The Magic Encyclopedia ™

The Magic Hypercube
(by Aale de Winkel)

The Magic Hypercube is defined as a hypercube, filled with integers [0,..,mⁿ-1] (analitic numberrange), [1,..,mⁿ] (regular numberrange). For the additional ('+') type of hypercube the numberrange doesn't matter, for the other types some operations don't work well with '0' so the regular numberrange needs to be used there. The normal '+' type hypercube the analitic numberrange this author prefers for analitic reasons. The constant Hypercube ⁿ1_m can simply be added to move to the regular numberrange (when needed)
Depending on the function used to combine numbers on a "magic line" several function types of magic hypercubes can be defined. Next to the addition magic hypercubes one thus can define subtraction, multiplication and division magic hypercubes, next to these of course this list can be expanded as suggested by the text below and the qualification article. Only the four types mentioned have been studied as they are defined below. Variations on the subtraction and division hypercube are also defined but currently also not studied.
Regularly the constant formula is observer on each 1-agonal and each n-agonal, leading to the magic constant

The Magic Hypercube
The hypercube is qualified as magic if all 1-agonals and all n-agonals sum to the same number
for a regular magic hypercube this number is 'the magic sum' S = m (mⁿ + 1) / 2

^mIN-Magic Hypercube {[_ji] ε [1..mⁿ] | C = f_n-1; f_i = function(x_i,f_i-1) i = 1..n-1; f₀ = x₀}
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: a-b 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!(a-b)!) at each step
(other functions definable)

for the remainder '+' and sum is assumed and we can move to:
[_ji] ε [0..mⁿ-1]

semi-magic the bodys n-agonals 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

pan-r-agonal magic all sums on all (broken) r-agonals sum up to the magic sum
panmagic
pandiagonal magic
pantriagonal magic
panquadragonal magic pan-n-agonal magic
pan-2-agonal magic
pan-3-agonal magic
pan-4-agonal magic

perfect pan-r-agonal for all r = 1 .. n

p-multimagic the hypercube sums up to a magic number when all number
are raised to the power q for all q = 1 .. p
bimagic
trimagic 2-multimagic
3-multimagic

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

most-perfect special magic condition defined by Kathleen Ollerenshaw and David Brée for squares
each order 2 Hypercube sums to 2 * (mⁿ+1) and each pair (n/2 apart) on
all (broken) n-agonals sum up to (mⁿ+1)

anti-magic 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 n-agonals

The Magic Hypercube
The hypercube is qualified as magic if all 1-agonals and all n-agonals sum to the same number for a regular magic hypercube this number is 'the magic sum' S = m (mⁿ + 1) / 2
^mIN-Magic Hypercube {[_ji] ε [1..mⁿ] \| C = f_n-1; f_i = function(x_i,f_i-1) i = 1..n-1; f₀ = x₀} 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: a-b 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!(a-b)!) at each step (other functions definable)
for the remainder '+' and sum is assumed and we can move to: [_ji] ε [0..mⁿ-1]
semi-magic	the bodys n-agonals 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
pan-r-agonal magic	all sums on all (broken) r-agonals sum up to the magic sum
pan-r-agonal magic	panmagic pandiagonal magic pantriagonal magic panquadragonal magic	pan-n-agonal magic pan-2-agonal magic pan-3-agonal magic pan-4-agonal magic
perfect	pan-r-agonal for all r = 1 .. n
p-multimagic	the hypercube sums up to a magic number when all number are raised to the power q for all q = 1 .. p
p-multimagic	bimagic trimagic	2-multimagic 3-multimagic
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
most-perfect	special magic condition defined by Kathleen Ollerenshaw and David Brée for squares each order 2 Hypercube sums to 2 * (mⁿ+1) and each pair (n/2 apart) on all (broken) n-agonals sum up to (mⁿ+1)
anti-magic	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 n-agonals