## Magic Square

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## Magic Square

# Magic Square

A magic square is a square table of numbers containing consecutive integers in arrangements so that the sum of numbers in any row, column, or diagonal are identical. Some magic square may have additional properties. Such squares were known approximately 4,000 years ago in China. They have no application in science and technology, but are purely recreational.

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

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:

X = 1\2n(n^{2}+1)

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

A close relative of the magic square is the sudoku puzzle, invented by American Howard Garns in 1979 but re-imported to the U.S. years later, achieving fad status in 2005. In a sudoku puzzle, all the numerals from 1 to 9 are present in each column, each row, and each 3 × 3 subsquare. The numbers do not generally sum to any single “magic” number, as in a magic square.

Jeanette Vass

## Magic Square

# 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.

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

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