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.
The first perfect cube of order 10 was constructed in 1988 (Li Wen, China).
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.
{|
|
{| class=wikitable style="text-align: center;"
|+ Level 1
| 32 || 5 || 52 || 41
|-
| 3 || 42 || 31 || 54
|-
| 61 || 24 || 33 || 12
|-
| 34 || 59 || 14 || 23
|}
| ||
{| class=wikitable style="text-align: center;"
|+ Level 2
| 10 || 35 || 22 || 63
|-
| 37 || 64 || 9 || 20
|-
| 27 || 2 || 55 || 46
|-
| 56 || 29 || 44 || 1
|}
| ||
{| class=wikitable style="text-align: center;"
|+ Level 3
| 49 || 28 || 45 || 8
|-
| 30 || 7 || 50 || 43
|-
| 36 || 57 || 16 || 21
|-
| 15 || 38 || 19 || 58
|}
| ||
{| class=wikitable style="text-align: center;"
|+ Level 4
| 39 || 62 || 11 || 18
|-
| 60 || 17 || 40 || 13
|-
| 6 || 47 || 26 || 51
|-
| 25 || 4 || 53 || 48
|}
|}
<!-- Missing image removed: File:First known perfect magic cube.jpg -->
2. Order 5 cube by Walter Trump and Christian Boyer, 2003-11-13; magic constant 315.
{|
|
{| class=wikitable style="text-align: center;"
|+ Level 1
| 25 || 16 || 80 || 104 || 90
|-
| 115 || 98 || 4 || 1 || 97
|-
| 42 || 111 || 85 || 2 || 75
|-
| 66 || 72 || 27 || 102 || 48
|-
| 67 || 18 || 119 || 106 || 5
|}
| ||
{| class=wikitable style="text-align: center;"
|+ Level 2
| 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
|}
| ||
{| class=wikitable style="text-align: center;"
|+ Level 3
| 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
|}
| ||
{| class=wikitable style="text-align: center;"
|+ Level 4
| 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
|}
| ||
{| class=wikitable style="text-align: center;"
|+ Level 5
| 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
- John R. Hendricks
- Magic cube classes
- Nasik magic hypercube
References
- 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
External links
- Christian Boyer: Perfect magic cubes
- MathWorld news: Perfect magic cube of order 5 discovered
- Harvey Heinz: Perfect Magic Hypercubes
- Aale de Winkel: The Magic Encyclopedia
- Impossibility Proof for doubly odd order Pandiagonal and Perfect hypercubes
- Most-perfect cube https://oeis.org/A270205