Arnoux Patterns

Introduction

On April 17th, 1887, the Frenchman Gabriel Arnoux deposited a perfect (i.e. pandiagonal and pantriagonal) magic cube of order 17 with the Académie des Sciences. It consists of 26 handwritten pages! As far as I have been able to determine, this is the first normal perfect magic cube ever constructed!
For more information on the Arnoux cube, see my Big Cubes page. You may also download the complete listing of the Arnoux cube by visiting my cube Downloads page.

Each of the 4913 cells in Arnoux’s order 17 cube is part of 13 different lines that sum to the constant.

However, there are many other patterns of 17 numbers that also sum to the constant.
These are sets of 17 numbers formed by moving through the cube similar to how a knight moves in chess. This page is a summary of an investigation of such lines. A multitude of these patterns can be created by varying the size of the jumps (I refer to them as steps).

Again I express my gratitude to Christian Boyer of Paris, France, for locating and obtaining a copy of this marvelous and historically significant cube, and making it available to me.

 Following is quoted from an email by Christian Boyer sent on July 7, 2003.

About the other announced characteristics, Gabriel Arnoux says
> that his cube is "hypermagic" : he means that all the
> miscellaneous lines are magic, not only the 90° lines and 45° diagonals.
> Choose randomly two numbers in the cube, draw the line going
> through these two numbers, and the magic sum of this line will be
> magic (excluding some rare exceptions).
> An incredible feature, far better that what you call a "perfect
> magic" cube. More perfect than perfect!
> Look at the "ArnouxExplications1.JPG" image, he gives an example
> with the line going through the 2 numbers 1759 and 3891.
> I will describe an easiest sample with for example the line going
> through the 2 numbers 4718 (high left corner of Arnoux01.JPG) and
> 3477 (line 2, column 3 of Arnoux02.JPG):
>    1,1 in Arnoux01 = 4718
>    2,3 in Arnoux02 = 3477
>    3,5 in Arnoux03 = 3443
>    4,7 in Arnoux04 = 3953
>    5,9 in Arnoux05 = 944
>    6,11 in Arnoux06 = 2916
>    7,13 in Arnoux07 = 1607
>    8,15 in Arnoux08 = 2780
>    9,17 in Arnoux09 = 4174
>    10,2 in Arnoux10 = 43
>    11,4 in Arnoux11 = 2304
>    12,6 in Arnoux12 = 2525
>    13,8 in Arnoux13 = 638
>    14,10 in Arnoux14 = 4565
>    15,12 in Arnoux15 = 1403
>    16,14 in Arnoux16 = 1879
>    17,16 in Arnoux17 = 400

In this example, Christian used jumps of 1 for coordinates Y and Z, jumps of 2 for the X coordinates.

General comments

Because of the very large number of possible patterns, I adopted the following strategy to make the number of tests required manageable. I did check both magic squares and magic cubes. Also, one plane each of the three orthogonal orientations of each cube tested.

·        Tests were limited to ODD orders of magic cubes and squares.

·        Step refers to the number of rows moved down as we move to the right (or left) to the adjacent column (X). For lines that move through 3 dimensions, I considered only moving 1 orthogonal plane (Z) at a time. So, in these tests, X and Z are limited to steps of 1.

·        For dimension Y (the rows), in most cases, I checked only for steps of 2, 3, or 4.
Other patterns obviously exist by using various steps for X, Y and Z.

With the following exceptions, all lines with steps from 2 to m-2 exist in odd order magic squares and cubes.
If the odd order square or cube is not prime, then steps that are factors or contain factors, of the order cannot exist. This is because the cycle cannot be of length m.

The sloping line with step m-1 is always a pandiagonal or pantriagonal (depending on whether the line goes through 2 or 3 dimensions.

Squares or cubes of even order can also contain Arnoux patterns if ‘step is not a factor of m. However, the tests summarized here only applied to odd order squares and cubes. I did check out two order 4 and two order 8 magic squares in the Sqr_9-OtherLines.xls spreadsheet.

For most tests, I checked only lines starting at each of the cells in columns 1 and 2 for lines moving down and to the right. I also checked lines starting at each of the cells in columns m and m-1 for lines sloping down to the left. I was confident that if all these lines are correct, then ALL lines in the magic square or cube are correct.

