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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 diagonal]]s, but also the [[cross section (geometry)|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 has been 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]]. 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

This same reasoning may be applied to [[Tesseract|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.

By the modern (Hendricks) 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.

'''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 Pandiagonaal Semi-magisch Magic Cube==
Thomas Krijgsman, 1982 March,  21 number 5 / link: http://www.pythagoras.nu/pyth/nummer.php?id=253

-Row 1 (4x4)- - - - - -Row 2 (4x4)- - - - - -Row 3 (4x4)- - - - - - Row 4 (4x4)

[32]-[05]-[52]-[41] = [10]-[35]-[22]-[63] = [49]-[28]-[45]-[08] = [39]-[62]-[11]-[18]

[03]-[42]-[31]-[54] = [37]-[64]-[09]-[20] = [30]-[07]-[50]-[43] = [60]-[17]-[40]-[13]

[61]-[24]-[33]-[12] = [27]-[02]-[55]-[46] = [36]-[57]-[16]-[21] = [06]-[47]-[26]-[51]

[34]-[59]-[14]-[23] = [56]-[29]-[44]-[01] = [15]-[38]-[19]-[58] = [25]-[04]-[53]-[48]

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

<!-- Missing image removed: [[Image:First known perfect magic cube.jpg]] -->
[[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.
<TABLE>
<TR> <TH> ''1° level'' </TH> <TH> - </TH>
<TH> ''2° level'' </TH> </TR>
<TR> <TH>
<math>
\begin{bmatrix}
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\\
\end{bmatrix}
</math>
</TH>
<TH>  </TH>
<TH>
<math>
\begin{bmatrix}
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\\
\end{bmatrix}
</math>
</TH>
</TR> 
<TR> <TH> ''3° level'' </TH> <TH> - </TH>
<TH> ''4° level'' </TH> </TR>
<TR> <TH>
<math>
\begin{bmatrix}
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\\
\end{bmatrix}
</math>
</TH>
<TH>  </TH>
<TH>
<math>
\begin{bmatrix}
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\\
\end{bmatrix}
</math>
</TH>
</TR>
<TR> <TH> ''5° level'' </TH> <TH> - </TH>
<TH>  </TH> </TR>
<TR> <TH>
<math>
\begin{bmatrix}
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\\
\end{bmatrix}
</math>
</TH>
<TH>  </TH>
<TH>
</TH>
</TR>
</TABLE>

==See also==

* [[Magic cube classes]]
* [[Nasik magic hypercube]]
* [[John R. Hendricks]]

==References==
* {{cite journal|last1=Frost |first1=A. H.
|title=On the General Properties of Nasik Cubes
|journal=Quart. J. Math.
|volume=15
|year=1878
|pages=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

== External links ==
* {{cite web| first1=G. |last1=Frankenstein|url=http://www.multimagie.com/English/Frankenstein.htm|year=1878|title=A big puzzle
}}
* {{cite web|first1=Walter|last1=Trump|url=http://www.trump.de/magic-squares/magic-cubes/cubes-1.html|title=Perfect magic cube of order 6 found}}
* [http://perso.club-internet.fr/cboyer/multimagie/English/Perfectcubes.htm Christian Boyer: Perfect magic cubes]
* [http://mathworld.wolfram.com/news/2003-11-18/magiccube/ MathWorld news: Perfect magic cube of order 5 discovered]
* [http://mathworld.wolfram.com/PerfectMagicCube.html MathWorld: Perfect magic cube]
* [http://members.shaw.ca/hdhcubes/cube_perfect.htm  Harvey Heinz: Perfect Magic Hypercubes]
* [http://www.magichypercubes.com/Encyclopedia/ Aale de Winkel: The Magic Encyclopedia]
* [http://members.shaw.ca/hdhcubes/cube_update-1.htm#Pandiagonal%20impossibility%20proof  Impossibility Proof for doubly odd order Pandiagonal and Perfect hypercubes]

[[Category:Magic squares]]