Editing Magic hypercube (section)

==Qualifications==

A hypercube <sup>n</sup>H<sub>m</sub> with numbers in the analytical numberrange [0..m<sup>n</sup>-1] has the magic sum:
 <sup>n</sup>S<sub>m</sub> = m (m<sup>n</sup> - 1) / 2.

Besides more specific qualifications the following are the most important, "summing" of course stands for "summing correctly to the magic sum"

*{'''r-agonal'''} : all main (unbroken) r-agonals are summing.
*{'''pan r-agonal'''} : all (unbroken and broken) r-agonals are summing.
*{'''magic'''} : {1-agonal n-agonal}
*{'''perfect'''} : {pan r-agonal; r = 1..n}

Note: This series doesn't start with 0 since a nill-agonal doesn't exist, the numbers correspond with the usual name-calling: 1-agonal = monagonal, 2-agonal = diagonal, 3-agonal = triagonal etc.. Aside from this the number correspond to the amount of "-1" and "1" in the corresponding pathfinder.

In case the hypercube also sum when all the numbers are raised to the power p one gets p-multimagic hypercubes. The above qualifiers are simply prepended onto the p-multimagic qualifier. This defines qualifications as {r-agonal 2-magic}. Here also "2-" is usually replaced by "bi",  "3-" by "tri" etc. ("1-magic" would be "monomagic" but "mono" is usually omitted). The sum for p-Multimagic hypercubes can be found by using [[Faulhaber's formula]] and divide it by m<sup>n-1</sup>.

Also "magic" (i.e. {1-agonal n-agonal}) is usually assumed, the [[perfect magic cube|Trump/Boyer {diagonal} cube]] is technically seen {1-agonal 2-agonal 3-agonal}.

[[Nasik magic hypercube]] gives arguments for using {'''nasik'''} as synonymous to {'''perfect'''}. The strange generalization of square 'perfect' to using it synonymous to {diagonal} in cubes is however also resolve by putting curly brackets around qualifiers, so {'''perfect'''} means {pan r-agonal; r = 1..n} (as mentioned above).

some minor qualifications are:
*{'''<sup>n</sup>compact'''} : {all order 2 subhyper cubes sum to 2<sup>n</sup> <sup>n</sup>S<sub>m</sub> / m}
*{'''<sup>n</sup>complete'''} : {all pairs halve an n-agonal apart sum equal (to (m<sup>n</sup> - 1)}

{'''<sup>n</sup>compact'''} might be put in notation as : '''<sub>(k)</sub>Σ [<sub>j</sub>i + <sub>k</sub>1] = 2<sup>n</sup> <sup>n</sup>S<sub>m</sub> / m'''. {'''<sup>n</sup>complete'''} can simply be written as: '''[<sub>j</sub>i] + [<sub>j</sub>i + <sub>k</sub>(m/2) ; #k=n ] = m<sup>n</sup> - 1''' where:
:<sub>(k)</sub>Σ is symbolic for summing all possible k's, there are 2<sup>n</sup> possibilities for <sub>k</sub>1.
:[<sub>j</sub>i + <sub>k</sub>1] expresses [<sub>j</sub>i] and all its r-agonal neighbors.
for {complete} the complement of [<sub>j</sub>i] is at position [<sub>j</sub>i + <sub>k</sub>(m/2) ; #k=n ].

for squares: {'''<sup>2</sup>compact <sup>2</sup>complete'''} is the "modern/alternative qualification" of what Dame [[Kathleen Ollerenshaw]] called [[most-perfect magic square]], {<sup>n</sup>compact <sup>n</sup>complete} is the qualifier for the feature in more than 2 dimensions.

Caution: some people seems to equate {compact} with {<sup>2</sup>compact} instead of {<sup>n</sup>compact}. Since this introductory article is not the place to discuss these kind of issues I put in the dimensional pre-superscript <sup>n</sup> to both these qualifiers (which are defined as shown) consequences of {<sup>n</sup>compact} is that several figures also sum since they can be formed by adding/subtracting order 2 sub-hyper cubes. Issues like these go beyond this articles scope.