 By the time I had started testing the order 17 Arnoux, I had established the fact that similar patterns starting on any column of a given plane of the cube contained the same m totals (but the series of totals start on a different line). Therefore if a column contains all correct sums, all occurrences of that pattern in the cube are correct. With this in mind, for the order 17 Arnoux cube and the order 13 Quadrant magic square, I tested only column 1 and m-1.

For the cube tests, I tested the 3-dimensional lines, then I tested the Top, Back and Left orthogonal planes for correct 2-dimensional lines.

To demonstrate that Arnoux example patterns with all 3 coordinates stepping by an amount greater then 1 is universal, I show here two other examples. In both cases I started at an arbitrary cell and used arbitrary steps for all 3 coordinates. And in both cases, my first attempt was successful! Of course, the fact that these two example patterns are correct does not guarantee that all patterns in the cube (with these particular steps) will sum correctly.

My two examples simply show that patterns with any particular combination of steps can exist in a particular cube.

Order 11 Seimiya Perfect:          Order 9 Hendricks Perfect
 X    Y    Z                       X   Y   Z
 1 - 11 -- 3 = 1279                4 - 7 - 1 =  501
 3 -- 6 -- 6 = 1056                9 - 9 - 3 =  228
 5 -- 1 -- 9 =  844                5 - 2 - 5 =  721
 7 -- 7 -- 1 =  500                1 - 4 - 7 =  133
 9 -- 2 -- 4 =  277                6 - 6 - 9 =  265
11 -- 8 -- 7 =   65                2 - 8 - 2 =    5
 2 -- 3 - 10 = 1173                7 - 1 - 4 =  461
 4 -- 9 -- 2 =  950                3 - 3 - 6 =  602
 6 -- 4 -- 5 =  606                8 - 5 - 8 =  369
 8 - 10 -- 8 =  394                Total = S = 3285
10 -- 5 - 11 =  171
   Total = S = 7315
X = step 2, Y = step 5,            X = step 5, Y = step 7, 
    and Z = step 3                     and Z = step 2
Both of these cubes are listed elsewhere on these cube pages.

X and Z step by 1?

I mentioned above that for the most part I have limited my tests to X and Z steps of 1, varying only the steps for Y. However, in my cube test spreadsheets, of which the Arnoux pattern test is just one sheet, I already have all the planes in the two vertical orientations. Because the row and column positions of these planes are exactly the same as those for the horizontal planes, it was simple to test these arrays also.

Without thinking the matter through carefully, I figured that, in effect, testing these different orientations was equivalent to varying the Y and Z coordinates. Now that I have virtually completed all the tests I wish to do, I have checked this out. I find that the steps of only the Z coordinate varies when I use the same formulae as I did when testing the horizontal planes. Testing the second orientation of the vertical planes produced only a variation of the same patterns the first orientation produced.

To better illustrate this discussion, here is an example I have extracted from Cube_7-Hendricks-JRM.xls on the basis of the X step = 3 (see the Cube Comparison chart).

For the cube:
I show a pattern generated from the horizontal planes. Then patterns generated from the two vertical orientations with the coordinates in the horizontal orientation where these values are found. The columns and rows (C-R) are the same for all three so I show them just once.  The Z, Y and X coordinates are also the same for all three orientations, but I show them next to the new horizontal orientation coordinate equivalents for convenience.

I have counted the planes (Z) as 1 on the top, 7 on the bottom of the cube. The rows (Y) as 1 on the top to 7 on the bottom (when the plane is shown as in my spreadsheet. The columns (X) are counted from the left.

