Hypercube Math

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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              .....  Ss = 2n-1 + nmn-1

  • Minimum sums required to be Nasik magic                ....   Sn = {(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(mn-1)

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

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

  • Edges in a n-dimensional magic hypercube                ….   E = n(2n-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/

Hypercubes –Minimum number of correct summations  - based on smallest order possible

Magic

Hypercube

Lowest

Order (m)

r-agonals

 

Min. Sums

This order first Built by

1

2

3

4

Total

Square

 

 

 

 

 

 

 

 

Simple

3

2m

2

----

----

2m + 2

8

?

Nasik  (Pandiagonal and Perfect)

4

2m

2m

----

----

4m

16

?

Cube

 

 

 

 

 

 

 

 

Simple

3

3m2

----

4

----

3m2 + 4

31

Hugel  - 1876

Diagonal   (Boyer/Trump Perfect)

5

3m2

6m

4

----

3m2+6m+4

109

Trump/Boyer - 2003

Pantriagonal

4

3m2

----

4m2

----

7m2

112

Frost - 1878

PantriagDiag

8?

3m2

6m

4m2

----

7m2+6m

496

Nakamura - 2005

Pandiagonal

7

3m2

6m2

4

----

9m2 + 4

445

Frost – 1866

Nasik (Hendricks Perfect)

8

3m2

6m2

4m2

----

13m2

832

Barnard 1888

Tesseract

 

 

 

 

 

 

 

 

Simple

3

4m3

----

----

8

4m3 + 8

116

Planck – 1905

Triagonal

4

4m3

----

16m

8

4m3 + 16m + 8

328

Nakamura - 2007

Diagonal

4

4m3

12m2

----

8

4m3 + 12m2 + 8

456

Nakamura – 2007

Diagonal + Triagonal

8?

4m3

12m2

16m

8

4m3 + 12m2 + 16m + 8

2,952

Nakamura – 2007

Panquadragonal

4

4m3

----

----

8m3

12m3

768

Hendricks – 1968

Triagonal + Pan4

4

4m3

----

16m

8m3

12m3 + 16m

832

Nakamura – 2007

Diagonal + Pan4

8?

4m3

12m2

----

8m3

12m3 + 12m2

6,912

Nakamura – 2007

Diagonal + Triagonal+Pan4

8?

4m3

12m2

16m

8m3

12m3 + 12m2 + 16m

7,040

Nakamura – 2007

Pandiagonal

9?

4m3

12m3

----

8

16m3 + 8

11,672

Nakamura – 2007

Triagonal + Pan2

16?

4m3

12m3

16m

8

16m3 + 16m + 8

65792

Nakamura – 2010

Pantriagonal

4

4m3

----

16m3

8

20m3 + 8

1,288

Nakamura – 2007

Diagonal + Pan3

8?

4m3

12m2

16m3

8

20m3 + 12m2 + 8

11016

Nakamura – 2010

Pan2 +Pan4

13?

4m3

12m3

----

8m3

24m3

52,728

Nakamura – 2007

Triagonal + Pan2 + Pan4

16?

4m3

12m3

16m

8m3

24m3 + 16m

98,560

Nakamura – 2007

Pan3 + Pan4

4

4m3

----

16m3

8m3

28m3

1,792

Nakamura – 2007

Diagonal + Pan3 + Pan4

8

4m3

12m2

16m3

8m3

28m3 + 12m2

15,104

Nakamura – 2007

Pan2 + Pan3

15?

4m3

12m3

16m3

8

32m3 + 8

108,008

Nakamura – 2007

Nasik  (Hendricks perfect)

16

4m3

12m3

16m3

8m3

40m3

163,840

Hendricks - 1998

Arnoux’s cube is the earliest normal perfect (nasik) magic cube that I have been able to locate. It preceded by one year the order 8 and two order 11 perfect cubes of F.A.P.Barnard. A.H. Frost  had published an order 9 perfect cube (but with non-consecutive numbers) in 1878. Of course, not the smallest possible.
Gabriel Arnoux, Cube Diabolique de Dix-Sept, Académie des Sciences, Paris, France, April 17, 1887.

I show Hendricks as the first to publish a perfect (nasik) 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).

