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A 21x21 Magic Square
Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (4641)
A Magic Square is a square of numbers in which every row, every column, and both diagonals add up to the same number. This number is often called the magic sum. The 21 by 21 magic square shown below has a magic sum equal to 4641.Though magic squares can be made with non-consecutive and non-regular sequences, they are usually seen made up of consecutive numbers. To the best of my knowledge, the middle number in the sequence must be in the center square, and the magic sum will be equal to this middle number times the number of rows in the square. (In the case of 3x3 squares, I have proved that the middle number MUST be equal to one-third of the sum. See proof here.)
All odd-size Magic Squares can be made with the method shown below, which can be summarized as "start in the middle of the top and keep moving up and to the right except when you get blocked, in which case drop down and continue." The more detailed directions about creating these squares can be found further below. There are other methods for creating magic squares, but this is the easiest to learn. (There are basically only two methods of creating a 3x3 square but the larger squares have a large number of irregular variants, and probably other regular methods.)
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A 21-by-21 Magic Square
(Magic Sum = 4641)
Books about
Magic Squares
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