Horizontal planes     Vertical B2F  Horz      Vertical L2R  Horz
C-R  Value  Z Y X     Val. Z Y X   Z Y X     Val. Z Y X   Z Y X
B6      98  1 1 1      98  1 1 1   1 1 1       1  1 1 1   1 7 1
C19    319  2 4 2     298  2 4 2   4 2 2     775  2 4 2   4 6 2
D32    197  3 7 3     204  3 7 3   7 3 3     342  3 7 3   7 5 3
E38    131  4 3 4     117  4 3 4   3 4 4     117  4 3 4   3 4 4
F51      9  5 6 5      23  5 6 5   6 5 5     284  5 6 5   6 3 5
G57    286  6 2 6     279  6 2 6   2 6 6      59  6 2 6   2 2 6
H70    164  7 5 7     185  7 5 7   5 7 7     226  7 5 7   5 1 7
Total 1204           1204                   1204

Conclusion: By testing the planes parallel to the front, I am in effect varying the Z coordinate step while limiting the X and Y coordinate steps to 1. The test of the planes parallel to the sides of the cubes, turns out to be redundant. Again, the X and Y steps have been a constant 1 with the Z step varied.

For the squares:
I show a pattern generated from the top horizontal plane. Then patterns generated from the back and the left vertical planes with the coordinates where these values are found in the cube. The results are the same as for the above. In effect, I have tested for variations in the steps of Y and Z (in the cube), but not for X.
Regardless, I feel that my tests have proven the general universality of these patterns within hypercubes of different orders and dimensions.

Top                       Back                Left
C_R  Value  Y X   Z Y X   Value  Y X   Z Y X   Value  Y X   Z Y X
B6      98  1 1   1 1 1      98  1 1   1 1 1       1  1 1   1 7 1
C9     264  4 2   1 4 2     257  4 2   4 1 2     132  4 2   4 6 1
D12    136  7 3   1 7 3     122  7 3   7 1 3     207  7 3   7 5 1
E8     302  3 4   1 3 4     330  3 4   3 1 4     331  3 4   3 4 1
F11    181  6 5   1 6 5     153  6 5   6 1 5      70  6 5   6 3 1
G7       4  2 6   1 2 6      18  2 6   2 1 6     194  2 6   2 2 1
H10    219  5 7   1 5 7     226  5 7   5 1 7     269  5 7   5 1 1
Total 1204                 1204                 1204

Files checked

Magic objects I tested for this feature are:

 The above cube spreadsheets are the original test spreadsheets with an additional worksheet appended at the end.
Following is a new and an existing spreadsheet to test this feature in magic squares.

The three order 5 cubes permitted a direct comparison between a cube with pantriagonals correct in all 4 directions, in 1 direction only, and in NO directions. This last cube is Walter Trump’s, which arrived at a late stage in my tests.
I have 8 different order 5 cubes and 1 order 7 cube by Trump. All have the same feature;  they do not contain all correct pantriagonals in at least one direction. These are the ONLY cubes of these two orders with this feature.

Because these 3 magic cubes do not appear on any of my other cube pages, I present the listings here.

Cube_5_Aale-1.xls      (Aale de Winkel)
Plane 1 - Top               II                          III
  1   50   69   88  107     110    4   48   67   86      89  108    2   46   70
125   19   38   57   76      79  123   17   36   60      58   77  121   20   39
 94  113    7   26   75      73   92  111   10   29      27   71   95  114    8
 63   82  101   25   44      42   61   85  104   23      21   45   64   83  102
 32   51  100  119   13      11   35   54   98  117     120   14   33   52   96
IV                          V - Bottom
 68   87  106    5   49      47   66   90  109    3
 37   56   80  124   18      16   40   59   78  122
  6   30   74   93  112     115    9   28   72   91
105   24   43   62   81      84  103   22   41   65
 99  118   12   31   55      53   97  116   15   34

Cube_5_Boyer-2.xls
Plane 1 - Top               II                          III
 54   31   13  115   97      10  117   99   51   33      96   53   30   12  119
 15  122   79   56   38      76   58   35   17  124      37   19  121   78   55
 81   63   40   22  104      42   24  101   83   60     103   80   62   44   21
 47    4  106   88   65     108   85   67   49    1      69   46    3  105   87
113   90   72   29    6      74   26    8  110   92       5  112   94   71   28
IV                          V - Bottom
 32   14  116   98   50     118   95   52   34   11
123   75   57   39   16      59   36   18  120   77
 64   41   23  100   82      20  102   84   61   43
  0  107   89   66   48      86   68   45    2  109
 91   73   25    7  114      27    9  111   93   70

