Magic Squares: 3×3

Magic Squares: 3×3

Magic Squares: Odd

The magic square, according to Chinese legend, was first constructed on the back of a turtle and is called the Lo Shu magic square, the only order of numbers in a 3×3 grid in which they all add up to 15.

English: The unique normal magic square of ord...
The unique normal magic square of order 3, also known as the Lo Shu Square. (Photo credit: Wikipedia)

By contrast, when there are 4 rows and columns, there are 880 different ways to lay out the numbers in a magic square. Constructing a magic square is different if there are an odd number of rows and columns than an even number. Let’s get started making your own!


  • Paper
  • Pen or pencil
  • Ruler or straight-edge


  • Put 1 in the middle of the top row and keep placing numbers diagonally, according to the following rules
  • Put the next number, 2, diagonally, up one square to the right
  • When a number falls outside the grid to the right diagonally above a column, wrap it around and put in the square at the bottom of that column. 2, for example, instead of going in a square above the upper right corner square, will go in the bottom right corner of the 3×3 grid.
  • When a number falls outside the grid to the right of a row, wrap it around and put it in the square on the left end of that row, thus 3, if put diagonally from 2 would be to the right of the middle row, wraps around to be placed in the left most square in the middle row
  • If a cell already has a number in it, put the number below the last number place. Thus 4, which can’t be place diagonally from 3 because there is already a 1 there, is placed below the 3, in the bottom left corner.
  • Continue through number 9.

What Should Happen?

  • When there is a number in every box, they should be laid out: first row – 8,1,6
  • Second row – 3, 5, 7
  • Third row – 4, 9, 2
  • Each row, column and the three-number diagonals should add up to 15.
  • These rules should work for all odd-numbered magic squares.
  • Click here to see an animation of how a 3×3 magic square is constructed and how the rules apply.
  • This is called the Siamese method of constructing a magic square, after a French ambassador to Siam (now Thailand), De La Loubere, who brought it back to France from Siam.

Why Is This Useful?

Most Sudoku puzzles are 9×9 grids, made up of 3×3 grids. Invented by Howard Garns and first published in May 1979, Sudoku puzzles are a variation of Latin squares, in which no number or symbol in a grid repeats. Players are challenged to put a number from 1 to 9 in each column such that it does not repeat within any 3×3 box or in any row or column. Two mathematicians, Bertram Felgenhauer and Frazer Jarvis estimated that there are 6 sextillion combinations of Sudoku puzzles.

Carol Covin, Granny-Guru

Author, “Who Gets to Name Grandma? The Wisdom of Mothers and Grandmothers”

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Comments (5 )

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  1. What a great tool. Love it. 🙂

    • carolcovin says:

      Thanks, Susan! I thought Sudoku puzzles were inscrutable, computer-generated mysteries. I love digging in to find the underlying logic.

  2. My wife likes puzzles like this. Not my thing though.

    • carolcovin says:

      My husband does Sudoku in the daily paper. I don't, but I liked learning about the patterns doing the research for this post and that you can make up your own. Thanks, Jon!