In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant.

Perfect magic cubes of order one are trivial; cubes of orders two to four can be proven not to exist, and cubes of orders five and six were first discovered by Walter Trump and Christian Boyer on November 13 and September 1 2003, respectively. A perfect magic cube of order seven was given by A. H. Frost in 1866, and on March 11 1875, an article was published in the Cincinnati Commercial newspaper on the discovery of a perfect magic cube of order 8 by Gustavus Frankenstein . Perfect magic cubes of orders nine and eleven have also been constructed, but none of order ten is known.

An alternate definition

In recent years, an alternate definition for the perfect magic cube has gradually come into use. It is based on the fact that a pandiagonal magic square has traditionally been called 'perfect', because all possible lines sum correctly. This is not the case with the above definition for the cube.

This same reasoning may be applied to [hypercubes] of any dimension. Simply stated; if all possible lines of m cells (m = order) sum correctly, the hypercube is perfect. All lower dimension hypercubes contained in this hypercube will also be perfect. This also is not the case with the original definition, which does not require that the planar and diagonal squares be pandiagonal.

The original definition is applicable only to magic cubes, not tesseracts, dimension 5 cubes, etc.

Example: A perfect magic cube of order 8 has 244 correct lines by the old definition, but 832 correct lines by this new definition.

Order 8 is the smallest possible perfect magic cube. 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 definition, there are actually six classes of magic cube; simple magic cube, pantriagonal magic cube, diagonal magic cube, pantriagdiag magic cube, pandiagonal magic cube, and perfect magic cube.

See also

• Magic cube classes

External links

• Walter Trump: Perfect magic cube of order 6 found
• Christian Boyer: Perfect magic cubes
• MathWorld news: Perfect magic cube of order 5 discovered
• MathWorld: Perfect magic cube
• Harvey Heinz: Perfect Magic Hypercubes
• Aale de Winkel: The Magic Encyclopedia
• Impossibility Proof for doubly-odd order Pandiagonal and Perfect hypercubes