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# Semiperfect Magic Cube

A semiperfect magic cube, sometimes also called an "Andrews cube" (Gardner 1976; Gardner 1988, p. 219) is a magic cube for which the cross section diagonals do not sum to the magic constant. Some care is needed with terminology, as some authors drop the "semiperfect" and refer to such cubes simple as "magic cubes" (e.g., Benson and Jacoby 1981, p. 4).

A semiperfect magic cube of order 3 has magic constant 42. It must be associative with opposite elements summing to (Andrews 1960, p. 65) and have as its center (Gardner 1976; Benson and Jacoby 1981, p. 4; Andrews 1960, p. 65). Hendricks (1972) proved that there are four distinct semiperfect magic cubes excluding rotations and reflections (Gardner 1976; Benson and Jacoby 1981, pp. 4 and 11-13), illustrated above. These cubes were described by Andrews (1960, pp. 66-70), although he seems not to have noted that they represent all distinct possibilities (Gardner 1976; Benson and Jacoby 1981, p. 4). Order three semiperfect magic cubes are also illustrated by Hunter and Madachy (1975, p. 31) and Ball and Coxeter (1987, p. 218)

The above semiperfect magic cube of order four (Ball and Coxeter 1987, p. 220) has magic constant 130.

Semiperfect cubes of odd order with and doubly even order can be constructed by extending the methods used for magic squares. Pandiagonal semiperfect cubes exist for all orders and all odd (Ball and Coxeter 1987).

Bimagic Cube, Magic Cube, Pandiagonal Semiperfect Magic Cube, Perfect Magic Cube

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## References

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.Benson, W. H. and Jacoby, O. Magic Cubes: New Recreations. New York: Dover, 1981.Gardner, M. "Mathematical Games: A Breakthrough in Magic Squares, and the First Perfect Magic Cube." Sci. Amer. 234, 118-123, Jan. 1976.Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 213-225, 1988.Hendricks, J. R. "The Third-Order Magic Cube Complete." J. Math. Recr. Math. 5, 43-50, Jan. 1972.Hunter, J. A. H. and Madachy, J. S. Mathematical Diversions. New York: Dover, 1975.

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Semiperfect Magic Cube

## Cite this as:

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