Bimagic Cube

A bimagic cube is a (normal) magic cube that remains magic when all its elements are squared. Of course, even a normal magic cubic becomes nonnormal (i.e., contains nonconsecutive elements) upon squaring.

Cazalas (1934) attempted but failed to construct a bimagic cube (Boyer). David M. Collison apparently constructed a bimagic cube of order 25 in an unpublished paper (Hendricks 1992), but it was not until the year 2000 that John Hendricks published an order 25 perfect magic cube whose square is a semiperfect magic cube.

On January 20, 2003, Christian Boyer discovered an order 16 bimagic cube (where the cube itself is perfect magic, but its square is only semiperfect magic). This was rapidly followed by another order 16 bimagic cube (where the base cube is perfect and its square semiperfect) on January 23, an order 32 bimagic cube (where both the base cube and its square are perfect) on January 27, and an order 27 bimagic cube (where the base cube is perfect but its square is semiperfect) on February 3, 2003.

Boyer's 16-cubes thus became the smallest known bimagic cube, and his order 32 cube became the first known perfect bimagic cube.

See also

Bimagic Square, Magic Cube, Multimagic Cube, Perfect Magic Cube, Semiperfect Magic Cube, Trimagic Cube

Portions of this entry contributed by Margherita Barile

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Boyer, C. "Les cubes magiques." Pour La Science. No. 311, pp. 90-95, Sept. 2003.Boyer, C. "Multimagic Cubes.", G. E. Carrés magiques au degré n. Paris: Hermann, 1934.Danielsson, H. "Printout of a Bimagic Cube: Order 25.", H. "Boyer's Bimagic 16 Cube.", H. "Multimagic Cubes.", J. R. "Notes--Towards the Bimagic Cube." In The Magic Square Course. Published by the author, p. 411, 1992.Hendricks, J. R. A Bimagic Cube of Order 25. Published by the author, 2000.Pickover, C. A. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press, p. 103, 2002.

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Bimagic Cube

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Bimagic Cube." From MathWorld--A Wolfram Web Resource.

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