Magic Encyclopedia (Qualification) magic square, magic cube, magic tesseract, magic hypercube,

The Magic Encyclopedia ™

Qualifications of objects
(by Aale de Winkel)

NOTE: Article currently rewritten, when finished this note wil be removed. Meanwhile I welcome sugestions, though notice these are definitions I use. Disputed terms I'll decide upon and make note as to its use with other meaning. Notification of this I also welcome.

Qualifying an object is one of the major tasks of the research in these fields, this article intents to define the qualifiers around more precisely and will be this authors reference in future uploads of either site pages or programs
The suggested notational effort (listed below the defined term) can easily be overridden by contextual definitions. The base letter (here listed as C) can simply be redefined in context to include certain investigated feature, In case one investigates several different features in the same text, features can be attributed onto different base letters, which then ought to be defined in the context.
the seperation between major and minor qualification is quite arbitrary, the major qualifications are suggested to be included in the base letter, while the minor qualifiers are suggested to be comma seperated between curly brackets (depending on context either might be appropriate)
At present notice that the qualifiers listed below isn't complete yet, features will be added when I come across them. Also the named compound hypercube qualifications will grow once I know them. The reader is invited to contribute to this listing

Many thanks to <John R Hendricks> This article as well as others benifited from email discussions.
Also to <Harvey Heinz> for his refinding of the definitions of the function types of magic squares in:
Spencer, D., Game Playing with Computers, Hayden Book Company, Hasbrouck Heights, NJ, 1975.
(here augmented to their hypercube variants, the binom function I added to these as a suggestion)

Recent discussion about the definitions "perfect" are about the old definition (Benson/Jacobi) and the new definition (Hendricks). While the first generalized the term for the square toward the cube by sticking to the planar view, while the latter asserts the magic summing of all the triagonals as well, which is easy to generalize toward the hypercube (see below). My use of the term is the same as Hendricks's (ie pan r-agonal for r = 1..n).
To accomodate the perfectness of lower dimensional subhypercubes I defined hereunder ^rperfect which might be used to assert the perfectness (Hendricks) of subhypercubes. The old use of perfect is thus equivalent to ²perfect. As said all my uses of "perfect" are equivalent with ⁿperfect

The magic Ratio came but recently into the picture as the term for specifying the fraction of magic item. This discussion was instigated by <Walter Trump> see also: magic-squares

Some terms I know the originator, and therefore added his/hers name, use of {(pan) r-agonal} as qualifiers are in general instigated by <John R Hendricks> with a few addition by others (pe this author).

Hypercube Qualifications
Various qualifications of the hypercube are around, some qualificatons changed in recent years
their meaning, ("perfect" changed from pan n-agonal to pan r-agonal 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 most-perfect only for squares. Presently the equivalent term ²compact ²complete
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) r-agonal}" 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 1-agonal and n-agonal
Thus defines the 'function magic constant' as C = f_n-1; f_i = function(x_i,f_i-1) i = 1..n-1; f₀ = x₀

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)

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 non-integer or doubly appearing numbers

major Qualifications
p-MultiMagic all 1-agonals and n-agonals sum to the same sum when all numbers are raised to
all powers from 1 to p
bimagic
trimagic
tetramagic
pentamagic 2-MultiMagic
3-MultiMagic
4-MultiMagic
5-MultiMagic
The p-Multimagic 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.

