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Perfect magic cube

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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. 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. See Nasik magic hypercube for an unambiguous alternative term.[7]

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 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.

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.[6]

By the modern (Hendricks) definition, there are actually six classes of magic cube; simple magic cube, pantriagonal magic cube, diagonal magic cube, pantriagonal diagonal magic cube, pandiagonal magic cube, and perfect magic cube.[7]

Nasik; A. H. Frost (1866) referred to all but the simple magic cube as Nasik! C. Planck (1905) redefined Nasik to mean magic hypercubes of any order or dimension in which all possible lines summed correctly.

i.e. Nasik is an alternative, and unambiguous term for the perfect class of any dimension of magic hypercube.

First known Perfect Pandiagonal Semi-magic Magic Cube

Thomas Krijgsman, 1982 March, 21 number 5 / link: http://www.pythagoras.nu/pyth/nummer.php?id=253[permanent dead link‍]

Row 1 (4x4)
32 5 52 41
3 42 31 54
61 24 33 12
34 59 14 23
   
Row 2 (4x4)
10 35 22 63
37 64 9 20
27 2 55 46
56 29 44 1
   
Row 3 (4x4)
49 28 45 8
30 7 50 43
36 57 16 21
15 38 19 58
   
Row 4 (4x4)
39 62 11 18
60 17 40 13
6 47 26 51
25 4 53 48

3D solution in my head, fill the numbers on graph paper, that all. |+

Walter Trump and Christian Boyer, 2003-11-13

This cube consists of all numbers from 1 to 125. The sum of the 5 numbers in each of the 25 rows, 25 columns, 25 pillars, 30 diagonals and 4 triagonals (space diagonals) equals the magic constant 315.

1° level
25 16 80 104 90
115 98 4 1 97
42 111 85 2 75
66 72 27 102 48
67 18 119 106 050
 
2° level
91 77 71 6 70
52 64 117 69 13
30 118 21 123 23
26 39 92 44 114
116 17 14 73 95
 
3° level
(47) (61) 45 (76) (86)
107 43 38 33 94
89 68 (63) 58 37
32 93 88 83 19
40 50 81 65 79
 
4° level
31 53 112 109 10
12 82 34 87 100
103 3 105 8 96
113 57 9 62 74
56 120 55 49 35
 
5° level
121 108 7 20 59
29 28 122 125 11
51 15 41 124 84
78 54 99 24 60
36 110 46 22 101

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 2016-12-04.{{cite web}}: CS1 maint: multiple names: authors list (link)
  2. ^ Alspach, Brian; Heinrich, Katherine. "Perfect Magic Cubes of Order 4m" (PDF). Retrieved December 3, 2016.
  3. ^ Weisstein, Eric W. (2002-12-12). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. ISBN 9781420035223.
  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. ISBN 978-1400841516.
  5. ^ "Perfect Magic Cubes". www.trump.de. Retrieved 2016-12-04.
  6. ^ a b "Magic Cube Timeline". www.magic-squares.net. Retrieved 2016-12-04.
  7. ^ a b "Magic Cubes Index Page". www.magic-squares.net. Retrieved 2016-12-04.
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