Cube_5_Trump-2.
Plane 1 - Top               II                          III
 34    6  115   35  120      11  122   92   47   38      78  108   96   23    5
 85  123   26   45   31     110   54   49   55   42      15   56   76   57  106
  8   91   20   94   97      81   74   73   72   10      80   63   62   61   44
 95   37   25  112   41      22   58   64   59  107      18   67   48   68  109
 88   53  124   24   21      86    2   32   77  113     119   16   28  101   46
IV                          V - Bottom
 84    3    7  105  111     103   71    0  100   36
 17   65   60   66  102      83   12   99   87   29
114   52   51   50   43      27   30  104   33  116
 82   69   75   70   14      93   79   98    1   39
 13  121  117   19   40       4  118    9   89   90

Order 16

As an afterthought, I decided to check out order 16 cubes because order 16 is 4², and also to investigate Arnoux patterns in the even orders.

As with the preceding tests, I tested X and Z steps of 1, Y steps of from 2 to m-1 and for X and Y steps of 1, Z steps of from 2 to m-1
For order 16, the possible steps are 3, 5, 7, 9, 11, and 13.
The even numbers of course, are factors (or contain factors) of 16 so cannot produce a full cycle of 16 numbers.

Here I will compare results of 4 different types of order 16 magic cubes.
In all cases, if any step produces all correct patterns, then all six possible steps result in m² correct patterns.
Therefore I will indicate only if all patterns (for all steps tested) or no patterns are correct.
Be aware that in many cases there will be some patterns correct for some steps tested. I do not indicate these situations.
The 2 Boyer cubes are Bimagic. Shown here are the degree 1 cube of each set, so contain the numbers from 0 to 4095.

The number of magic squares shown in the table are the orthogonal squares included in the cube .

Conclusion

The following notes and table summarize the findings from my tests.

For magic cubes

For X and Z steps of 1, Y steps of from 2 to m-1 and for X and Y steps of 1, Z steps of from 2 to m-1

·        Patterns starting in column m-1 will always be equivalent to a pantriagonal

·        This feature is NOT limited to the Arnoux perfect order 17 cube!
      It is common to all cubes to a lesser or greater degree

o       It seems to be dependent to some degree on correct pantriagonals
Notice that Trumps order 5 with ALL pantriagonals correct in none of the 4 directions has NO steps with ALL solutions correct!

o       Notice also the Hendricks lowly order 7 pantriagonal cube has ALL solutions correct for steps 3 and 4! (This cube, by definition,  has all triagonals correct in ALL 4 directions!)

o       Frost’s order 7 pandiagonal cube also has all correct solutions for steps 3 and 4.

·        This feature is NOT limited to magic hypercubes whose order is a prime number. However, only steps that are relatively prime to the order can possibly work. So, for example, for order 9, steps of 3 and 6 will not work. That is because it is impossible to cycle through a series of 9 numbers.

·        I tested two other cubes with all three coordinates using steps other then 1!

For magic squares

For X steps of 1, Y steps of from 2 to m-1

·        Patterns with a step for X of 1 and Y of 1 or m-1 will always be equivalent to a pandiagonal

·        This feature is very common in magic squares and orthogonal arrays of magic cubes

·        Hendricks order 7 pantriagonal cube (no magic squares) – ALL solutions for steps 2, 3, 4, and 5

·        Arnoux order 17 perfect cube (orthogonal planes) - ALL solutions are correct for steps 2, and 3 (and step 4 for one , step 5 for two orientations).

·        Pandiagonal magic order 13 square (my quadrant magic square) – All solutions for steps 2, 5, 6. 7, 8, 11, and 12. For this square, I also tested patterns with X confined to a step of 1 and Y with steps 2, 3, 4, and 5. They show exactly the same results as the normal steps

·        This feature is NOT dependent on pandiagonals because I tested it on two order 5 pandiagonal magic squares, and none of the steps had ALL solutions correct (except step 4 which is m-1)

Summary

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Harvey Heinz   harveyheinz@shaw.ca
This page last updated October 14, 2009
Copyright © 2003 by Harvey D. Heinz