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Inmathematics, amagic hypercube is thek-dimensional generalization ofmagic squares andmagic cubes, that is, ann ×n ×n × ... ×n array ofintegers such that the sums of the numbers on each pillar (along any axis) as well as on the mainspace diagonals are all the same. The common sum is called themagic constant of the hypercube, and is sometimes denotedMk(n). If a magic hypercube consists of the numbers 1, 2, ...,nk, then it has magic number
Fork = 4, a magic hypercube may be called amagic tesseract, with sequence of magic numbers given byOEIS: A021003.
The side-lengthn of the magic hypercube is called itsorder. Four-, five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed byJ. R. Hendricks.
Marian Trenkler proved the following theorem:Ap-dimensional magic hypercube of ordern exists if and only ifp > 1 andn is different from 2 orp = 1. A construction of a magic hypercube follows from the proof.
TheR programming language includes a module,library(magic), that will create magic hypercubes of any dimension withn a multiple of 4.
If, in addition, the numbers on everycross section diagonal also sum up to the hypercube's magic number, the hypercube is called aperfect magic hypercube; otherwise, it is called asemiperfect magic hypercube. The numbern is called the order of the magic hypercube.
This definition of "perfect" assumes that one of the older definitions for perfect magic cubes is used. TheUniversal Classification System for Hypercubes (John R. Hendricks) requires that for any dimension hypercube,all possible lines sum correctly for the hypercube to be consideredperfect magic. Because of the confusion with the termperfect,nasik is now the preferred term forany magic hypercube whereall possible lines sum toS. Nasik was defined in this manner by C. Planck in 1905. A nasik magic hypercube has1/2(3n − 1) lines ofm numbers passing through each of themn cells.
ANasik magic hypercube is a magic hypercube with the added restriction that all possible lines through each cell sum correctly toS =m(mn+1)/2 whereS is the magic constant,m the order andn the dimension of the hypercube.
Or, to put it more concisely, all pan-r-agonals sum correctly forr = 1...n. This definition is the same as the Hendricks definition ofperfect, but different from the Boyer/Trump definition.
The termnasik would apply to all dimensions of magic hypercubes in which the number of correctly summing paths (lines) through any cell of the hypercube isP =3n − 1/2.
Apandiagonal magic square then would be anasik square because 4 magic line pass through each of them2 cells. This was A.H. Frost’s original definition of nasik. Anasik magic cube would have 13 magic lines passing through each of itsm3 cells. (This cube also contains 9m pandiagonal magic squares of orderm.) Anasik magic tesseract would have 40 lines passing through each of itsm4 cells, and so on.
In 1866 and 1878, Rev. A. H. Frost coined the termNasik for the type of magic square we commonly callpandiagonal and often callperfect. He then demonstrated the concept with an order-7 cube we now class aspandiagonal, and an order-8 cube we class aspantriagonal.[1][2] In another 1878 paper he showed anotherpandiagonal magic cube and a cube where all 13m lines sum correctly[3] i.e. Hendricksperfect.[4]He referred to all of these cubes asnasik as a respect to the great Indian MathematicianD R Kaprekar who hails fromDeolali inNasik District inMaharashtra,India.In 1905 Dr. Planck expanded on the nasik idea in his Theory of Paths Nasik. In the introductory to his paper, he wrote;
Analogy suggest that in the higher dimensions we ought to employ the term nasik as implying the existence of magic summations parallel to any diagonal, and not restrict it to diagonals in sections parallel to the plane faces. The term is used in this wider sense throughout the present paper.
— C. Planck, M.A., M.R.C.S., The Theory of Paths Nasik, 1905[5]
In 1917, Dr. Planck wrote again on this subject.
It is not difficult to perceive that if we push the Nasik analogy to higher dimensions the number of magic directions through any cell of a k-fold must be ½(3k-1).
— W. S. Andrews, Magic Squares and Cubes, Dover Publ., 1917, page 366[6]
In 1939, B. Rosser and R. J. Walker published a series of papers on diabolic (perfect) magic squares and cubes. They specifically mentioned that these cubes contained 13m2 correctly summing lines. They also had 3m pandiagonal magic squares parallel to the faces of the cube, and 6m pandiagonal magic squares parallel to thespace-diagonal planes.[7]
in order to keep things in hand a special notation was developed:
Note: The notation for position can also be used for the value on that position. Then, where it is appropriate, dimension and order can be added to it, thus forming:n[ki]m
As is indicatedk runs through the dimensions, while the coordinatei runs through all possible values, when valuesi are outside the range it is simply moved back into the range by adding or subtracting appropriate multiples ofm, as the magic hypercube resides in n-dimensional modular space.
There can be multiplek between brackets, these cannot have the same value, though in undetermined order, which explains the equality of:
Of course givenk also one valuei is referred to.
When a specific coordinate value is mentioned the other values can be taken as 0, which is especially the case when the amount of 'k's are limited using pe. #k = 1 as in:
("axial"-neighbor of )
(#j=n-1 can be left unspecified) j now runs through all the values in [0..k-1,k+1..n-1].
