Magic cube classes

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

Contents

This new system is more precise in defining magic cubes. But possibly of more importance, it is consistent for all orders and all dimensions of magic hypercubes.

Minimum requirements for a cube to be magic are: All rows, columns, pillars, and 4 triagonals must sum to the same value.

The six classes

The minimum requirements for a magic cube are: All rows, columns, pillars, and 4 triagonals must sum to the same value. A Simple magic cube contains no magic squares or not enough to qualify for the next class.
The smallest normal simple magic cube is order 3. Minimum correct summations required = 3m2 + 4

It is conjectured that, for integers n ≥ 1 and m ≥ 1, simple n-dimensional magic hypercube of order m exists if and only if either m ≠ 2 or n = 1 (or both)

Each of the 3m planar arrays must be a simple magic square. The 6 oblique squares are also simple magic. The smallest normal diagonal magic cube is order 5.
These squares were referred to as ‘Perfect’ by Gardner and others! At the same time he referred to Langman’s 1962 pandiagonal cube also as ‘Perfect’.
Christian Boyer and Walter Trump now consider this and the next two classes to be Perfect. (See Alternate Perfect below).
A. H. Frost referred to all but the simple class as Nasik cubes.
The smallest normal diagonal magic cube is order 5. See Diagonal magic cube. Minimum correct summations required = 3m2 + 6m + 4

It is conjectured that, for integers n ≥ 1 and m ≥ 1, diagonal n-dimensional magic hypercube of order m exists if and only if either m = 1 or m ≥ 2n−1+1

All 4m2 pantriagonals must sum correctly (that is 4 one-segment, 12(m-1) two-segment, and 4(m-2)(m-1) three-segment). There may be some simple AND/OR pandiagonal magic squares, but not enough to satisfy any other classification.
The smallest normal pantriagonal magic cube is order 4. See Pantriagonal magic cube.
Minimum correct summations required = 7m2. All pan-r-agonals sum correctly for r = 1 and 3.

A cube of this class was first constructed in late 2004 by Mitsutoshi Nakamura. This cube is a combination Pantriagonal magic cube and Diagonal magic cube. Therefore, all main and broken triagonals sum correctly, and it contains 3m planar simple magic squares. In addition, all 6 oblique squares are pandiagonal magic squares. The only such cube constructed so far is order 8. It is not known what other orders are possible. See Pantriagdiag magic cube. Minimum correct summations required = 7m2 + 6m

ALL 3m planar arrays must be pandiagonal magic squares. The 6 oblique squares are always magic (usually simple magic). Several of them MAY be pandiagonal magic. Gardner also called this (Langman’s pandiagonal) a ‘perfect’ cube, presumably not realizing it was a higher class then Myer’s cube. See previous note re Boyer and Trump.
The smallest normal pandiagonal magic cube is order 7. See Pandiagonal magic cube.
Minimum correct summations required = 9m2 + 4. All pan-r-agonals sum correctly for r = 1 and 2.

ALL 3m planar arrays must be pandiagonal magic squares. In addition, ALL pantriagonals must sum correctly. These two conditions combine to provide a total of 9m pandiagonal magic squares.
The smallest normal perfect magic cube is order 8. See Perfect magic cube.

It is conjectured that, for integers n ≥ 1 and m ≥ 1, perfect n-dimensional magic hypercube of order m exists if and only if either m = 1 or m ≥ 2n and m is either odd number or divisible by 2n

Nasik; A. H. Frost (1866) referred to all but the simple magic cube as Nasik!
C. Planck (1905) redefined Nasik to mean magic hypercubes of any order or dimension in which all possible lines summed correctly.
i.e. Nasik is a preferred alternate, and less ambiguous term for the perfect class.
Minimum correct summations required = 13m2. All pan-r-agonals sum correctly for r = 1, 2 and 3.

Alternate Perfect Note that the above is a relatively new definition of perfect. Until about 1995 there was much confusion about what constituted a perfect magic cube (see the discussion under diagonal:)
. Included below are references and links to discussions of the old definition
With the popularity of personal computers it became easier to examine the finer details of magic cubes. Also more and more work was being done with higher dimension magic Hypercubes. For example, John Hendricks constructed the world's first Nasik magic tesseract in 2000. Classed as a perfect magic tesseract by Hendricks definition.