{r-agonal} the unbroken r-agonal result in the constant

{pan r-agonal} all r-agonals (broken and unbroken) result in the constant
Although the hypercubes monagonal are not broken {pan 1-agonal} 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 r-agonals results
in the certain constant (which depends on type, numberranges and more such factors
The "major qualifiers" are abbreviations of the r-agonal constants. The below listed
define the abbreviations used by this author.
although in principle unneccesairy, the term is explained more thoroughly

magic {1-agonal n-agonal}
All 1-agonals and unbroken n-agonals

semi-magic {1-agonal}
All 1-agonals and at least one of the
unbroken n-agonals is NOT resulting in the constant

pan magic {1-agonal pan n-agonal}
All 1-agonals and broken and unbroken n-agonals

strictly-magic {r-agonal for r = 1 .. n}
This term is currently not in use yet, stating with a simple term that all unbroken
r-agonals are resulting in the constant might come in handy

perfect {pan r-agonal 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

s-pan r-agonal every s broken r-agonals are summing to the magic sum
pan r-agonal
pan
s-pan 1-pan r-agonal
1-pan n-agonal
s-pan n-agonal

semi-pan r-agonal opposite short r-agonals are summing to the magic sum
opposite short r-agonals lines // the hypercubes r-agonals of length m/2 (m even)
or of length (m-1)/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)

semi-pan semi-pan n-agonals

^rperfect pan q-agonal for all q = 1 .. r
perfect ⁿperfect (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

^rcompact_s r-dimensional order s subhypercube corners is summing to the same sum: 2^r-1 * (m^r+1)
^rcompact
compact ^rcompact₂
ⁿcompact₂

see also Compact (c-section)

^rcomplete r-dimensional order 2 subhypercube r-agonal m/2 apart cells are summing to the same sum: (m^r+1)
complete ⁿcomplete

Ratio The fraction of magic items within the total set of items
the relaevant items needs to be specified
r-agonal Ratio the fraction of magic r-agonal 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^n-1 * (mⁿ+1) and
each pair (n/2 apart) on all (broken) n-agonals sum up to (mⁿ+1)
most-perfect term defined by Kathleen Ollerenshaw and David Brée for squares
for the squares, equivalent to: ²compact ²complete
(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 non-trivial number.
functions
+
-
*
/
binom
.... Plausible interesting functions:
additional magic: a+b performed on each pair (a,b)
subtractional magic: a-b 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!(a-b)!) 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 n-Point and apropriate n-Vectors
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 one-another (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 n-Points resp. n-Vectors contains n numbers in the range [0,..,n-1].

qualified hypercubes
ⁿC_m{q1,..qp} the various qualifications are listed between curly brackets

subhypercubes
ⁿC_m{..} [
c₁{..}^[..]<..>,..,
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 n-Point, and might be oriented
by listing their necessairy n-Vectors to orient the subhypercube

Hypercube Qualifications
This table show just a few fictitious samples of the above defined suggested notation.

²C₆ [
²C₄{pan}^[1,1]
²C₃^[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)

³C₈{perfect} [
²C₅{quadrant("+")}^{[1,2,3]<1,0,0><0,1,0>} ] A perfect order 8 cubes with a plus-magic 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:

ⁿC_m1.m2{perfect} == ⁿC_m1{perfect} * ⁿC_m2{perfect} == ⁿC_m1{perfect} ([ⁿC_m2{perfect}] - 1) m1ⁿ ==
ⁿC_m1.m2{perfect} [ⁿC_m1{perfect}^[j^i]; i = 0..m2-1 ; j = 0..n-1 ]

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)

