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Magic Squares Theory

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Magic square is known for mathematical recreation giving entertainment and an interesting outlet for creating mathematical knowledge.

An nth-order magic square is a square array of n2 distinct integers in which the sum of the n numbers in each row, column, and diagonal is the same.Magic squares history started around 2200 B.C. from China to India, then to the Arab countries.

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The first known mathematical use of magic squares in India was by Thakkura Pheru in his work Ganitasara (ca.

1315 A.D.).

Pheru gave a method for constructing odd magic squares, that is to say, squares, where, n is an odd integer. We begin by putting the number 1 in the bottom cell of the central column (as illustrated below). Where by to arrive at the next cell above it, add n + 1, getting n + 2. And the next cell up n + 2, add n + 1 again, getting 2n + 3.

Continue to add in this way to arrive at the cell values in the central column results in an arithmetic progression with a common difference of n + 1.

Continue adding n + 1 until arriving at the central column’s top cell, of the value n2.ÂThe first steps in Pheru’s method for constructing odd-order magic squaresOther cells in the square are derived by starting from the numbers in the central column. The diagram above illustrates Pheru’s method.

When making a 9-by-9 magic square, hence n = 9. Take any number in the central column, say, 1. Add n to 1, obtaining9 + 1 = 10. Then move as a knight in chess would, starting at 1 and moving one cell to the left, then two cells up.

In this cell, place the 10. From this cell, repeat the same process. Add 10 + 9 to get 19, complete the knight move, and put 19 in the resulting cell. Further this process by arriving at the cell with a number of 37.

Add 9 and completing the next process puts 46 outside of the original 9-by-9 square. To solve this situation, assume you have 9-by-9 squares on each side and corner of the original 9-by-9 square. You will observe that the cell where 46 is present is in the outside square on top the original square and off to the left-hand corner. Simplifying futher move 46 to the corresponding cell in the original 9-by-9 square.


  1. http://illuminations.nctm.org/Lessons.aspx ( Visited 24 Novemeber, 2007)

Cite this Magic Squares Theory

Magic Squares Theory. (2017, Mar 20). Retrieved from https://graduateway.com/magic-squares/

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