The Magic Hypercube  

The hypercube is qualified as magic if all 1agonals and all nagonals sum to the same number for a regular magic hypercube this number is 'the magic sum' S = m (m^{n} + 1) / 2 

^{m}INMagic Hypercube {[_{j}i] ε [1..m^{n}] 
C = f_{n1}; f_{i} = function(x_{i},f_{i1}) i = 1..n1; f_{0} = x_{0}} x_{i} are the number on each 'magic line'. C is called the 'magic constant'. 

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) 
for the remainder '+' and sum is assumed and we can move to: [_{j}i] ε [0..m^{n}1] 

semimagic  the bodys nagonals do not sum up to the magic sum  
blockwise magic 
the sums of lines parrallel to the axis are blockwise off in a linear fashion these blocks are all of half the hyper cubes order (usually the order is even) all direct left multipliction with the order 2 regular hypercube fall in this category 

panragonal magic  all sums on all (broken) ragonals sum up to the magic sum  
panmagic pandiagonal magic pantriagonal magic panquadragonal magic 
pannagonal magic pan2agonal magic pan3agonal magic pan4agonal magic 

perfect  panragonal for all r = 1 .. n  
pmultimagic 
the hypercube sums up to a magic number when all number are raised to the power q for all q = 1 .. p 

bimagic trimagic 
2multimagic 3multimagic 

Magic Square  
the two dimensional hypercube is still the most investigated. due to the limited amount of numbers a few subjects are only for the square although they can be defined on the higher dimensional figures 

mostperfect 
special magic condition defined by
Kathleen Ollerenshaw and David Brée
for squares each order 2 Hypercube sums to 2 * (m^{n}+1) and each pair (n/2 apart) on all (broken) nagonals sum up to (m^{n}+1) 

antimagic  all sums range form a set of 2m+2 consecutive integers  
quadrant magic 
a square is said to be quadrant magic for certain patterns if those patterns of m numbers defined in one quadrant is repeated in all four other quadrants and all these patterns sum to the magic sum 

Serrated Magic Hypercube  
This is a hypercube tilted by 45^{o} such that a corner is on top In practise this figure consists of a tilted hypercube of order m placed within the gaps of a tilted hypercube of order m + 1, thus forming a hypercube of order 2 m + 1 (as far as the author knows) the figure is called magic if it has 3 sums: S_{m+1} the sum of the m + 1 hypercube S_{m} the sum of the m hypercube S = S_{m} + S_{m+1} the sum of the hypercubes nagonals 