Further: without restrictions specified 'k' as well as 'i' run through all possible values, in combinations same letters assume same values. Thus makes it possible to specify a particular line within the hypercube (see r-agonal in pathfinder section)
Note: as far as I know this notation is not in general use yet(?), Hypercubes are not generally analyzed in this particular manner.
Further: "perm(0..n-1)" specifies apermutation of the n numbers 0..n-1.
Besides more specific constructions two more general construction method are noticeable:
This construction generalizes the movement of the chessboard horses (vectors) to more general movements (vectors). The method starts at the position P0 and further numbers are sequentially placed at positions further until (after m steps) a position is reached that is already occupied, a further vector is needed to find the next free position. Thus the method is specified by the n by n+1 matrix:
This positions the number 'k' at position:
C. Planck gives in his 1905 article"The theory of Path Nasiks" conditions to create with this method "Path Nasik" (or modern {perfect}) hypercubes.
(modular equations).This method is also specified by an n by n+1 matrix. However this time it multiplies the n+1 vector [x0,..,xn-1,1], After this multiplication the result is taken modulus m to achieve the n (Latin) hypercubes:
LPk = (l=0Σn-1 LPk,l xl + LPk,n ) % m
of radix m numbers (also called "digits"). On these LPk's "digit changing" (?i.e. Basic manipulation) are generally applied before these LPk's are combined into the hypercube:
nHm =k=0Σn-1 LPk mk
J.R.Hendricks often uses modular equation, conditions to make hypercubes of various quality can be found onhttp://www.magichypercubes.com/Encyclopedia at several places (especially p-section)
Both methods fill the hypercube with numbers, the knight-jump guarantees (given appropriate vectors) that every number is present. The Latin prescription only if the components are orthogonal (no two digits occupying the same position)
Amongst the various ways of compounding, the multiplication[8] can be considered as the most basic of these methods. Thebasic multiplication is given by:
nHm1 *nHm2 :n[ki]m1m2 =n[ [[ki \ m2]m1m1n]m2 + [ki % m2]m2]m1m2
Most compounding methods can be viewed as variations of the above, As most qualifiers are invariant under multiplication one can for example place any aspectual variant ofnHm2 in the above equation, besides that on the result one can apply a manipulation to improve quality. Thus one can specify pe the J. R. Hendricks / M. Trenklar doubling. These things go beyond the scope of this article.
A hypercube knowsn! 2n Aspectial variants, which are obtained by coordinate reflection ([ki] --> [k(-i)]) and coordinate permutations ([ki] --> [perm[k]i]) effectively giving the Aspectial variant:
nHm~R perm(0..n-1); R =k=0Σn-1 ((reflect(k)) ? 2k : 0) ; perm(0..n-1) a permutation of 0..n-1
Where reflect(k) true iff coordinate k is being reflected, only then 2k is added to R.As is easy to see, only n coordinates can be reflected explaining 2n, the n! permutation of n coordinates explains the other factor to the total amount of "Aspectial variants"!
Aspectial variants are generally seen as being equal. Thus any hypercube can be represented shown in"normal position" by:
[k0] = min([kθ ; θ ε {-1,0}]) (by reflection)[k1 ; #k=1] < [k+11 ; #k=1] ; k = 0..n-2 (by coordinate permutation)(explicitly stated here: [k0] the minimum of all corner points. The axial neighbour sequentially based on axial number)
Besides more specific manipulations, the following are of more general nature
Note: '#', '^', '_' and '=' are essential part of the notation and used as manipulation selectors.
Defined as the exchange of components, thus varying the factor mk in mperm(k), because there are n component hypercubes the permutation is over these n components
The exchange of coordinate [ki] into [perm(k)i], because of n coordinates a permutation over these n directions is required. The termtranspose (usually denoted byt) is used with two dimensional matrices, in general though perhaps "coordinate permutation" might be preferable.
Defined as the change of [ki] into [kperm(i)] alongside the given "axial"-direction. Equal permutation along various axes can be combined by adding the factors 2axis. Thus defining all kinds of r-agonal permutations for any r. Easy to see that all possibilities are given by the corresponding permutation of m numbers.
Noted be thatreflection is the special case:
~R = _R[n-1,..,0]
Further when all the axes undergo the same permutation (R = 2n-1) ann-agonal permutation is achieved, In this special case the 'R' is usually omitted so:
_[perm(0..n-1)] = _(2n-1)[perm(0..n-1)]
Usually being applied at component level and can be seen as given by[ki] inperm([ki]) since a component is filled with radix m digits, a permutation over m numbers is an appropriate manner to denote these.
J. R. Hendricks called the directions within a hypercubes "pathfinders", these directions are simplest denoted in a ternary number system as:
Pfp where: p =k=0Σn-1 (ki + 1) 3k <==> <ki> ; i ε {-1,0,1}This gives 3n directions. since every direction is traversed both ways one can limit to the upper half [(3n-1)/2,..,3n-1)] of the full range.
