Magic Square Calculator (2024)

Sprinkle some magic in your math with our magic square calculator! In this calculator, we will lead you in an exploration of this interesting mathematical puzzle.

Here you will learn:

• What is a magic square?
• The history of magic squares;
• The properties of magic squares;
• How to calculate a magic square (with a step by step guide on the calculation of some of them);
• Where is the 2 by 2 magic square; and
• How to shuffle a magic square.

And much more, from the weird types of magic squares to how many 3x3 magic squares are there. Abracadabra!

A magic square is a square grid of integer numbers arranged in such a way that their positions respect three rules:

• No numbers repeat;
• The summation the numbers in each row and column returns the same value; and
• The sum of the values in both diagonals is equal to the values of summed rows and columns.

The magic square is not you everyday square; for that one, visit the square calculator! And if you are looking for perfect but not magic squares, we've got you: try the perfect square calculator😉

However, the magic doesn't end here. It is possible to arrange numbers in such a way to satisfy other rules - this generates different types of magic squares. We then have:

• Pandiagonal magic squares - Where the sum of the numbers on a broken diagonal is the same as the sums of the rows and columns;
• Associative magic squares - Where cells symmetrical with respect to the center sum to the value of $n^2-1$n2−1, where $n$n is the order of the square; and
• Most-perfect magic squares - This square is both pandiagonal and:
  • Each 2 by 2 subsquare has a total sum equal to a quarter of the value of its rows divided by $n$n; and
  • Cells separated by $\tfrac{n}{2}$2n​ on a diagonal sum to $n^2+1$n2+1.

Why stop at squares, though? Mathematicians have sprinkled magic over circles, triangles, hexagons, and many other shapes. Triangular numbers are not magic, but still interesting: discover them at our triangular numbers calculator.

Generalizing a magic square in higher dimensions gives us the magic hypercubes, where the sums are generalized over the various faces of the solid and the spatial diagonals. But this is definitely too complicated. Back to squares one! 😉

If something goes wrong in physics or math, it's likely that a factor two went missing somewhere. That hideous number decided to ruin magic squares for everyone, too.

The value $n=2$n=2 is, in fact, the only one for which we can't build a magic square. You can try to understand why by yourself (it's pretty easy), or you can read below!

Building a magic square is not difficult! There are different algorithms that you can use: here we will explain how to build every possible magic square!

Odd magic squares

This is probably the easiest magic square to build on a piece of paper. Take a pen and follow the steps! We will start calculating a 3 by 3 magic square:

• Chose the cell in the middle of a side (we chose the top one), and place the number $1$1 there.
• Move one square up, and one square left, and place the next number, $2$2.
  • We are out of the square: imagine to "tile" around it with its copies, and place the number $2$2 in the corresponding place of the original square.
• Proceed in this fashion until the movement brings you to an occupied square. In that case, move down one square.

After nine iterations, you should have filled the square!

For magic squares with even order the process is slightly more complex. We need to identify two possible behaviors:

• Magic squares with singly even order, which means $n\ \text{mod}\ 4=2$nmod4=2; and
• Magic squares with doubly even order, that is, multiples of $4$4, or $n\ \text{mod}\ 4=0$nmod4=0.

We will begin with doubly even magic squares.

Doubly even magic squares

To fill a doubly even magic square, you must follow the diagonals!

1. Start by placing the number $1$1 in the upper left corner.
2. Moving as if you were reading, identify the next cell lying on one of the diagonals.
3. Fill the cell with the value $(i-1) \cdot n + j$(i−1)⋅n+j where $n$n is the order of the square, $i$i the index of the row, and $j$j the index of the column of the cell (in this case, $(1-1)\cdot(4)+4=4$(1−1)⋅(4)+4=4.
4. Proceed as if you were "reading" the square, filling the diagonals. The last number will be $16$16 in the lower right corner.
5. Return to the topmost row and fill the empty squares with the remaining numbers in descending order (in the case of $n=4$n=4, $15$15, $14$14, $12$12,...).

Singly even magic squares

The last algorithm we use in our magic square calculator allows you to fill singly even magic squares.