Generalized for All Dimensions

A magic hypercube of dimension n is perfect if all pan-n-agonals sum correctly. Then all lower dimension hypercubes contained in it are also perfect.
For dimension 2, The Pandiagonal Magic Square has been called perfect for many years. This is consistent with the perfect (nasik) definitions given above for the cube. In this dimension, there is no ambiguity because there are only two classes of magic square, simple and perfect.
In the case of 4 dimensions, the magic tesseract, Mitsutoshi Nakamura has determined that there are 18 classes. He has determined their characteristics and constructed examples of each. And in this dimension also, the Perfect (nasik) magic tesseract has all possible lines summing correctly and all cubes and squares contained in it are also nasik magic.

Another definition and a table

Proper: A Proper magic cube is a magic cube belonging to one of the six classes of magic cube, but containing exactly the minimum requirements for that class of cube. i.e. a proper simple or pantriagonal magic cube would contain no magic squares, a proper diagonal magic cube would contain exactly 3m + 6 simple magic squares, etc. This term was coined by Mitsutoshi Nakamura in April, 2004.

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Notes for table

  1. For the diagonal or pandiagonal classes, one or possibly 2 of the 6 oblique magic squares may be pandiagonal magic. All but 6 of the oblique squares are ‘broken’. This is analogous to the broken diagonals in a pandiagonal magic square. i.e. Broken diagonals are 1-D in a 2_D square; broken oblique squares are 2-D in a 3-D cube.
  2. The table shows the minimum lines or squares required for each class (i.e. Proper). Usually there are more, but not enough of one type to qualify for the next class.

See also

Related Research Articles

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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

Tesseract Four-dimensional analogue of the cube

In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

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 .

Magic square Sums of each row, column, and main diagonals are equal

In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The order of the magic square is the number of integers along one side (n), and the constant sum is called the magic constant. If the array includes just the positive integers , the magic square is said to be normal. Some authors take magic square to mean normal magic square.

Squaring the square

Squaring the square is the problem of tiling an integral square using only other integral squares. The name was coined in a humorous analogy with squaring the circle. Squaring the square is an easy task unless additional conditions are set. The most studied restriction is that the squaring be perfect, meaning the sizes of the smaller squares are all different. A related problem is squaring the plane, which can be done even with the restriction that each natural number occurs exactly once as a size of a square in the tiling. The order of a squared square is its number of constituent squares.

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 mathematics, a magic hypercube is the k-dimensional generalization of magic squares, magic cubes and magic tesseracts; that is, a number of integers arranged in an n × n × n × ... × n pattern such that the sum of the numbers on each pillar as well as the main space diagonals is equal to a single number, the so-called magic constant of the hypercube, denoted Mk(n). It can be shown that if a magic hypercube consists of the numbers 1, 2, ..., nk, then it has magic number

Magic constant

The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. In general where is the side length of the square.

Most-perfect magic square

A most-perfect magic square of doubly even order n = 4k is a pan-diagonal magic square containing the numbers 1 to n2 with three additional properties:

  1. Each 2×2 subsquare, including wrap-round, sums to s/k, where s = n(n2 + 1)/2 is the magic sum.
  2. All pairs of integers distant n/2 along any diagonal are complementary.

John Robert Hendricks was a Canadian amateur mathematician specializing in magic squares and hypercubes. He published many articles in the Journal of Recreational Mathematics as well as other journals.

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.

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.

In mathematics, a diagonal 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 recreational mathematics, a pandiagonal magic cube is a magic cube with the additional property that all broken diagonals have the same sum as each other. Pandiagonal magic cubes are extensions of diagonal magic cubes and generalize pandiagonal magic squares to three dimensions.

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

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.

Simple magic cube

A simple magic cube is the lowest of six basic classes of magic cube. These classes are based on extra features required.

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.

In mathematics, an orthogonal array is a "table" (array) whose entries come from a fixed finite set of symbols, arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of the symbols, formed by taking the entries in each row restricted to these columns, appear the same number of times. The number t is called the strength of the orthogonal array. Here is a simple example of an orthogonal array with symbol set {1,2} and strength 2:

References

Further reading

Cube classes

Perfect Cube

Tesseract Classes