With these pathfinders any line to be summed over (or r-agonal) can be specified:
[j0kplq ; #j=1 #k=r-1 ; k > j ] <j1kθl0 ; θ ε {-1,1} > ; p,q ε [0,..,m-1]which specifies all (broken) r-agonals, p and q ranges could be omitted from this description. The main (unbroken) r-agonals are thus given by the slight modification of the above:
[j0k0l-1sp ; #j=1 #k+#l=r-1 ; k,l > j ] <j1k1l-1s0 >
A hypercubenHm with numbers in the analytical numberrange [0..mn-1] has the magic sum:
nSm = m (mn - 1) / 2.
Besides more specific qualifications the following are the most important, "summing" of course stands for "summing correctly to the magic sum"
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 usingFaulhaber's formula and divide it by mn-1.
Also "magic" (i.e. {1-agonal n-agonal}) is usually assumed, theTrump/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} might be put in notation as :(k)Σ [ji +k1] = 2nnSm / m. {ncomplete} can simply be written as:[ji] + [ji +k(m/2) ; #k=n ] = mn - 1 where:
for {complete} the complement of [ji] is at position [ji +k(m/2) ; #k=n ].
for squares: {2compact2complete} is the "modern/alternative qualification" of what DameKathleen Ollerenshaw calledmost-perfect magic square, {ncompactncomplete} 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-superscriptn 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.
Amagic hyperbeam (n-dimensional magic rectangle) is a variation on a magic hypercube where the orders along each direction may be different. As such amagic hyperbeam generalises the two dimensionalmagic rectangle and the three dimensionalmagic beam, a series that mimics the seriesmagic square,magic cube and magic hypercube. This article will mimic themagic hypercubes article in close detail, and just as that article serves merely as an introduction to the topic.
It is customary to denote thedimension with the letter 'n' and theorders of a hyperbeam with the letter 'm' (appended with the subscripted number of the direction it applies to).
Further: In this article the analytical number range [0..k=0Πn-1mk-1] is being used.
in order to keep things in hand a special notation was developed:
Note: The notation for position can also be used for the value on that position. There where it is appropriate dimension and orders can be added to it thus forming:n[ki]m0,..,mn-1
Description of more general methods might be put here, I don't often create hyperbeams, so I don't know whether Knightjump or Latin Prescription work here.Other more adhoc methods suffice on occasion I need a hyperbeam.
Amongst the various ways of compounding, the multiplication[9] can be considered as the most basic of these methods. Thebasic multiplication is given by:
(m..) abbreviates: m0,..,mn-1. (m..)1(m..)2 abbreviates: m01m02,..,mn-11mn-12.
A fact that can be easily seen since the magic sums are:
When any of the orders mk is even, the product is even and thus the only way Sk turns out integer is when all mk are even. Thus suffices: all mk are either even or odd.
This is with the exception of mk=1 of course, which allows for general identities like:
Which goes beyond the scope of this introductory article
since any number has but one complement only one of the directions can have mk = 2.
A hyperbeam knows2n Aspectial variants, which are obtained by coördinate reflection ([ki] → [k(-i)]) effectively giving the Aspectial variant:
nB(m0..mn-1)~R ; R =k=0Σn-1 ((reflect(k)) ? 2k : 0) ;
Where reflect(k) true if and only if coordinate k is being reflected, only then 2k is added to R.
In case one views different orientations of the beam as equal one could view the number of aspectsn! 2n just as with themagic hypercubes, directions with equal orders contribute factors depending on the hyperbeam's orders. This goes beyond the scope of this article.
Besides more specific manipulations, the following are of more general nature
Note: '^' and '_' are essential part of the notation and used as manipulation selectors.
The exchange of coördinaat [ki] into [perm(k)i], because of n coördinates a permutation over these n directions is required. The termtranspose (usually denoted byt) is used with two dimensional matrices, in general though perhaps "coördinaatpermutation" might be preferable.
Defined as the change of [ki] into [kperm(i)] alongside the given "axial"-direction. Equal permutation along various axes with equal orders can be combined by adding the factors 2axis. Thus defining all kinds of r-agonal permutations for any r. Easy to see that all possibilities are given by the corresponding permutation of m numbers.
In case no restrictions are considered on the n-agonals a magic hyperbeam can be represented shown in"normal position" by:
Qualifying the hyperbeam is less developed then it is on themagic hypercubes in fact only the k'th monagonal direction need to sum to:
for all k = 0..n-1 for the hyperbeam to be qualified {magic}
When the orders are not relatively prime the n-agonal sum can be restricted to:
with all orders relatively prime this reaches its maximum:
The following hyperbeams serve special purposes:
This hyperbeam can be seen as the source of all numbers. A procedure called"Dynamic numbering" makes use of theisomorphism of every hyperbeam with this normal, changing the source, changes the hyperbeam. Basic multiplications of normal hyperbeams play a special role with the"Dynamic numbering" ofmagic hypercubes of orderk=0Πn-1 mk.
The hyperbeam that is usually added to change the here used "analytic" number range into the "regular" number range. Other constant hyperbeams are of course multiples of this one.
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