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Magic Cube


A magic cube is an n×n×n version of a magic square in which the n^2 rows, n^2 columns, n^2 pillars, and four space diagonals each sum to a single number M_3(n) known as the cube's magic constant. Magic cubes are most commonly assumed to be "normal," i.e., to have elements that are the consecutive integers 1, 2, ..., n^3. However, this requirement is dropped (as it must be) in the consideration of so-called multimagic cubes.

If it exists, a normal magic cube has magic constant

 M(n)=1/2n(n^3+1).

For n=1, 2, ..., the magic constants are given by 1, 9, 42, 130, 315, 651, ... (OEIS A027441).

If only rows, columns, pillars, and space diagonals sum to M_3(n), a magic cube is called a semiperfect magic cube, or sometimes an Andrews cube (Gardner 1988, p. 219). If, in addition, the diagonals of each n×n orthogonal slice sum to M_3(n), then the magic cube is called a perfect magic cube. If a perfect or semiperfect magic cube is magic not only along the main space diagonals, but also on the broken space diagonals, it is known as a pandiagonal magic cube.

There is a trivial perfect magic cube of order one, but no perfect cubes exist for orders 2-4. While normal perfect magic cubes of orders 7 and 9 have been known since the late 1800s, it was long not known if perfect magic cubes of orders 5 or 6 could exist. A 5×5×5 perfect magic cube was subsequently discovered by C. Boyer and W. Trump on Nov. 14 2003.

A perfect or semiperfect magic cube that yields another magic cube of the same type when its elements are squared is known as a bimagic cube. Similarly, a magic cube that remains magic when its elements are both squared and cubed is known as a trimagic cube.

The smallest known multiplication magic cube is 4×4×4 with largest term 416 and magic product 8648640, or 13!/6! (Boyer 2006).


See also

Bimagic Cube, Magic Constant, Magic Graph, Magic Hexagon, Magic Square, Magic Tesseract, Multimagic Cube, Perfect Magic Cube, Semiperfect Magic Cube, Trimagic Cube

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References

Adler, A. and Li, S.-Y. R. "Magic Cubes and Prouhet Sequences." Amer. Math. Monthly 84, 618-627, 1977.Andrews, W. S. Magic Squares and Cubes, 2nd rev. ed. New York: Dover, 1960.Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 216-224, 1987.Barnard, F. A. P. "Theory of Magic Squares and Cubes." Mem. Nat. Acad. Sci. 4, 209-270, 1888.Benson, W. H. and Jacoby, O. Magic Cubes: New Recreations. New York: Dover, 1981.Boyer, C. "Les cubes magiques." Pour La Science. No. 311, pp. 90-95, Sept. 2003.Boyer, C. "Multimagie News." Apr. 4, 2006. http://www.multimagie.com/English/News0604.htm.Cazalas, G. E. Carrés magiques au degré n. Paris: Hermann, 1934.Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 213-225, 1988.Heinz, H. "Magic Cubes-Introduction." http://members.shaw.ca/hdhcubes/.Hirayama, A. and Abe, G. Researches in Magic Squares. Osaka, Japan: Osaka Kyoikutosho, 1983.Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, p. 31, 1975.Update a linkLei, A. "Magic Cube and Hypercube." http://www.cs.ust.hk/~philipl/magic/mcube2.htmlMadachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 99-100, 1979.Pappas, T. "A Magic Cube." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 77, 1989.Pickover, C. A. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press, 2002.Rosser, J. B. and Walker, R. J. "The Algebraic Theory of Diabolical Squares." Duke Math. J. 5, 705-728, 1939.Sloane, N. J. A. Sequence A027441 in "The On-Line Encyclopedia of Integer Sequences."Trenkler, M. "A Construction of Magic Cubes." Math. Gaz. 84, 36-41, 2000.

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Magic Cube

Cite this as:

Weisstein, Eric W. "Magic Cube." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MagicCube.html

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