Arnoux's Perfect Magic Cube

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On this page I will discuss an order 15 perfect and an order 15 composition cube. Then two order 16 magic cubes and finally Arnoux’s historical order 17 perfect magic cube of 1887. 

Introduction

Order 15 Perfect

Order 15 Composition

Pantriagonal Order 16

Perfect Order 16

Arnoux Perfect Order 17

 Introduction

 On this page I will discuss an order 15 perfect and an order 15 composition cube. Then two order 16 magic cubes and finally Arnoux’s historical order 17 perfect magic cube of 1887.

Three other composition cubes are discussed on my composition cubes page. Two other order 16 cubes are discussed on the Multimagic cube page. They are Christian Boyer’s amazing bi-magic cubes.
The first one contains 32 simple magic squares and is classed as a ‘simple' magic cube.
The second of Boyer’s cubes contains 48 simple magic squares and because all planar squares have diagonals that sum correctly, the tube is classed as ‘diagonal’ magic.

For comparison with the cubes below, the simple cube has all order 9 sub-cube corners summing correctly.
The diagonal cube has no order of sub-cubes where all corners sum correctly.
Neither cube has any Arnoux type patterns where all of a pattern sum correctly.

For each of the five cubes discussed, because of their size, I will include the list for only the top horizontal plane. Any interested person can obtain the complete listing for any, or all, of these cubes by contacting me.

 

 Order 15 Perfect

 Guenter Sternenbrink (Germany) sent me this cube as an email attachment on Nov. 11, 2003. This was the first order 15 cube I had seen.

This cube was described by Dr C. Planck in 1905 [1]. He probably did not construct the actual cube, because he claimed the cube was associated (which it is not). He based this statement on the fact that the generating magic rectangles were associated.
This cube is classed as ‘perfect’ because it contains 45 planar, 6 oblique, and 84 oblique two-segment order 15 pandiagonal magic squares.

Looking at the definition for perfect another way; this cube has the features of a pandiagonal magic cube (all planar arrays are pandiagonal magic) AND a pantriagonal magic cube (all pantriagonals sum correctly)!

Some Arnoux Patterns are also correct. (More on Arnoux patterns here.)

Following are the results when I checked patterns with X step of 1. Y or Z with steps of 2, 4, or 7 and the other (Y or Z) step of 1.
Orders 2, 4, and 7, and also 11 and 13 are the only possible steps for an order 15 cube (others are multiples of factors of 15).
If all coordinate steps are 1 or all are 14, a pantriagonal is produced.
For the 2 dimensional arrays, I checked only the top, back, and left side planes.

Arnoux Patterns in the Planck/Stertenbrink Order 15 Cube
              Step 2     Step 4     Step 7
Cube   Hor.   N   A      1   A      N   A
       Vert.  A   N      A   A      A   A
Plane  Top    N   A      A   A      A   1
       Back   1   A      1   A      A   A
       Left   A   A      1   A      A   1
N = None of this pattern in the cube (or square) 
A = All patterns in the cube (or square) are correct  
1 = 1 pattern only (in each column) is correct
Top horizontal plane (because of space restraints, I will show only this plane)
 242   961  1690  2412  3133  3367  1209   723   484  1926  1455  2894  2651  2168     5
1943  1355  2717  2536  2140   162   433  1117  1659  2298  2959  3276  1230   869   626
2444  3101  3293  1130   692   511  1915  1512  2908  2692  2109    48   259  1026  1680
2601  2130   194   401  1043  1580  2267  2986  3265  1287   883   667  1884  1398  2734
1173   709   576  1905  1544  2876  2618  2030    17   286  1015  1737  2458  3142  3234
 442   984  1623  2284  3051  3255  1319   851   593  1805  1367  2761  2590  2187   208
1962  1558  2917  2559  2073    34   351  1005  1769  2426  3068  3155  1142   736   565
2311  3040  3312  1333   892   534  1848  1384  2826  2580  2219   176   368   905  1592
2480  2042    61   340  1062  1783  2467  3009  3198  1159   801   555  1994  1526  2843
1301   818   455  1817  1411  2815  2637  2233   217   309   948  1609  2376  3030  3344
 330  1094  1751  2393  2930  3167  1186   790   612  2008  1567  2784  2523  2059   126
1834  1476  2805  2669  2201   143   230   917  1636  2365  3087  3358  1342   759   498
2334  2973  3184  1251   780   644  1976  1493  2705  2492  2086   115   387  1108  1792
2683  2242    84   273   934  1701  2355  3119  3326  1268   680   467  1861  1465  2862
1240   837   658  2017  1434  2748  2509  2151   105   419  1076  1718  2255  2942  3211

A few days earlier, Guenter sent me an order 4 pantriagonal magic cube that had the integers arranged to form a closed knight tour. That cube may be seen on my Unusual Cubes page. Good work Guenter! Thanks.

[1] Dr. C. Planck, Theory of Paths Nasik, Printed for private circulation by A. J. Lawrence, Printer, Rugby.

 Order 15 Composition

This order 15 was constructed simply and quickly by myself on November 25, 2003.
It consists of 27 order 5 magic cubes placed as per the numbers in an order 3 magic cube and so is a composition cube.
It is classified as a simple magic cube, because all orthogonal planes are not magic squares and/or all pantriagonals do not sum correctly.
More cubes and a more complete construction explanation are at my Composition Cubes page.

