Hypercube Qualifications  

Various qualifications of the hypercube are around, some qualificatons changed in recent years their meaning, ("perfect" changed from pan nagonal to pan ragonal for all r recently by Hendricks) this author introduced step pan magic into his program, and defined "blockwise" to replace his second use of "semi" which confused with the regular use of semi (defined below), Ollerenshaw and Brée defined the term mostperfect only for squares. Presently the equivalent term ^{2}compact ^{2}complete is moved onto because of its obvious generalisation on higher dimansioned hypercubes 

Future enhanchements of articles might hold these kind of fields, it'll hold qualification strings in the manner outlined in this article. The use of common abbreviations is outlined in the encyclopedia's introduction page (a few others might be added to that list over time) for short te qualification strings consist of strings of "{(pan) ragonal}" terms, which prefixes the multimagic qualifiers it describes, the string starts with the higest powers and ends with a string refering to the assumed power "1". (notice my use of the abbreviation "magic") 

Function types of magic  
The function type defines the function which generate the magic sums on every magic line Every number x_{i} if 'functioned' with the result of the previous result on each 1agonal and nagonal Thus defines the 'function magic constant' as C = f_{n1}; f_{i} = function(x_{i},f_{i1}) i = 1..n1; f_{0} = x_{0} 

functions +  * / binom .... 
'constant term: Sum Remainder Product Quotient Binom (?) (???) 
Plausible interesting functions: addition magic: a+b performed at each step subtraction magic: ab performed at each step multiplication (multiply) magic: a*b performed at each step division magic: a/b performed at each step binom magic: (^{a}_{b}) = a!/(b!(ab)!) at each step (other functions definable) 
The addition type forms the regularly studied set of magic hypercube '*' '/' '' seem to have been studied between 1950 and 1955 The binom magic I defined in this article, other types might be interesting also The nonsymmetric functions like '/' and '' give rise to a few more possibilities pe. function(a,b) > function(b,a) or function(max(a,b),min(a,b)) or function(min(a,b),max(a,b)) 

Number ranges  
Regular  The normal range of numbers is used  
Analytical  Regular number range shifted to start with '0'  
Irregular  There are gaps in the range of used numbers  
Prime  Numbers are prime numbers  
Palindromic  Numbers are palindromic numbers  
Arithmetic  Used numbers are in arithmetic progression (the default)  
Geometric  Used numbers are in geometric progression  
Generalized 
No condition posed on the numbers. There might be noninteger or doubly appearing numbers 

major Qualifications  
pMultiMagic 
all 1agonals and nagonals sum to the same sum when all numbers are raised to all powers from 1 to p 

bimagic trimagic tetramagic pentamagic 
2MultiMagic 3MultiMagic 4MultiMagic 5MultiMagic 

The pMultimagic qualifier serves as a splitter of the qualifying string the qualifiers prior to these refer to the hypercube with numbers raised to the power p. 

{ragonal}  the unbroken ragonal result in the constant  
{pan ragonal}  all ragonals (broken and unbroken) result in the constant  
Although the hypercubes monagonal are not broken {pan 1agonal} is often used as it becomes convenient to do so, though the "pan prefix" is redundant in this case. Also hencheforth I'll dispense with the "magic postfix" in these kind of qualifying strings and asume its meaning by the curly brackets 

The main portion of qualifying a hypercube is determining wheter the ragonals results in the certain constant (which depends on type, numberranges and more such factors The "major qualifiers" are abbreviations of the ragonal constants. The below listed define the abbreviations used by this author. although in principle unneccesairy, the term is explained more thoroughly 

magic  {1agonal nagonal}  
All 1agonals and unbroken nagonals  
semimagic  {1agonal}  
All 1agonals and at least one of the unbroken nagonals is NOT resulting in the constant 

pan magic  {1agonal pan nagonal}  
All 1agonals and broken and unbroken nagonals  
strictlymagic  {ragonal for r = 1 .. n}  
This term is currently not in use yet, stating with a simple term that all unbroken ragonals are resulting in the constant might come in handy 

perfect  {pan ragonal for r = 1 .. n}  
This term is highly disputed as the old definition was mere {diagonal} and some people want to stick with old definitions. The above was defined by <John R Hendricks> and makes more sense to me when discussing the hypercube 

minor Qualifications  
The qualifiers below are some modifications to the major qualifiers, these are used when the feature of the major qualifier is present in a subset of the hypercube 

span ragonal  every s broken ragonals are summing to the magic sum  
pan ragonal pan span 
1pan ragonal 1pan nagonal span nagonal 

semipan ragonal  opposite short ragonals are summing to the magic sum  
opposite short ragonals 
lines // the hypercubes ragonals of length m/2 (m even) or of length (m1)/2 or (m+1)/2 (m odd), the center of the hypercube is added or subtracted in this case) (future upload might hold better deription) 

semipan  semipan nagonals  
^{r}perfect  pan qagonal for all q = 1 .. r  
perfect  ^{n}perfect (J.R. Hendricks)  
note: some people maintain to use the old definition of Benson/Jacobi which is synonymous to pandiagonal (so some caution is warrented) to deal with this confusion I added the superscripted dimension 

^{r}compact_{s}  rdimensional order s subhypercube corners is summing to the same sum: 2^{r1} * (m^{r}+1)  
^{r}compact compact 
^{r}compact_{2} ^{n}compact_{2} 

see also  Compact (csection)  
^{r}complete  rdimensional order 2 subhypercube ragonal m/2 apart cells are summing to the same sum: (m^{r}+1)  
complete  ^{n}complete  
Ratio 
The fraction of magic items within the total set of items the relaevant items needs to be specified 

ragonal Ratio  the fraction of magic ragonal lines  
notable Qualifiers and other notes  
surface magic 
All the surface squares of the hypercubes are magic squares <Walter Trump> 

compact complete 
special magic condition defined by
each order 2 Hypercube sums to 2^{n1} * (m^{n}+1) and each pair (n/2 apart) on all (broken) nagonals sum up to (m^{n}+1) 

mostperfect 
term defined by
Kathleen Ollerenshaw and David Brée
for squares for the squares, equivalent to: ^{2}compact ^{2}complete (ie compact complete for n=2) 

