Editing Perfect magic cube

{{Short description|Magic cube with extra constraints}}
{{Use dmy dates|date=September 2023}}
{{original research|date=November 2016}}

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]].<ref>{{Cite web|url=http://mathworld.wolfram.com/PerfectMagicCube.html|title=Perfect Magic Cube|last=W.|first=Weisstein, Eric|website=mathworld.wolfram.com|language=en|access-date=2016-12-04}}</ref><ref>{{Cite web|url=http://www.fq.math.ca/Scanned/19-2/alspach.pdf|title=Perfect Magic Cubes of Order 4m|last1=Alspach|first1=Brian|author1-link=Brian Alspach|last2=Heinrich|first2=Katherine|author2-link= Katherine Heinrich |date=|website=|publisher=|access-date=December 3, 2016}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=D_XKBQAAQBAJ&q=perfect+magic+cube&pg=PA2203|title=CRC Concise Encyclopedia of Mathematics, Second Edition|last=Weisstein|first=Eric W.|date=2002-12-12|publisher=CRC Press|isbn=9781420035223|language=en}}</ref>

Perfect magic cubes of order one are trivial; cubes of orders two to four can be [[mathematical proof|proven]] not to exist,<ref>{{Cite book|url=https://books.google.com/books?id=-mu8O8RMG6QC&q=perfect+magic+cube&pg=PA101|title=The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions|last=Pickover|first=Clifford A.|date=2011-11-28|publisher=Princeton University Press|isbn=978-1400841516|language=en}}</ref> and cubes of orders five and six were first discovered by [[Walter Trump]] and [[Christian Boyer]] on November 13 and September 1, 2003, respectively.<ref>{{Cite web|url=http://www.trump.de/magic-squares/magic-cubes/cubes-1.html|title=Perfect Magic Cubes|website=www.trump.de|access-date=2016-12-04}}</ref> 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).<ref name=":1" />

==An alternative definition==

In recent years{{when|date=June 2021}}, 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.<ref name=":0">{{Cite web|url=http://www.magic-squares.net/magic_cubes_index.htm|title=Magic Cubes Index Page|website=www.magic-squares.net|access-date=2016-12-04}}</ref>

This same reasoning may be applied to [[hypercube]]s 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.<ref name=":1">{{Cite web|url=http://www.magic-squares.net/c-t-htm/C_timeline.htm|title=Magic Cube Timeline|website=www.magic-squares.net|access-date=2016-12-04}}</ref>

By the modern (given by [[John R. Hendricks|J.R. Hendricks]]) definition, there are actually six [[magic cube class|classes of magic cube]]; [[simple magic cube]]s, [[pantriagonal magic cube]]s, [[diagonal magic cube]]s, pantriagonal diagonal magic cubes, [[pandiagonal magic cube]]s, and perfect magic cubes.<ref name=":0" />

==Examples==
1. Order 4 cube by Thomas Krijgsman,&thinsp;1982; magic constant 130.<ref>{{Cite web |url=http://www.pythagoras.nu/pyth/nummer.php?id=253 |title=Archived copy |access-date=28 January 2012 |archive-date=4 March 2016 |archive-url=https://web.archive.org/web/20160304034747/http://www.pythagoras.nu/pyth/nummer.php?id=253 |url-status=dead }}</ref>
{|
|
{| class=wikitable style="text-align: center;"
|+ Level 1
| 32 ||  5 || 52 || 41
|-
|  3 || 42 || 31 || 54
|-
| 61 || 24 || 33 || 12
|-
| 34 || 59 || 14 || 23
|}
| {{nbsp|3}} ||
{| class=wikitable style="text-align: center;"
|+ Level 2
| 10 || 35 || 22 || 63
|-
| 37 || 64 ||  9 || 20
|-
| 27 ||  2 || 55 || 46
|-
| 56 || 29 || 44 ||  1
|}
| {{nbsp|3}} ||
{| class=wikitable style="text-align: center;"
|+ Level 3
| 49 || 28 || 45 ||  8
|-
| 30 ||  7 || 50 || 43
|-
| 36 || 57 || 16 || 21
|-
| 15 || 38 || 19 || 58
|}
| {{nbsp|3}} ||
{| 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 || {{0|0}}5{{0|0}}
|}
| {{nbsp}} || 
{| 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
|}
| {{nbsp}} || 
{| class=wikitable style="text-align: center;"
|+ Level 3
| {{0|(}}47{{0|)}} || {{0|(}}61{{0|)}} || 45 || {{0|(}}76{{0|)}} || {{0|(}}86{{0|)}}
|-
| 107 || 43 || 38 || 33 || 94 
|-
| 89 || 68 || 63 || 58 || 37 
|-
| 32 || 93 || 88 || 83 || 19 
|-
| 40 || 50 || 81 || 65 || 79
|}
| {{nbsp}} || 
{| 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
|}
| {{nbsp}} || 
{| 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==

* [[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
{{Reflist}}

== 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}}
* [https://web.archive.org/web/20040224223239/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://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]
* Most-perfect cube https://oeis.org/A270205

[[Category:Magic squares]]