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Magic Square

Magic square is an unusual numerical configuration containing consecutive integers in arrangements so that the sum of numbers in any row, column, or diagonal are identical. Such squares were known approximately 4,000 years ago in China.

The basic magic square is a square containing consecutive integers starting with number 1. Three of the basic magic squares are shown in Table 1.

TABLE 1. THREE BASIC MAGIC SQUARES
2 9 4   12 7 9 6   9 2 25 18 11
7 5 3   13 2 16 3   3 21 19 12 10
  8 11 5 10   22 20 13 6 4
6 1 8   1 14 4 15   16 14 7 5 23
    15 8 1 24 17

Other magic squares can be constructed by starting with one of the basic squares shown above and adding the same whole integers to each integer; equals added to equals, the sums are equivalent. Likewise subtracting the same value from each integer can result in other magic squares. In a similar manner, multiplication or division can be used to create other magic squares.

A general equation for constructing basic magic squares is:

where X equals the sum of integers in any row, column, or diagonal, and n equals the number of rows.

Jeanette Vass

Additional topics

Science EncyclopediaScience & Philosophy: Macrofauna to Mathematics