Quadrant magic quadrant(pattern) 
In each of the squares quadrant a pattern of numbers is summing to the magic sum  
plusmagic crosmagic diammagic lringmagic sringmagic 
pattern is '+' shaped pattern is 'x' shaped pattern is diamant shaped pattern is a large ring pattern is a small ring 

Functional magic Functional(function) 
Every pair of adjoining numbers 'function(highest,lowest)' is integral multiple of nontrivial number.  
functions +  * / binom .... 
Plausible interesting functions: additional magic: a+b performed on each pair (a,b) subtractional magic: ab performed on each pair (a,b) multiplicational magic: a*b performed on each pair (a,b) divisional magic: a/b performed on each pair (a,b) binomial magic: (^{a}_{b}) = a!/(b!(ab)!) performed on each pair (a,b) (other functions definable) 

Notice in contrast to the function types this is an added feature to an hypercube of function type  
Bent diagonal 
The bent diagonal (bent over at hypercubes center) is summing to the magic sum [Franklin] 

Named Compound Hypercube Qualifications  
some historic names got associated with certain studied features  
Franklin squares  Semi magic squares with bent diagonals.  
Structural Qualifications  
The following qualifiers denote structural parts of a given hypercube. when appropriate they might be seperately qualified and in compound qualifiers exactly positioned and oriented by a nPoint and apropriate nVectors most of these originate from work by <John R Hendricks> 

Inlaid  The Hypercube contains notable subhypercubes of smaller order  
Bordered 
The Hypercube contains a notable subhypercubes of smaller order surrounded on all side by a band of equal width 

Composition 
The Hypercube consists of equal ordered subhypercubes juxtapposed to oneanother (pe results of hypercube multipication) 

Qualification Status  
Conveying the status of a qualification might be a good idea, the here defined {!] and [?] serve this purpose. If neither [!] nor [?] are present it simply means that the investigator did not search further, this and [?] is an invite to the reader to take a shot at it, also [!] in a single hypercube qualifier might be discarded and augmented by the reader. proper is the first adjunct I added onto this list (perhaps others are around) 

proper  exactly the minumum requirements are met  
A usefull adjunct to have around, since every other term place no restriction on the other lines in a hypercube it is a bit stronger then the meaning of '!' <Mitsutoshi Nakamura> 

[!]  Qualification complete, or further qualification not concidered  
when used in the qualifier of a peculiar hypercube it means the qualification is complete, or "other (known) features where not found", in case of a bundle of hypercubes it means other features not concidered so the construction of pandiagonal hypercubes those hypercubes qualify as (pandiagonal monagonal [!]) 

[?]  Qualification incomplete  
This is a definite statement that other features are present but could not be fitted into a qualifier by the investigator. 

Compound Hypercube Qualifications  
The suggested notation below is most flexible as well as exact The basis of this notation is the "qualified hypercube" which holds the qualification of each listed hypercube, either in it base letter or in a summation of features between curly brackets, added to the base letter. Between square brackets each subhypercube likewise qualified can be listed but of course needs to be positioned [..]. Subhypercubes of lower dimensions needs also be oriented <..> Both nPoints resp. nVectors contains n numbers in the range [0,..,n1]. 

qualified hypercubes ^{n}C_{m}{q1,..qp} 
the various qualifications are listed between curly brackets  
subhypercubes ^{n}C_{m}{..} [ c_{1}{..}^{[..]<..>},.., c_{r}{..}^{[..]<..>}] 
A given hypercube can have subhypercubes of certain qualifications the qualified hypercube is followed by an array of qualified subhypercubes each subhypercubes ought to be positioned with an nPoint, and might be oriented by listing their necessairy nVectors to orient the subhypercube 
Hypercube Qualifications  

This table show just a few fictitious samples of the above defined suggested notation.  
^{2}C_{6} [ ^{2}C_{4}{pan}^{[1,1]} ^{2}C_{3}^{[3*i,3*j]} ] 
an order 6 square consisting of four order 3 squares (assumed magic by default) the order 6 square is a bordered order 4 panmagic square. i and j are assumed to take the values 0 and 1 (the possible values) 
^{3}C_{8}{perfect} [ ^{2}C_{5}{quadrant("+")}^{[1,2,3]<1,0,0><0,1,0>} ] 
A perfect order 8 cubes with a plusmagic order 5 square square which starts at cubes position [1,2,3] parallel with the cubes front face. 
Hypercube Equalities  
The above defined notational effort can also be used to state general equations.  
Multiplication of two perfect hypercubes:  
^{n}C_{m1.m2}{perfect} ==
^{n}C_{m1}{perfect} * ^{n}C_{m2}{perfect} ==
^{n}C_{m1}{perfect} ([^{n}C_{m2}{perfect}]  1) m1^{n} == ^{n}C_{m1.m2}{perfect} [^{n}C_{m1}{perfect}^{[}j^{i]}; i = 0..m21 ; j = 0..n1 ] 

The above defines that we are defining a perfect hypercube of order m1.m2 as a product of the two perfect hypercubes of orders m1 and m2 which is explicated by the definition of the basic multiplication. The defined product is further investigated by showing the composition hypercube explicitely, the ranges of the hypercube locator is explicitely stated (which might be viewed redundant in this case, but show the possibilities of the notation) 