using 1 to 9 the centre is always 5.

Then look for any row, column or diagonal with only one gap. That gap is not a guess: it is the magic sum minus the numbers already there. Fill it, and new lines with one gap appear. When none is left, try a number in the most constrained gap and see whether everything still adds up.

That is exactly how the solver works, which is why it finishes a worksheet instantly. Type in the numbers you have, leave the rest empty, and press Fill in the rest. It shows the answer and how many answers there are, so you can tell whether a puzzle has one solution or several.

Make a magic square of any size

The Make one tab builds a square of any size from 3 to 20 with the numbers 1 to n squared. Odd sizes use the Siamese method: start in the top middle, move up and right, and drop down a row when the way is blocked. Sizes divisible by four count straight through and swap each number on the diagonals of every 4 by 4 block for its partner. The awkward sizes, 6, 10, 14 and so on, are four smaller odd squares put together with a few columns exchanged, a method due to Ralph Strachey. Every square the page hands out is checked by actually adding up every line.

For a particular magic sum, enter it. Adding the same number to every cell keeps every line equal and the numbers all different, so a 3 by 3 can be made to add up to any multiple of 3, 12 included.

Ramanujan's birthday magic square

Srinivasa Ramanujan built a 4 by 4 square whose top row is his date of birth: 22, 12, 18, 87. Every row, column and diagonal adds up to 139, and so do the four corners, the four in the middle and each of the four 2 by 2 corners of the square. The same pattern works for any date, which is what the Birthday square tab does with yours.

How many magic squares are there?

With the numbers 1 to 9 there is exactly one 3 by 3 magic square, the Lo Shu, or eight if you count it turned and flipped. With 1 to 16 there are 880 of order four, or 7,040 counted that way. Both numbers come out of this page's own search, not out of a book. Order five has 275,305,224, counted by Richard Schroeppel in 1973, far too many to list here, which is why the solver stops counting at a point and says "at least".

More on magic squares

The Lo Shu, Durer's 1514 square and why there is only one 3 by 3 are in magic squares explained. If what you like is numbers that have to fit together exactly, Cube Alchemy is the same itch in solid form, and the rest of the range is in mechanical puzzles. There are more solvers on the puzzle tools page.