Magic Square 3 X 3

magic square 3 x3

Magic squares have intrigued people from ancient times. A magic square is a square with equal number of rows and columns filled with distinct positive numbers 1, 2, …, arranged in such a way that the sum of the numbers in any horizontal, vertical, or diagonal line is always the same. A 3 X 3 magic square would have 9 cells, a 4 X 4 square would have 16 cells and so on.


We can easily construct the Odd magic squares using the extended Pyramid method or diagonals. But magic squares for even order (like 6 X 6, 8 X 8 ) are harder to construct and need extra algorithms.

Let’ try to find a 3 X 3 magic square with numbers ranging from 1,2,3……9 as there are 9 cells. We can also fill the square with any arithmetic progression series, meaning, starting with any number and maintaining the difference in the series ( eg series with a constant difference of 3) like , 2,5,8,11 and so on. We would be using the math concept of transpose and apply. In mathematics, transpose means to transfer a number to the other side of an equation. Meaning, a+ b = c can be transposed to a = c – b.


Create a line across one of the diagonal of the square.

Increase the center row and column with one more cell on each four side. ( as shown in the picture below ). Draw two lines parallel to the diagonal line cutting through the outermost cells on both sides. ( as in the picture).

Maintaining a particular direction, let’s say North-east, start filling out the cells sequentially, through which the diagonal lines are passing.

From the above picture, we notice that four cells inside the square are empty and outside four cells are filled with numbers 1,3,7 and 9. Applying the concept of transpose, we would move the number on the left side to the right, right side to left, numbers in the up cells to the down and down to up. So, we get

This is a magic square, where the sum of all the numbers in a row, column and diagonal is same, called the magic constant. So, for a 3 X 3 magic square, with numbers from 1-9, the magic constant is 15.

Try creating magic squares with different arithmetic progression series. Notice, on how easily it can be done instead of using trial and correction method.

Stay tuned to learn an easy method to solve a 4 X 4 and higher order magic squares.

Book Recommendations

There are variation of Magic square problems which are asked in competitive exams like Math Olympiad and Math Kangaroo. If your kids are preparing for any of these exams then here is GREAT books with practice problems. There are brief solutions at the end of the each book.

Here are some math puzzles and my latest posts to keep you entertained

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