Magic hypercube

Last updated

In mathematics, a magic hypercube is the k-dimensional generalization of magic squares and magic cubes, that is, an n × n × n × ... × n array of integers such that the sums of the numbers on each pillar (along any axis) as well as on the main space diagonals are all the same. The common sum is called the magic constant of the hypercube, and is sometimes denoted Mk(n). If a magic hypercube consists of the numbers 1, 2, ..., nk, then it has magic number



For k = 4, a magic hypercube may be called a magic tesseract, with sequence of magic numbers given by OEIS:  A021003 .

The side-length n of the magic hypercube is called its order. Four-, five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by J. R. Hendricks.

Marian Trenkler proved the following theorem: A p-dimensional magic hypercube of order n exists if and only if p > 1 and n is different from 2 or p = 1. A construction of a magic hypercube follows from the proof.

The R programming language includes a module, library(magic), that will create magic hypercubes of any dimension with n a multiple of 4.

Perfect and Nasik magic hypercubes

If, in addition, the numbers on every cross section diagonal also sum up to the hypercube's magic number, the hypercube is called a perfect magic hypercube; otherwise, it is called a semiperfect magic hypercube. The number n is called the order of the magic hypercube.

The above definition of "perfect" assumes that one of the older definitions for perfect magic cubes is used. See Magic Cube Classes. The Universal Classification System for Hypercubes (John R. Hendricks) requires that for any dimension hypercube, all possible lines sum correctly for the hypercube to be considered perfect magic. Because of the confusion with the term perfect, nasik is now the preferred term for any magic hypercube where all possible lines sum to S. Nasik was defined in this manner by C. Planck in 1905. A nasik magic hypercube has 1/2(3n 1) lines of m numbers passing through each of the mn cells.


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 indicated 'k' runs through the dimensions, while the coordinate 'i' runs through all possible values, when values 'i' are outside the range it is simply moved back into the range by adding or subtracting appropriate multiples of m, as the magic hypercube resides in n-dimensional modular space.

There can be multiple 'k' between bracket, these can't have the same value, though in undetermined order, which explains the equality of:

Of course given 'k' also one value 'i' 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 a permutation of the n numbers 0..n-1.


Besides more specific constructions two more general construction method are noticeable:

KnightJump construction

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.

Latin prescription construction

(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 on 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 [1] can be considered as the most basic of these methods. The basic 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 aspectial variant of nHm2 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 knows n! 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)

Basic manipulations

Besides more specific manipulations, the following are of more general nature

Note: '#', '^', '_' and '=' are essential part of the notation and used as manipulation selectors.

Component permutation

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

Coordinate permutation

The exchange of coordinate [ki] into [perm(k)i], because of n coordinates a permutation over these n directions is required.
The term transpose (usually denoted by t) is used with two dimensional matrices, in general though perhaps "coordinate permutation" might be preferable.

Monagonal permutation

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 that reflection is the special case:

~R = _R[n-1,..,0]

Further when all the axes undergo the same ;permutation (R = 2n-1) an n-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] in perm([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:

[ j0 kp lq ; #j=1 #k=r-1 ; k > j ] <j1 kθ l0 ; 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:

[ j0 k0 l-1 sp ; #j=1 #k+#l=r-1 ; k,l > j ] <j1 k1 l-1 s0 >


A hypercube nHm with numbers in the analytical numberrange [] 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 using Faulhaber's formula and divide it by mn-1.

Also "magic" (i.e. {1-agonal n-agonal}) is usually assumed, the Trump/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.
(k)Σ is symbolic for summing all possible k's, there are 2n possibilities for k1.
[ji + k1] expresses [ji] and all its r-agonal neighbors.
for {complete} the complement of [ji] is at position [ji + k(m/2) ; #k=n ].

for squares: {2compact 2complete} is the "modern/alternative qualification" of what Dame Kathleen Ollerenshaw called most-perfect magic square, {ncompact ncomplete} 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-superscript n 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.

See also

Related Research Articles

In mathematics, the geometric algebra (GA) of a vector space with a quadratic form is an algebra over a field, the Clifford algebra of a vector space with a quadratic form with its multiplication operation called the geometric product. The algebra elements are called multivectors, which contains both the scalars and the vector space .

In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality in many mathematical fields, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics.

Hypercube Convex polytope, the n-dimensional analogue of a square and a cube

In geometry, a hypercube is an n-dimensional analogue of a square and a cube. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to .

In quantum mechanics, a density matrix is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.

In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant.

Magic cube

In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal to the same number, the so-called magic constant of the cube, denoted M3(n). It can be shown that if a magic cube consists of the numbers 1, 2, ..., n3, then it has magic constant

In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars, and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant.

In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons. Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity.

The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch–Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart, who developed the formalism as a link between the symmetry transformation groups of space and the laws of conservation of energy, momentum, and angular momentum.

In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations. The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory.

Space diagonal

In geometry, a space diagonal of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with face diagonals, which connect vertices on the same face as each other.

Every magic cube may be assigned to one of six magic cube classes, based on the cube characteristics.

A pantriagonal magic cube is a magic cube where all 4m2 pantriagonals sum correctly. There are 4 one-segment, 12(m − 1) two-segment, and 4(m − 2)(m − 1) three-segment pantriagonals. This class of magic cubes may contain some simple magic squares and/or pandiagonal magic squares, but not enough to satisfy any other classifications.

The Peres–Horodecki criterion is a necessary condition, for the joint density matrix of two quantum mechanical systems and , to be separable. It is also called the PPT criterion, for positive partial transpose. In the 2x2 and 2x3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply.

In quantum field theory and statistical mechanics, the Mermin–Wagner theorem states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2. Intuitively, this means that long-range fluctuations can be created with little energy cost and since they increase the entropy they are favored.

In combinatorics, the Eulerian numberA(n, m) is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element. They are the coefficients of the Eulerian polynomials:

Hypercube graph

In graph theory, the hypercube graphQn is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cubical graph Q3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Qn has 2n vertices, 2n−1n edges, and is a regular graph with n edges touching each vertex.

A Nasik magic hypercube is a magic hypercube with the added restriction that all possible lines through each cell sum correctly to where S = the magic constant, m = the order and n = the dimension, of the hypercube.

A magic hyperbeam is a variation on a magic hypercube where the orders along each direction may be different. As such a magic hyperbeam generalises the two dimensional magic rectangle and the three dimensional magic beam, a series that mimics the series magic square, magic cube and magic hypercube. This article will mimic the magic hypercubes article in close detail, and just as that article serves merely as an introduction to the topic.


The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.


  1. this is a n-dimensional version of (pe.): Alan Adler magic square multiplication

Further reading