Editing Magic hypercube (section)

==Construction==

Besides more specific constructions two more general construction method are noticeable:

===KnightJump construction===
This construction generalizes the movement of the chessboard horses (vectors <math>\langle 1,2 \rangle, \langle 1,-2 \rangle, \langle -1,2 \rangle, \langle -1,-2 \rangle </math>) to more general movements (vectors <math>\langle {}_k i \rangle</math>). The method starts at the position P<sub>0</sub> and further numbers are sequentially placed at positions <math>V_0</math> further until (after m steps) a position is reached that is already occupied, a further vector is needed to find the next free position. Thus the method is specified by the n by n+1 matrix:
:<math>[P_0, V_0 \dots V_{n-1}]</math>
This positions the number 'k' at position:
:<math>P_k = P_0 + \sum_{l=0}^{n-1}((k\backslash m^l)\ \%\ m) V_l;\quad k = 0 \dots m^n-1.</math>
'''C. Planck''' gives in his 1905 article [http://www.magichypercubes.com/Encyclopedia/k/PathNasiks.zip "'''The theory of Path Nasiks'''"] conditions to create with this method "Path Nasik" (or modern {perfect}) hypercubes.

===Latin prescription construction===
(modular equations).
This method is also specified by an n by n+1 matrix. However this time it multiplies the n+1 vector  [x<sub>0</sub>,..,x<sub>n-1</sub>,1], After this multiplication the result is taken modulus m to achieve the n (Latin) hypercubes:
 LP<sub>k</sub> = ( <sub>l=0</sub>Σ<sup>n-1</sup> LP<sub>k,l</sub> x<sub>l</sub> + LP<sub>k,n</sub> ) % m
of radix m numbers (also called "'''digits'''"). On these LP<sub>k</sub>'s "'''digit changing'''" (?i.e. Basic manipulation) are generally applied before these LP<sub>k</sub>'s are combined into the hypercube:
 <sup>n</sup>H<sub>m</sub> = <sub>k=0</sub>Σ<sup>n-1</sup> LP<sub>k</sub> m<sup>k</sup>

'''J.R.Hendricks''' often uses modular equation, conditions to make hypercubes of various quality can be found on [http://www.magichypercubes.com/Encyclopedia http://www.magichypercubes.com/Encyclopedia] at several places (especially p-section)

Both methods fill the hypercube with numbers, the knight-jump guarantees (given appropriate vectors) that every number is present. The Latin prescription only if the components are orthogonal (no two digits occupying the same position)

===Multiplication===
Amongst the various ways of compounding, the multiplication<ref>this is a n-dimensional version of (pe.): [http://mathforum.org/alejandre/magic.square/adler/product.html Alan Adler magic square multiplication]</ref> can be considered as the most basic of these methods. The '''basic multiplication''' is given by:
 <sup>n</sup>H<sub>m<sub>1</sub></sub> * <sup>n</sup>H<sub>m<sub>2</sub></sub> : <sup>n</sup>[<sub>k</sub>i]<sub>m<sub>1</sub>m<sub>2</sub></sub> = <sup>n</sup>[ [<nowiki>[</nowiki><sub>k</sub>i \ m<sub>2</sub>]<sub>m<sub>1</sub></sub>m<sub>1</sub><sup>n</sup>]<sub>m<sub>2</sub></sub> + [<sub>k</sub>i % m<sub>2</sub>]<sub>m<sub>2</sub></sub>]<sub>m<sub>1</sub>m<sub>2</sub></sub>

Most compounding methods can be viewed as variations of the above, As most qualifiers are invariant under multiplication one can for example place any aspectual variant of <sup>n</sup>H<sub>m<sub>2</sub></sub> in the above equation, besides that on the result one can apply a manipulation to improve quality. Thus one can specify pe the  J. R. Hendricks / M. Trenklar doubling. These things go beyond the scope of this article.