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Other Magic Square Sizes: 3 * 5 * 7 * 9 * 11 * 13 * 15 * 17 * 19 *
21 * 23 * 25 * 27 * 29 * 31 * 33 * 35 * 37 * 39 * 41 * 43 * 45 * 47 * 49 * 51 * 53 * 55 * 57 * 59 *
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A 45x45 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (45585)

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 45 by 45 magic square shown below has a magic sum equal to 45585.

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.)

Read more about Magic Squares

If you'd like to read more about magic squares, click one of these links to look for books about Magic Squares at Amazon.com, at Amazon.canada, and at Amazon.co.UK.

A 45-by-45 Magic Square

(Magic Sum = 45585)

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Magic Squares
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How to Make Your Own Odd-Size Magic Square

  1. Draw a square with any odd number of rows and columns.
  2. Put your first number in the MIDDLE square of the TOP row.
  3. Move UP and to the RIGHT to find the position for your next number, but ...
    • if the next "position" is ABOVE a column, go to BOTTOM of that column (you can see a sample of this movement by looking at how we moved from 1 to 2 in the square above).
    • if the next "position" is to the RIGHT of a row, go to LEFT of that row (you can see this movement between (23) and (24) above).
    • if the next cell is already filled in, drop down to the cell below the number you just filled in for the next number (you can see this when you look at how we moved from 45 to the next number above).
    • if the next cell is both ABOVE and to the RIGHT of the entire square, drop down to the cell below the number you just filled in for your next number (you can see this if you look at (1035) and the next number).
  4. Print your NEXT number into the new square you just moved to.
  5. Repeat steps 3-4 until the entire square is full.
  6. You can see how it works by printing this page and drawing arrows to see how it worked when I created this magic square
This Magic Square was created by John Knoderer, a webmaster and computer programmer who is available to telecommute to your location.

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