Perfect magic cube

An alternative definition

In recent years, an alternative definition for the perfect magic cube was proposed by John R. Hendricks. By this definition, a perfect magic cube is one in which all possible lines through each cell sum to the magic constant. The name Nasik magic hypercube is another, unambiguous, name for such a cube. This definition is based on the fact that a pandiagonal magic square has traditionally been called 'perfect', because all possible lines sum correctly.

This same reasoning may be applied to hypercubes of any dimension. Simply stated; in an order m magic hypercube, if all possible lines of m cells sum to the magic constant, the hypercube is perfect. All lower dimension hypercubes contained in this hypercube will then also be perfect. This is not the case with the original definition, which does not require that the planar and diagonal squares be a pandiagonal magic cube. For example, a magic cube of order 8 has 244 correct lines by the old definition of "perfect", but 832 correct lines by this new definition.

The smallest perfect magic cube has order 8, and none can exist for double odd orders.

Gabriel Arnoux constructed an order 17 perfect magic cube in 1887. F.A.P.Barnard published order 8 and order 11 perfect cubes in 1888.

By the modern (given by J.R. Hendricks) definition, there are actually six classes of magic cube; simple magic cubes, pantriagonal magic cubes, diagonal magic cubes, pantriagonal diagonal magic cubes, pandiagonal magic cubes, and perfect magic cubes.

Examples

1. Order 4 cube by Thomas Krijgsman, 1982; magic constant 130.

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2. Order 5 cube by Walter Trump and Christian Boyer, 2003-11-13; magic constant 315.

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References

• {{cite journal|last1=Frost |first1=A. H.
• Planck, C., The Theory of Paths Nasik, Printed for private circulation, A.J. Lawrence, Printer, Rugby,(England), 1905
• H.D, Heinz & J.R. Hendricks, Magic Square Lexicon: Illustrated, hdh, 2000, 0-9687985-0-0

References

1. W., Weisstein, Eric. "Perfect Magic Cube".
2. "Perfect Magic Cubes of Order 4m".
3. Weisstein, Eric W.. (2002-12-12). "CRC Concise Encyclopedia of Mathematics, Second Edition". CRC Press.
4. Pickover, Clifford A.. (2011-11-28). "The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions". Princeton University Press.
5. "Perfect Magic Cubes".
6. "Magic Cubes Index Page".
7. "Magic Cube Timeline".
8. "Archived copy".