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Antimagic Square


AntimagicSquare

An antimagic square is an n×n array of integers from 1 to n^2 such that each row, column, and main diagonal produces a different sum such that these sums form a sequence of consecutive integers. It is therefore a special case of a heterosquare. It was defined by Lindon (1962) and appeared in Madachy's collection of puzzles (Madachy 1979, p. 103), originally published in 1966. Antimagic squares of orders 4-9 are illustrated above (Madachy 1979). For the 4×4 square, the sums are 30, 31, 32, ..., 39; for the 5×5 square they are 59, 60, 61, ..., 70; and so on.

Let an antimagic square of order n have entries 0, 1, ..., n^2-2, n^2-1, and let

 M(n)=1/2n(n^2+1)

be the magic constant. Then if an antimagic square of order n exists, it is either positive with sums [M(n)-n,M(n)+n+1], or negative with sums [M(n)-n-1,M(n)+n] (Madachy 1979).

Antimagic squares of orders one, two, and three are impossible. In the case of the 3×3 square, there is no known method of proof of this fact except by case analysis or enumeration by computer. There are 18 families of antimagic squares of order four. The total number of antimagic squares of orders 1, 2, ... modulo the full group of symmetries (reflection, rotation, complementation, and exchanges) are 0, 0, 0, 299710, ... (OEIS A050257; Cormie).

Madachy (1979) and Abe (1994) asked for methods of constructing antimagic squares of every order. Recently, J. Cormie and V. Linek have developed general constructions for squares of order n for all n>3, as well as for bordering antimagic squares.


See also

Antimagic Graph, Heterosquare, Magic Square, Talisman Square

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References

Abe, G. "Unsolved Problems on Magic Squares." Disc. Math. 127, 3-13, 1994.Cormie, J. "The Anti-Magic Square Project." http://www.uwinnipeg.ca/~vlinek/jcormie/.Heinz, H. "Anti-Magic Squares." http://www.magic-squares.net/anti_ms.htm.Linek, V. "The Anti-Magic Square Project." http://io.uwinnipeg.ca/~vlinek/jcormie/.Lindon, J. A. "Anti-Magic Squares." Recr. Math. Mag., No. 7, 16-19, Feb. 1962.Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in Madachy's Mathematical Recreations. New York: Dover, pp. 103-113, 1979.Pickover, C. A. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press, 2002.Sloane, N. J. A. Sequence A050257 in "The On-Line Encyclopedia of Integer Sequences."

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Antimagic Square

Cite this as:

Weisstein, Eric W. "Antimagic Square." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AntimagicSquare.html

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