DATA STRUCT

Magic Square

A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, …,arranged such that the sum of thenumbers in any horizontal, vertical, or main diagonal line is always the same number (Kraitchik 1942, p. 142; Andrews 1960, p. 1; Gardner 1961, p. 130; Madachy 1979, p. 84; Benson and Jacoby 1981, p. 3; Ball and Coxeter 1987, p. 193), known as the magic constant

If every number in a magic square is subtracted from , another magic square is obtained called the complementary magic square. A square consisting of consecutive numbers starting with 1 is sometimes known as a “normal” magic square.

The unique normal square of order three was known to the ancient Chinese, who called it the Lo Shu. A version of the order-4 magic square with the numbers 15 and 14 in adjacent middle columns in the bottom row is called Dürer’s magic square. Magic squares of order 3 through 8 are shown above.

The magic constant for an th order general magic square starting with an integerand with entries in an increasing arithmetic series with differencebetween terms is

(Hunter and Madachy 1975).

It is an unsolved problem to determine the number of magic squares of an arbitrary order, but the number of distinct magic squares (excluding those obtained by rotation and reflection) of order , 2, … are 1, 0, 1, 880, 275305224, … (Sloane’s A006052; Madachy 1979, p. 87). The 880 squares of order four were enumerated by Frénicle de Bessy in 1693, and are illustrated in Berlekamp et al. (1982, pp. 778-783). The number ofmagic squares was computed by R. Schroeppel in 1973. The number ofsquares is not known, but Pinn and Wieczerkowski (1998) estimated it to beusing Monte Carlo simulation and methods from statistical mechanics. Methods for enumerating magic squares are discussed by Berlekamp et al. (1982) and on the MathPages website.

A square that fails to be magic only because one or both of the main diagonal sums do not equal the magic constant is called a semimagic square. If all diagonals (including those obtained by wrapping around) of a magic square sum to the magic constant, the square is said to be a panmagic square (also called a diabolic square or pandiagonal square). If replacing each numberby its squareproduces another magic square, the square is said to be a bimagic square (or doubly magic square). If a square is magic for , , and , it is called a trimagic square (or trebly magic square). If all pairs of numbers symmetrically opposite the center sum to , the square is said to be an associative magic square.

Squares that are magic under multiplication instead of addition can be constructed and are known as multiplication magic squares. In addition, squares that are magic under both addition and multiplication can be constructed and are known as addition-multiplication magic squares (Hunter and Madachy 1975).

Kraitchik (1942) gives general techniques of constructing even and odd squares of order . Forodd, a very straightforward technique known as the Siamese method can be used, as illustrated above (Kraitchik 1942, pp. 148-149). It begins by placing a 1 in any location (in the center square of the top row in the above example), then incrementally placing subsequent numbers in the square one unit above and to the right. The counting is wrapped around, so that falling off the top returns on the bottom and falling off the right returns on the left. When a square is encountered that is already filled, the next number is instead placed below the previous one and the method continues as before. The method, also called de la Loubere’s method, is purported to have been first reported in the West when de la Loubere returned to France after serving as ambassador to Siam.

A generalization of this method uses an “ordinary vector”that gives the offset for each noncolliding move and a “break vector”that gives the offset to introduce upon a collision. The standard Siamese method therefore has ordinary vector (1,and break vector (0, 1). In order for this to produce a magic square, each break move must end up on an unfilled cell. Special classes of magic squares can be constructed by considering the absolute sums , , , and . Call the set of these numbers the sumdiffs (sums and differences). If all sumdiffs are relatively prime toand the square is a magic square, then the square is also a panmagic square. This theory originated with de la Hire. The following table gives the sumdiffs for particular choices of ordinary and break vectors. ordinary vectorbreak vectorsumdiffsmagic squarespanmagic squares

(1, )(0, 1)(1, 3)none

(1, )(0, 2)(0, 2)none

(2, 1)(1, )(1, 2, 3, 4)none

(2, 1)(1, )(0, 1, 2, 3)

(2, 1)(1, 0)(0, 1, 2)none

(2, 1)(1, 2)(0, 1, 2, 3)none

A second method for generating magic squares of odd order has been discussed by J. H. Conway under the name of the “lozenge” method. As illustrated above, in this method, the odd numbers are built up along diagonal lines in the shape of a diamond in the central part of the square. The even numbers that were missed are then added sequentially along the continuation of the diagonal obtained by wrapping around the square until the wrapped diagonal reaches its initial point. In the above square, the first diagonal therefore fills in 1, 3, 5, 2, 4, the second diagonal fills in 7, 9, 6, 8, 10, and so on.

An elegant method for constructing magic squares of doubly even orderis to draw s through eachsubsquare and fill all squares in sequence. Then replace each entryon a crossed-off diagonal byor, equivalently, reverse the order of the crossed-out entries. Thus in the above example for , the crossed-out numbers are originally 1, 4, …, 61, 64, so entry 1 is replaced with 64, 4 with 61, etc.

A very elegant method for constructing magic squares of singly even orderwith(there is no magic square of order 2) is due to J. H. Conway, who calls it the “LUX” method. Create an array consisting ofrows of s, 1 row of Us, androws of s, all of length . Interchange the middle U with the L above it. Now generate the magic square of orderusing the Siamese method centered on the array of letters (starting in the center square of the top row), but fill each set of four squares surrounding a letter sequentially according to the order prescribed by the letter. That order is illustrated on the left side of the above figure, and the completed square is illustrated to the right. The “shapes” of the letters L, U, and X naturally suggest the filling order, hence the name of the algorithm.

Variations on magic squares can also be constructed using letters (either in defining the square or as entries in it), such as the alphamagic square and templar magic square.

Various numerological properties have also been associated with magic squares. Pivari associates the squares illustrated above with Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon, respectively. Attractive patterns are obtained by connecting consecutive numbers in each of the squares (with the exception of the Sun magic square).