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).
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.
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. + Level 1 32 5 52 41 3 42 31 54 61 24 33 12 34 59 14 23 + Level 2 10 35 22 63 37 64 9 20 27 2 55 46 56 29 44 1 + Level 3 49 28 45 8 30 7 50 43 36 57 16 21 15 38 19 58 + Level 4 39 62 11 18 60 17 40 13 6 47 26 51 25 4 53 48
2. Order 5 cube by Walter Trump and Christian Boyer, 2003-11-13; magic constant 315.
+ 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 + 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 + 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 + 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 + 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