Magic square

A square $ n \times n $ array $ \| a _ {ij} \| $ composed of the integers from 1 up to $ n ^ {2} $ and satisfying the following conditions:

$$ \tag{* } \sum_{i=1} ^ { n } a _ {ij} = \ \sum_{j=1} ^ { n } a _ {ij} = \ \sum_{i=1} ^ { n } a _ {ii} = \ \sum_{i=1} ^ { n } a _ {i , n + 1 - i } = s , $$

where $ s = n ( n ^ {2} + 1 ) / 2 $. There are also more general magic squares, in which $ 1 \leq a _ {ij} \leq n ^ {2} $ is not required.

Any number $ a $, $ 1 \leq a \leq n ^ {2} $, is uniquely characterized by a pair of residues $ ( \alpha , \beta ) $ $ \mathop{\rm mod} n $( the digits to base $ n $ of $ a - 1 $), that is, by the points of the two-dimensional space $ ( \mathbf Z / n ) ^ {2} $ over the ring $ \mathbf Z / n $ of residues modulo $ n $. Since the coordinates $ ( i , j ) $ of the cells of the square may also be regarded as the elements of $ ( \mathbf Z / n ) ^ {2} $, it follows that any distribution of the numbers from 1 up to $ n ^ {2} $ in an array $ \| a _ {ij} \| $ is given by a mapping

$$ ( \mathbf Z / n ) ^ {2} \rightarrow \ ( \mathbf Z / n ) ^ {2} , $$

that is, by a pair of functions $ \alpha = \alpha ( i , j ) \in \mathbf Z / n $, $ \beta = \beta ( i , j ) \in \mathbf Z / n $, where $ i , j \in \mathbf Z / n $. The problem is to investigate those pairs that give magic squares. In general this has been done (see [1]) only under the additional assumption of linearity of $ \alpha $ and $ \beta $. It turns out, in particular, that magic squares with linear $ \alpha $ and $ \beta $ exist for odd $ n $ only.

Already in the Middle Ages a number of algorithms for constructing magic squares of odd order $ n $ had been found. Each such algorithm is characterized by six residues $ i _ {0} $, $ j _ {0} $, $ p $, $ q $, $ \overline{p}\; $, $ \overline{q}\; $, and is described by the following rules: 1) the number 1 is put into the cell $ ( i _ {0} , j _ {0} ) $; and 2) if $ a $ was put into $ ( i , j ) $, then $ a + 1 $ is put into $ ( i + p , j + q ) $ if that cell is still empty or into $ ( i + \overline{p}\; , j + \overline{q}\; ) $ if $ ( i + p , j + q ) $ is occupied.

The residues $ i _ {0} , j _ {0} , p , q , \overline{p}\; , \overline{q}\; $ are not arbitrary but must satisfy certain conditions to ensure not only that (*) holds, but also that the algorithm is feasible, that is, that $ ( i + \overline{p}\; , j + \overline{q}\; ) $ is empty when $ ( i + p , j + q ) $ is occupied. These conditions are easily found (see [1]). Moreover, it turns out that a magic square can be constructed by an algorithm of this type if and only if the functions $ \alpha $ and $ \beta $ describing the square are linear.

Many algorithms for constructing magic squares are known (resulting in squares with non-linear $ \alpha $ and $ \beta $), but there is no general theory for them (1989). Even the number of magic squares of order $ n $ is unknown (for $ n \geq 5 $; for $ n = 3 $ there is, up to obvious symmetries, only one magic square, whereas for $ n = 4 $ there are 880 magic squares).

Magic squares having additional symmetry have also been investigated, again only in very special circumstances (for example, for $ n \leq 5 $; see [2]).

References

[1]	M.M. Postnikov, "Magic squares" , Moscow (1964) (In Russian)
[2]	E.Ya. Gurevich, "The secret of the Ancient Talisman" , Moscow (1969) (In Russian)

Comments

Magic squares have been considered since ancient times. For instance, the magic square of order 3 was known in China around 2000 B.C.. Dürer's famous "Melancholy" shows a magic square of order 4.

There is a close connection between (pairs of orthogonal) Latin squares (cf. Latin square; Orthogonal Latin squares) and magic squares, which has been studied since L. Euler (see [a1] and [a2]). See also [a3] and the references given there.

References

[a1]	L. Euler, "De quadratis magicis" G. Kowalewski (ed.) , Opera Omnia Ser. 1; opera mat. , 7 , Teubner (1923) pp. 441–457
[a2]	L. Euler, "Recherches sur une nouvelle espèce de quarrés magiques" G. Kowalewski (ed.) , Opera Omnia Ser. 1; opera mat. , 7 , Teubner (1923) pp. 291–392
[a3]	J. Dénes, A.D. Keedwell, "Latin squares and their applications" , English Univ. Press (1974) Zbl 0283.05014