Hypercubes - Correct Summations Required

Magic Square
Simple

Magic Cube
Simple

Magic Tesseract
Simple

m rows

m2 rows

m3 rows

m columns

m2 columns

m3 columns

2 diagonals

m2 pillars

m3 pillars

 

4   3-agonals

m3 files

 

 

8    4-agonals

Total = 2m + 2

Total = 3m2 + 4

Total = 4m3 + 8

Nasik perfect

Nasik perfect

Nasik perfect

m rows

m2 rows

m3 rows

m columns

m2 columns

m3 columns

2m diagonals

m2 pillars

m3 pillars

 

4m2   3-agonals

m3 files

 

6m2   2-agonals

8m3    4-agonals

 

 

12m3    3-agonals

 

 

16m3    2-agonals

Total = 4m

Total = 13m2

Total = 40m3

Comparing Order-3 Hypercube Dimension Facts

Dimension

Correct lines

Number of Basic

Aspects

2

8

1

8

3

31

4

48

4

116

58

384

5

421

2992

3840

6

1490

543328

46080

Keh Ying Lin, Cubes and Hypercubes of Order Three, Discrete Mathematics, 58, 1986,
pp 159-166
J. R. Hendricks, Magic Square Course, self-published 1991
J. R. Hendricks, All Third Order Magic Tesseracts, self-published 1999, 0-9684700-2-5
H. D.  Heinz & J. R. Hendricks, Magic Square Lexicon: Illustrated, , HDH, 2000, 0-9687985-0-0, page 93

Notes for table on left:
J. R. Hendricks, Perfect n-Dimensional Magic Hypercubes of Order 2n,  Self-published, 1999, 0-9684700-4-1.
H. D.  Heinz & J. R. Hendricks, Magic Square Lexicon: Illustrated, , HDH, 2000, 0-9687985-0-0, page 90

An n-agonal is a line going from 1 corner, through the center to the opposite corner, of a magic hypercube.

For each continuous n-agonal, there are a number of broken parallel lines, depending upon the order of the hypercube. There are 2 continuous diagonals in a square, 4 continuous triagonals in a cube, and 8 continuous quadragonals in a tesseract.

Number of broken n-agonals for each continuous one

Total pan-n-agonals

n

2 segment

3 segments

4 segments

Total

1 segment

Total

2

m–1

0

0

m

2

2m

3

3(m-1)

(m-1)(m-2)

0

m2

4

4m2

4

2(5m-8)

2(2m2-7m+7)

(m-1)(m-2)(m-3)

m3

8

8m3

H. D.  Heinz & J. R. Hendricks, Magic Square Lexicon: Illustrated, , HDH, 2000, 0-9687985-0-0, page 99 (edited)

Because there are four triagonals in a magic cube,  the above figures must be multiplied by four to obtain the actual number of triagonals in the cube.

H. D.  Heinz & J. R. Hendricks, Magic Square Lexicon: Illustrated, , HDH, 2000,
 0-9687985-0-0, page 170 (edited)

 

Triagonals in one direction of a cube

Order

1 segments

2 segments

3 segments

Total

3

1

6

2

9

4

1

9

6

16

5

1

12

12

25

6

1

15

20

36

7

1

18

30

49

8

1

21

42

64

9

1

24

56

81

10

1

27

72

100

Number of Hyperplanes Within a Hypercube

Magic Hypercube

i-rows
(1-agonals)

Squares

Cubes

Tesseracts

5-D
Hypercubes

Squares

2m

1

0

0

0

Cubes

3m2

3m

1

0

0

Tesseracts

4m3

6m2

4m

1

0

5-D Hypercubes

5m4

10m3

10m2

5m

1

6-D Hypercubes

6m5

15m4

20m3

15m2

6m

7-D Hypercubes

7m6

21m5

35m4

35m3

21m2

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.

n-Dimensional Magic Hypercubes – Statistical Information

Dim

Hyper-
cube

# of Corners

# of Edges

Bounded by

Magic Sum

Paths through any cell

Minimum Sums required for magic

Minimum Sums required for Nasik Perfect

# of Viewing Aspects

0

Point

1

0

0

1

1

0

1

1

Line seg.

2

1

2 points

S = {m + 1)} / 2

 0

1

1

2

2

Square

4

4

4 line segments

S = {m(m2 + 1)} / 2

4

2m + 2

4m

8

3

Cube

8

12

6 squares

S = {m(m3 + 1)} / 2

13

3m2 + 4

13m2

48

4

Tesseract

16

32

8 cubes

S = {m(m4 + 1)} / 2

40

4m3 + 8

40m3

384

n

Hypercube

2n

n(2n-1)

2n hypercubes
(of n-1)

S = {m(mn + 1)} / 2

P = (3n – 1) / 2

nmn-1 + 2n-1

Sn = {(3n – 1)mn-1} / 2

2nn!

 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. 

This page was originally posted November 2007
It was last updated February 25, 2013
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz