Editing Magic cube classes

{{Short description|Categories of number cubes}}
{{original research|date=June 2012}}
{{inline |date=April 2024}}
In [[mathematics]], a [[magic cube]] of order <math>n</math> is an <math>n\times n \times n</math> grid of [[natural numbers]] satisying the property that the numbers in the same row, the same column, the same pillar or the same length-<math>n</math> [[space diagonal|diagonal]] add up to the same number. It is a <math>3</math>-dimensional generalisation of the [[magic square]]. A magic cube can be assigned to one of six '''magic cube classes''', based on the cube characteristics. A benefit of this classification is that it is consistent for all orders and all dimensions of [[magic hypercube]]s.

== The six classes==
* '''Simple:'''
The minimum requirements for a magic cube are: all rows, columns, pillars, and 4 space diagonals must sum to the same value. A [[simple magic cube]] contains no magic squares or not enough to qualify for the next class. <br>The smallest [[magic square|normal]] simple magic cube is order 3. Minimum correct summations required = 3''m''<sup>2</sup> + 4

* '''Diagonal:'''
Each of the 3''m'' planar arrays must be a [[Magic_square#Classification_of_magic_squares|simple magic square]]. The 6 oblique squares are also simple magic. The smallest normal diagonal magic cube is order 5.<br>
These squares were referred to as 'Perfect' by Gardner and others. At the same time he referred to Langman’s 1962 [[pandiagonal magic cube|pandiagonal cube]] also as 'Perfect'.<br>
Christian Boyer and Walter Trump now consider this ''and'' the next two classes to be ''Perfect''.  (See ''Alternate Perfect'' below).<br>A. H. Frost referred to all but the simple class as '''Nasik''' cubes. <br>The smallest normal diagonal magic cube is order 5; see [[Diagonal magic cube]]. Minimum correct summations required = 3''m''<sup>2</sup> + 6''m'' + 4

* '''Pantriagonal:'''
All 4''m''<sup>2</sup> pantriagonals must sum correctly (that is 4 one-segment, 12(''m''−1) two-segment, and 4(''m''−2)(''m''−1) three-segment). There may be some simple AND/OR pandiagonal magic squares, but not enough to satisfy any other classification. <br>The smallest normal pantriagonal magic cube is order 4; see [[Pantriagonal magic cube]]. <br>Minimum correct summations required = 7''m''<sup>2</sup>. All pan-''r''-agonals sum correctly for ''r''&nbsp;=&thinsp;1 and 3.

* '''PantriagDiag:'''
A cube of this class was first constructed in late 2004 by Mitsutoshi Nakamura. This cube is a combination [[pantriagonal magic cube]] and [[diagonal magic cube]]. Therefore, all main and broken space diagonals sum correctly, and it contains 3''m'' planar [[Magic_square#Classification_of_magic_squares|simple magic squares]]. In addition, all 6 oblique squares are [[pandiagonal magic square]]s. The only such cube constructed so far is order 8. It is not known what other orders are possible; see [[Pantriagdiag magic cube]]. Minimum correct summations required = 7''m''<sup>2</sup> + 6''m''

* '''Pandiagonal:'''
All 3''m'' planar arrays must be [[pandiagonal magic square]]s. The 6 oblique squares are always magic (usually simple magic). Several of them ''may'' be pandiagonal magic.
Gardner also called this (Langman’s pandiagonal) a 'perfect' cube, presumably not realizing it was a higher class then Myer’s cube. See previous note re Boyer and Trump. <br>The smallest normal pandiagonal magic cube is order 7; see [[Pandiagonal magic cube]].<br>Minimum correct summations required = 9''m''<sup>2</sup> + 4. All pan-''r''-agonals sum correctly for ''r''&nbsp;=&thinsp;1 and 2. 

* '''Perfect:'''
All 3''m'' planar arrays must be [[pandiagonal magic square]]s. In addition, all pantriagonals must sum correctly. These two conditions combine to provide a total of 9''m'' pandiagonal magic squares. <br>The smallest normal perfect magic cube is order 8; see [[Perfect magic cube]].

'''Nasik;'''
A. H. Frost (1866) referred to all but the simple magic cube as Nasik!<br>
C. Planck (1905) redefined ''Nasik'' to mean magic hypercubes of any order or dimension in which all possible lines summed correctly.<br> i.e. '''''Nasik''''' is a '''preferred alternate''', and less ambiguous term for the ''perfect'' class.<br>Minimum correct summations required =&thinsp;13''m''<sup>2</sup>. All pan-''r''-agonals sum correctly for ''r''&nbsp;=&thinsp;1, 2 and 3. 

'''Alternate Perfect'''
Note that the above is a relatively new definition of ''perfect''. Until about 1995 there was much confusion about what  constituted a ''perfect'' magic cube (see the discussion under '''Diagonal''').<br> Included below are references and links to discussions of the old definition<br>
With the popularity of personal computers it became easier to examine the finer details of magic cubes. Also more and more work was being done with higher-dimension [[magic hypercube]]s. For example, John Hendricks constructed the world's first '''Nasik''' [[magic tesseract]] in 2000. Classed as a [[perfect magic tesseract]] by Hendricks definition.

