Hypercube Math

Introduction     Hypercube equations     Statistical tables     Modular equations     Comparative rarity

Introduction    

This page is a collection of mathematical expressions and tables that are involved with magic squares and cubes.  Emphasis will be put on relationships between magic hypercubes of different dimensions.The terms magic square, magic cube, magic tesseract, etc. will be used for specific dimensions.
The term magic hypercube will be used to indicate a magic rectilinear object in any dimension. 

Much of this material is taken from John R. Hendricks books which, unfortunately, are now out of print.
Some are available for free download at John Hendricks memorial website.
A small interactive spreadsheet program that shows statistics based on 3 input variables is available for download here. 
The Magic Square Lexicon is still available here.
Mitsutoshi Nakamura seems to be the only person doing extensive work with magic tesseracts at this time. He contributed much of the information on tesseract classes. His website is here.

It is appropriate to mention here, the different types of mathematics that may be involved in the investigation of magic hypercubes.

• ·        Arithmetic

• ·        Algebra

• ·        Geometry (in regard to coordinates, used in modular equations and paths).

• ·        Modular arithmetic, congruences

• ·        Different number systems (where radix = m)

• ·        Matrix arithmetic

 Variables used on this page

• m  =  order

• n  =  dimension

• r  =  represents all agonals from 1-agonal to n-agonals. i.e all lines in the hypercube.

• S  =  magic constant

• Nasik will be used to denote hypercubes where all lines through each cell sum correctly. This is an unambiguous term that
  avoids the confusion between Hendricks perfect and Boyer’s perfect. C. Planck set the precedent for this in his 1905 paper.

 C. Planck, The Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence, Printer, Rugby (England),1905
         (Available from The University Library, Cambridge).See a quotation here.
  W. S. Andrews, Magic Squares and Cubes. Open Court Publ.,1917. Pages 365,366, by Dr. C. Planck.
         Re-published by Dover Publ., 1960 (no ISBN); Dover Publ., 2000, 0486206580; Cosimo Classics, Inc., 2004, 1596050373

Hypercube equations

• Magic constant or Sum of a hypercube                        ….  S = {m(mⁿ + 1)} / 2

• Minimum sums required to be Simple magic              .....  S_s = 2^n-1 + nm^n-1

• Minimum sums required to be Nasik magic                ....   S_n = {(3ⁿ – 1)mⁿ-1} / 2

• Smallest order for a Nasik hypercube                          ....   2ⁿ  ....   2ⁿ + 1 if associated

• Paths (lines) through any cell of a hypercube               ….   P = (3ⁿ – 1) / 2

• Number of aspects (views) of a hypercube                  ….   A = 2ⁿ n!

• 1-agonals (orthogonal lines, i-rows) in a hypercube    ….  O = n(m^n-1)

• There are 2ⁿ corners and 2^n-1  n-agonals in a magic hypercube 

• Squares in a n-dimensional hypercube of order m     ….    N = {n(n – 1) / 2} m^n-2
  Of the above,   ….  n(n-1)2^n-3  are boundary squares.

• Edges in a n-dimensional magic hypercube                ….   E = n(2^n-1)

Hypercube tables

Hypercubes –Minimum number of correct summations

This table provides the minimum requirements for each category. Usually, there are some extra lines which may sum the magic sum, but not a complete set so as to change the category.
In this table I have replaced the term perfect (Hendricks) with Nasik.

This table is taken from The Magic Cube Lexicon, but edited with added tesseract material supplied by Mitsutoshi Nakamura Sept. 20, 2007.

First person shown to construct each minimum order cube and tesseract is to the best of my knowledge. If you have different information, please let me know. Minimum order for some of the tesseract classes shown has not yet been established. In that case I show the first constructer for the class.
John Hendricks was the first to publish all 58 order-3 magic tesseracts in (Magic Square Course, 2nd edition, 1992).

NOTE: I cannot testify as to the correctness of these tesseracts, as I have not had a chance to check out their features.

 H. D.  Heinz & J. R. Hendricks, Magic Square Lexicon: Illustrated, , HDH, 2000, 0-9687985-0-0, page 165 (edited)
 Mitsutoshi Nakamura’s tesseracts at http://homepage2.nifty.com/googol/magcube/en/

I show Hendricks as the first to publish a perfect magic tesseract. However, C. Planck showed 1 plane of a perfect order 16 octahedroid in his 1905 paper.
C. Planck, The Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence, Printer, Rugby (England),1905  (Available from The University Library, Cambridge).

Not all of these hyperplanes are magic unless the Hypercube is Nasik (Hendricks perfect).

J. R. Hendricks, Perfect n-Dimensional Magic Hypercubes of Order 2ⁿ, Self-published, 1999, 0-9684700-4-1 page 5.

 J. R. Hendricks, Magic Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9  page 134 (edited)

Digital equations

This is one method of finding solutions to magic hypercubes. This was a favored method of the late John R. Hendricks.

If the digits of a number can be expressed as a function of their coordinate location, then the equation(s) describing the relationship can be called digital equations. They are sometimes referred to as congruence equations or modular equations.

For example:  To solve the order 3 magic square (the Luo-shu).
If at coordinate location (1, 3) we wish to find the number and it is known that:
        D2 Ξ x + y        (mod 3)
And D1 Ξ 2x + y + 1   (mod 3

then the two digits D2 and D1 can be found.

        D2 Ξ 1 + 3 Ξ 4 Ξ 1 (mod 3)
And D1 Ξ 2 + 3 + 1 Ξ 6 Ξ 0 (mod 3)

So the number 10 (mod 3) is located at (1, 3).
To convert to a decimal number in the range of 1 to m;  3 * 1 + 0 + 1 = 4

J. R. Hendricks, Magic Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9  pp. 10-13
H. D.  Heinz & J. R. Hendricks, Magic Square Lexicon: Illustrated, , HDH, 2000, 0-9687985-0-0, page 35

Comparative rarity

A quick study reveals the futility of attempting to construct a magic hypercube by simply arranging the numbers randomly. That is, without using mathematical methods.

To illustrate this point, consider the easiest of all magic hypercubes to construct, the order 3 magic square.
The array uses just nine integers, which can be arranged in 9 factorial ways. 9 factorial (written 9!) = 1x2x3x4x5x6x7x8x9 = 362,880. There is 1 basic order 3 magic square, but it may be shown in 8 aspects (due to rotations and /or reflections). So the chance of stumbling on one of these 8 variations is 8/9! Or 1 chance in 45,360.

The next smallest hypercube is the order 4 magic square. There are 880 basic squares of this order, times the 8 variations gives a total of 7040 squares. They use the integers from 1 to 16 so the relative rarity is 7040/16!, or 1 chance in 2,971,987,200.

By the time we get to an order 8 magic square, we will find the number of possible combinations to try is 64 factorial. This is greater then the number of atoms in the universe!

Consider then, the rarity of an order 3 tesseract, which uses the numbers from 1 to 81. Or an order 8 tesseract where the number of possibilities is factorial 4096. 

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Harvey Heinz   harveyheinz@shaw.ca
This page originated November, 2007
This page last updated October 14, 2009
Copyright © 2007 by Harvey D. Heinz