We resort again to a subdivision in smaller magic squares, but this time of odd order. Follow our steps:

1. Divide the magic square with order $N$N into four odd order magic squares with order $\tfrac{N}{2}$2N​, name them (in reading order) $S_1$S1​, $S_3$S3​, $S_4$S4​, and $S_2$S2​.
2. Use the algorithm for odd order magic squares to fill them, then sum the number in each square to $(i-1)\cdot n^2$(i−1)⋅n2, where $i$i is the index of the square. For a square with $N=6$N=6:
   • $S_1$S1​ remains unchanged;
   • $S_2=S_1 + 9$S2​=S1​+9;
   • $S_3=S_1 + 18$S3​=S1​+18; and
   • $S_4=S_1 + 27$S4​=S1​+27.

Let's take a look at the square now!

It looks neat, doesn't it. Well, let's check the sum of the columns:

• $8+3+4+35+30+31=111$8+3+4+35+30+31=111;
• $1+5+9+28+32+36=111$1+5+9+28+32+36=111;

And so on.

Now check the rows:

• $8+1+6+26+19+24=84$8+1+6+26+19+24=84;

What?

• $35+28+33+17+10+15=138$35+28+33+17+10+15=138.

Oh no! We need to shuffle the numbers, but just a little.

To correctly fill singly even magic square follow these three final rules.

1. Swap the elements of the upper and lower squares in the first and last $k-1$k−1 columns of the square.
2. Swap all but the central element of the column $k$k of the squares $S_1$S1​ and $S_4$S4​.
3. Swap only the central element of the $(k+1)^{\text{th}}$(k+1)th column between the upper and lower square.

Take a look at the rearranged square:

Let's check the sums of the rows that were wrong before.

• $8+28+33+17+10+15=111$8+28+33+17+10+15=111;
• $35+1+6+26+19+24=111$35+1+6+26+19+24=111.

You can check the others, but trust us: the magic is back!

Depending on the order of the square, there can be many possible combinations that fulfill the magic requirements. Some of them are "trivial", others a bit more complex.

Let's take a closer look at some of them.

There is a single way to calculate a 3 by 3 magic square with the algorithm we've already seen. We can rotate and reflect it, but the relative positions of the number wouldn't change. We say that the 3x3 magic square is unique.

The story is different for higher orders. For $n>3$n>3, we can transform a magic square into another by:

• Rotation: four possible magic squares.
• Reflection: by mirroring, you can create three magic squares.
• Combination of rotations and reflections (you may find duplicates)!
• Exchange of columns and rows. We can identify:
  • The exchange of a pair of rows and columns intersecting on a diagonal. We swap the columns $x$x and $n-x+1$n−x+1 followed by the swap of the same index rows. In an even magic square there are $\tfrac{n}{2}$2n​ such columns, in an odd one $\tfrac{n-1}{2}$2n−1​: the resulting combinations are respectively $2^{\tfrac{n}{2}}$22n​ and $2^{\tfrac{n-1}{2}}$22n−1​.
  • The exchange of a pair of rows and columns on the same side of the center with the corresponding pairs on the other side. The squares we can obtain are $2^{\tfrac{n\cdot(n-2)}{8}}$28n⋅(n−2)​ for even orders and $2^{\tfrac{(n-1)\cdot(n-3)}{8}}$28(n−1)⋅(n−3)​ for odd orders.
  • A generalized exchange, where we swap any pair of non-central rows (columns) $x$x and $y$y, alongside a swap of the rows (columns) $n-x+1$n−x+1 and $n-y+1$n−y+1. If $x$x and $y$y are on the same side (and $x \neq y$x=y). This is the second case we've seen. If $x=n-y+1$x=n−y+1 we have a "diagonal" exchange.
• Dividing the square into four sub-squares and swapping them with their opposite across the center.

Our magic square calculator allows you to create a magic square of any (possible) size. You only have to choose the size of the square, say the magic word ("math!"). We will generate the square, and if the size is too big, we will ask you if you really want to visualize it.

In any case, we give you the possibility of downloading the magic square as a text file - if you know how to code and you want to play a little with magic, we can give you the raw materials!

You can fill a 3×3 magic square using an algorithm or by intuition. The sum of rows, columns, and diagonals is 15.

1. The center is 5 (it appears in four sums).
2. Since the number in the center is odd, and 15 is odd too, each pair of numbers on its side has same evenness.
3. All the pairs of numbers encircling the center sum to 10. There are four combinations: 1 + 9, 2 + 8, 3 + 7, and 4 + 6.
4. Fill the square by starting with a pair and checking the sums!