Perfect magic cube

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.[1][2][3]

Perfect magic cubes of order one are trivial; cubes of orders two to four can be proven not to exist,[4] and cubes of orders five and six were first discovered by Walter Trump and Christian Boyer on November 13 and September 1, 2003, respectively.[5] 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. The first perfect cube of order 10 was constructed in 1988 (Li Wen, China).[6]

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.[7]

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.[6]

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.[7]

Examples

1. Order 4 cube by Thomas Krijgsman, 1982; magic constant 130.[8]

Level 1
3255241
3423154
61243312
34591423
    
Level 2
10352263
3764920
2725546
5629441
    
Level 3
4928458
3075043
36571621
15381958
    
Level 4
39621118
60174013
6472651
2545348


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

Level 1
25168010490
115984197
4211185275
66722710248
6718119106050
  
Level 2
917771670
52641176913
301182112323
26399244114
11617147395
  
Level 3
(47)(61)45(76)(86)
10743383394
8968635837
3293888319
4050816579
  
Level 4
315311210910
12823487100
1033105896
1135796274
56120554935
  
Level 5
12110872059
292812212511
51154112484
7854992460
361104622101

See also

References
  • Frost, A. H. (1878). "On the General Properties of Nasik Cubes". Quart. J. Math. 15: 93–123.
  • 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
  1. W., Weisstein, Eric. "Perfect Magic Cube". mathworld.wolfram.com. Retrieved 4 December 2016.{{cite web}}: CS1 maint: multiple names: authors list (link)
  2. Alspach, Brian; Heinrich, Katherine. "Perfect Magic Cubes of Order 4m" (PDF). Retrieved 3 December 2016.
  3. Weisstein, Eric W. (12 December 2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. ISBN 9781420035223.
  4. Pickover, Clifford A. (28 November 2011). The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions. Princeton University Press. ISBN 978-1400841516.
  5. "Perfect Magic Cubes". www.trump.de. Retrieved 4 December 2016.
  6. "Magic Cube Timeline". www.magic-squares.net. Retrieved 4 December 2016.
  7. "Magic Cubes Index Page". www.magic-squares.net. Retrieved 4 December 2016.
  8. "Archived copy". Archived from the original on 4 March 2016. Retrieved 28 January 2012.{{cite web}}: CS1 maint: archived copy as title (link)
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