Magic Hypercube - Qualifications

Qualifications

A hypercube nH_m with numbers in the analytical numberrange has the magic sum:

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 mn-1.

Also "magic" (i.e. {1-agonal n-agonal}) is usually assumed, the 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:

• {ncompact} : {all order 2 subhyper cubes sum to 2n nS_m / m}
• {ncomplete} : {all pairs halve an n-agonal apart sum equal (to (mn - 1)}

{ncompact} might be put in notation as : _(k)∑ = 2n nS_m / m.
{ncomplete} can simply written as: + = mn - 1.
Where:
_(k)∑ is symbolic for summing all possible k's, there are 2n possibilities for _k1.
expresses and all its r-agonal neighbors.
for {complete} the complement of is at position .

for squares: {2compact 2complete} is the "modern/alternative qualification" of what Dame Kathleen Ollerenshaw called most-perfect magic square, {ncompact ncomplete} is the qualifier for the feature in more than 2 dimensions
Caution: some people seems to equate {compact} with {2compact} instead of {ncompact}. Since this introductory article is not the place to discuss these kind of issues I put in the dimensional pre-superscript n to both these qualifiers (which are defined as shown)
consequences of {ncompact} 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.