Hypercube Qualifications
Various qualifications of the hypercube are around, some qualificatons changed in recent years their meaning, ("perfect" changed from pan n-agonal to pan r-agonal 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 most-perfect only for squares. Presently the equivalent term ²compact ²complete 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) r-agonal}" 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 1-agonal and n-agonal Thus defines the 'function magic constant' as C = f_n-1; f_i = function(x_i,f_i-1) i = 1..n-1; f₀ = x₀
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)
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
Regular	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 non-integer or doubly appearing numbers
major Qualifications
p-MultiMagic	all 1-agonals and n-agonals sum to the same sum when all numbers are raised to all powers from 1 to p
	bimagic trimagic tetramagic pentamagic	2-MultiMagic 3-MultiMagic 4-MultiMagic 5-MultiMagic
	The p-Multimagic 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.
{r-agonal}	the unbroken r-agonal result in the constant
{pan r-agonal}	all r-agonals (broken and unbroken) result in the constant
{pan r-agonal}	Although the hypercubes monagonal are not broken {pan 1-agonal} 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 r-agonals results in the certain constant (which depends on type, numberranges and more such factors The "major qualifiers" are abbreviations of the r-agonal constants. The below listed define the abbreviations used by this author. although in principle unneccesairy, the term is explained more thoroughly
magic	{1-agonal n-agonal}
magic	All 1-agonals and unbroken n-agonals
semi-magic	{1-agonal}
semi-magic	All 1-agonals and at least one of the unbroken n-agonals is NOT resulting in the constant
pan magic	{1-agonal pan n-agonal}
pan magic	All 1-agonals and broken and unbroken n-agonals
strictly-magic	{r-agonal for r = 1 .. n}
strictly-magic	This term is currently not in use yet, stating with a simple term that all unbroken r-agonals are resulting in the constant might come in handy
perfect	{pan r-agonal for r = 1 .. n}
perfect	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
s-pan r-agonal	every s broken r-agonals are summing to the magic sum
s-pan r-agonal	pan r-agonal pan s-pan	1-pan r-agonal 1-pan n-agonal s-pan n-agonal
semi-pan r-agonal	opposite short r-agonals are summing to the magic sum
	opposite short r-agonals	lines // the hypercubes r-agonals of length m/2 (m even) or of length (m-1)/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)
	semi-pan	semi-pan n-agonals
^rperfect	pan q-agonal for all q = 1 .. r
	perfect	ⁿperfect (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
^rcompact_s	r-dimensional order s subhypercube corners is summing to the same sum: 2^r-1 * (m^r+1)
	^rcompact compact	^rcompact₂ ⁿcompact₂
	see also	Compact (c-section)
^rcomplete	r-dimensional order 2 subhypercube r-agonal m/2 apart cells are summing to the same sum: (m^r+1)
^rcomplete	complete	ⁿcomplete
Ratio	The fraction of magic items within the total set of items the relaevant items needs to be specified
Ratio	r-agonal Ratio	the fraction of magic r-agonal 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^n-1 * (mⁿ+1) and each pair (n/2 apart) on all (broken) n-agonals sum up to (mⁿ+1)
compact complete	most-perfect	term defined by Kathleen Ollerenshaw and David Brée for squares for the squares, equivalent to: ²compact ²complete (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
Quadrant magic quadrant(pattern)	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 non-trivial number.
	functions + - * / binom ....	Plausible interesting functions: additional magic: a+b performed on each pair (a,b) subtractional magic: a-b 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!(a-b)!) 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 n-Point and apropriate n-Vectors 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 one-another (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
proper	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 n-Points resp. n-Vectors contains n numbers in the range [0,..,n-1].
qualified hypercubes ⁿC_m{q1,..qp}	the various qualifications are listed between curly brackets
subhypercubes ⁿC_m{..} [ c₁{..}^[..]<..>,.., 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 n-Point, and might be oriented by listing their necessairy n-Vectors to orient the subhypercube

Hypercube Qualifications
This table show just a few fictitious samples of the above defined suggested notation.
²C₆ [ ²C₄{pan}^[1,1] ²C₃^[3i,3j] ]	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)
³C₈{perfect} [ ²C₅{quadrant("+")}^{[1,2,3]<1,0,0><0,1,0>} ]	A perfect order 8 cubes with a plus-magic 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:
ⁿC_m1.m2{perfect} == ⁿC_m1{perfect} * ⁿC_m2{perfect} == ⁿC_m1{perfect} ([ⁿC_m2{perfect}] - 1) m1ⁿ == ⁿC_m1.m2{perfect} [ⁿC_m1{perfect}^[j^i]; i = 0..m2-1 ; j = 0..n-1 ]
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)