==Generalized for all dimensions==
A magic hypercube of dimension ''n'' is perfect if all pan-''n''-agonals sum correctly. Then all lower-dimension hypercubes contained in it are also perfect.<br>
For dimension 2, The Pandiagonal Magic Square has been called ''perfect'' for many years. This is consistent with the perfect (Nasik) definitions given above for the cube. In this dimension, there is no ambiguity because there are only two classes of magic square, simple and perfect. <br>
In the case of 4 dimensions, the magic tesseract, Mitsutoshi Nakamura has determined that there are 18 classes. He has determined their characteristics and constructed examples of each.
And in this dimension also, the ''Perfect'' (''Nasik'') magic tesseract has all possible lines summing correctly and all cubes and squares contained in it are also Nasik magic.

==Another definition and a table==

'''Proper:'''
A proper magic cube is a magic cube belonging to one of the six classes of magic cube, but containing exactly the minimum requirements for that class of cube. i.e. a proper simple or pantriagonal magic cube would contain no magic squares, a proper diagonal magic cube would contain exactly 3''m'' + 6 simple magic squares, etc. This term was coined by Mitsutoshi Nakamura in April, 2004.

{| class="wikitable"
|+ Minimum lines (and magic squares) required for each class of magic cube
|-
! Class of magic cube
! Smallest<br />possible<br />order
! colspan=4 | Lines summing correctly to S<br />(m(m<sup>3</sup>+1)) / 2
! colspan=2 | Simple<br />magic squares
! colspan=2 | Pandiagonal<br />(Nasik)<br />magic squares
|-
! <br />(''r''-agonal) ||   || Ortho.<br />1 || Diag.<br />2 || Triag.<br />3 || Total || Planar || Oblique || Planar || Unique
|-
| Simple || 3 || 3m<sup>2</sup> || — || 4 || 3m<sup>2</sup> + 4 || — || — || — || —
|-
| Diagonal || 5 || 3m<sup>2</sup> || 6m || 4 || 3m<sup>2</sup> + 6m + 4 || 3m || 6 || — || —
|-
| Pantriagonal || 4 || 3m<sup>2</sup> || — || 4m<sup>2</sup> || 7m<sup>2</sup> || — || — || — || — 
|-
| PantriagDiag || 8 ? || 3m<sup>2</sup> || 6m || 4m<sup>2</sup> || 7m<sup>2</sup> + 6m || 3m || 0 || — || 6
|-
| Pandiagonal || 7 || 3m<sup>2</sup> || 6m<sup>2</sup> || 4 || 9m<sup>2</sup> + 4 || — || 6 || 3m || —
|-
| Perfect (Nasik) || 8 || 3m<sup>2</sup> || 6m<sup>2</sup> || 4m<sup>2</sup> || 13m<sup>2</sup> || — || — || 3m || 6m
|}

'''Notes for the table'''
# For the diagonal or pandiagonal classes, one or possibly 2 of the 6 oblique magic squares may be pandiagonal magic. All but 6 of the oblique squares are 'broken'. This is analogous to the [[broken diagonal]]s in a pandiagonal magic square. i.e. Broken diagonals are 1-D in a 2-D square; broken oblique squares are 2-D in a 3-D cube.
# The table shows the minimum lines or squares required for each class (i.e. proper). Usually there are more, but not enough of one type to qualify for the next class.

== See also ==
* [[Magic hypercube]]
* [[Nasik magic hypercube]]
* [[Panmagic square]]
* [[Space diagonal]]
* [[John R. Hendricks]]

==Further reading ==
* Frost, Dr. A. H., On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93–123
* Planck, C., The Theory of Paths Nasik, Printed for private circulation, A.J. Lawrence, Printer, Rugby,(England), 1905
* Heinz, H.D. and Hendricks, J. R., Magic Square Lexicon: Illustrated. Self-published, 2000, 0-9687985-0-0.
* Hendricks, John R., The Pan-4-agonal Magic Tesseract, The American Mathematical Monthly, Vol. 75, No. 4, April 1968, p.&nbsp;384.
* Hendricks, John R., The Pan-3-agonal Magic Cube, Journal of Recreational Mathematics, 5:1, 1972, pp51–52
* Hendricks, John R., The Pan-3-agonal Magic Cube of Order-5, JRM, 5:3, 1972, pp 205–206
* Hendricks, John R., Magic Squares to Tesseracts by Computer, Self-published 1999. 0-9684700-0-9
* Hendricks, John R., Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published 1999. 0-9684700-4-1
* [[Clifford A. Pickover]] (2002). ''The Zen of Magic Squares, Circles and Stars''. Princeton Univ. Press, 2002, 0-691-07041-5. pp 101–121

== External links ==
Cube classes
* [https://web.archive.org/web/20050205173640/http://cboyer.club.fr/multimagie/index.htm  Christian Boyer: Perfect Magic Cubes]
* [http://members.shaw.ca/hdhcubes/cube_perfect.htm  Harvey Heinz: Perfect Magic Hypercubes]
* [http://members.shaw.ca/hdhcubes/index.htm#  Harvey Heinz: 6 Classes of Cubes]
* [http://www.trump.de/magic-squares/magic-cubes/cubes-1.html  Walter Trump: Search for Smallest ]
* [https://oeis.org/A270205 Most perfect cube]
Perfect Cube
* [http://www.magichypercubes.com/Encyclopedia/  Aale de Winkel: Magic Encyclopedia]
* [http://members.shaw.ca/hdhcubes/cube_define.htm#Theory%20of%20Paths%20Nasik  A long quote from C. Plank (1917) on the subject of ''nasik'' as a substitute term for ''perfect''.]
Tesseract Classes
* [http://members.shaw.ca/tesseracts/t_classes.htm  The Square, Cube, and Tesseract Classes]
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[[Category:Magic squares]]