To save space, I will show only the top horizontal layer.

 317   268   369   356   255  1567  1518  1619  1606  1505  3192  3143  3244  3231  3130
 366   267   264   323   345  1616  1517  1514  1573  1595  3241  3142  3139  3198  3220
 290   300   331   315   329  1540  1550  1581  1565  1579  3165  3175  3206  3190  3204
 306   370   305   299   285  1556  1620  1555  1549  1535  3181  3245  3180  3174  3160
 286   360   296   272   351  1536  1610  1546  1522  1601  3161  3235  3171  3147  3226
2817  2768  2869  2856  2755  1067  1018  1119  1106  1005  1192  1143  1244  1231  1130
2866  2767  2764  2823  2845  1116  1017  1014  1073  1095  1241  1142  1139  1198  1220
2790  2800  2831  2815  2829  1040  1050  1081  1065  1079  1165  1175  1206  1190  1204
2806  2870  2805  2799  2785  1056  1120  1055  1049  1035  1181  1245  1180  1174  1160
2786  2860  2796  2772  2851  1036  1110  1046  1022  1101  1161  1235  1171  1147  1226
1942  1893  1994  1981  1880  2442  2393  2494  2481  2380   692   643   744   731   630
1991  1892  1889  1948  1970  2491  2392  2389  2448  2470   741   642   639   698   720
1915  1925  1956  1940  1954  2415  2425  2456  2440  2454   665   675   706   690   704
1931  1995  1930  1924  1910  2431  2495  2430  2424  2410   681   745   680   674   660
1911  1985  1921  1897  1976  2411  2485  2421  2397  2476   661   735   671   647   726

The order 5 cube is not associated. Neither are any of it's 21 magic squares.
The order 3 cube is associated. The resulting order 15 cube is not associated.
Each plane of the order 15 cube contains 9 order 5 simple magic squares.
Each group of 5 planes contain 9 order 5 diagonal .magic cubes
The central 5 planes in each orientation are order 15 simple magic squares.
The central order 5 and order 15 magic squares (in each orientation) are associated.
Only a very few scattered Arnoux patterns are correct.
The magic constant of the order 15 cube is 25320.

Following is the order 5 diagonal cube that was discovered by Walter Trump and Christian Boyer on November 12, 2003. All 30 planar diagonals sum correctly, thus forming 15 planar magic squares. By the old definition (and the one still used by Boyer and Trump), it would be called a perfect cube (although it contains only simple magic squares).
More information is available on my order 5 cubes page.
This cube is repeated 27 times, using the consecutive numbers from 1 to 3375 to form the order 15 cube.

Horizontal plane 1 - Top    Plane 2                    Plane 3
 67   18  119  106    5     66   72   27  102   48     42  111   85    2   75
116   17   14   73   95     26   39   92   44  114     30  118   21  123   23
 40   50   81   65   79     32   93   88   83   19     89   68   63   58   37
 56  120   55   49   35    113   57    9   62   74    103    3  105    8   96
 36  110   46   22  101     78   54   99   24   60     51   15   41  124   84
Plane 4                     Plane 5 - Bottom
 25   16   80  104   90      3   13   26    3   13
 91   77   71    6   70     23    9   10   23    9
 47   61   45   76   86     16   20    6   16   20
 31   53  112  109   10      3   13   26    3   13
121  108    7   20   59     23    9   10   23    9

This is index # 2 of only 4 order 3 basic cubes.
It is used as a multiplier to find the successive series of 125 numbers for the 27 order 5 cubes. It is also used as a pattern for placing the order 5 cubes into the order 15 composition cube.

Plane 1 - Top    Plane 2          Plane 3 - Bottom
 3  13  26       17  21   4       22   8  12
23   9  10        1  14  27       18  19   5
16  20   6       24   7  11        2  15  25

 Pantriagonal Order 16

 This order 16 cube was reconstructed in September 2003 from information supplied by Aale de Winkel.

It uses the numbers from 1 to 4096 so has a magic constant of 32776. It contains no magic squares, but does have many other magic patterns besides the basic orthogonal lines and the 4 main triagonals.

Because it is a pantriagonal cube, all 4m2  pantriagonals are correct. These pantriagonals consist of 4 one-segment, 36 two-segment, and 24 three-segment lines of 16 numbers.

CompactPlus patterns

Consider 8 numbers located within the cube so that they form the corners of a sub-cube. Furthermore, consider that the top left back corner of this sub-cube may be located on any of the 163 cells. For all sub-cubes of each order 2, 4, 6, 8, 9, 10, 12, 14, and 16, these 8 cells sum to 16388, or 8/16 of the magic constant.
Of course, because of wrap-around, many of these combinations of 8 numbers will be identical (only in a different order).
If all positions of ALL the possible orders of sub-cubes sum correctly, I call the feature “compactplus”. (Several writers have referred to the feature in magic squares where the 4 corner numbers in ANY 2x2 square sum correctly as “compact”. [1])

[1] See more recent information on compact and compactplus on cube_update-6 page (Feb. 2010).

Arnoux patterns

Gabriel Arnoux (see entry below) suggested a type of pattern that I have not found mentioned by anyone else, either before or since 1887. These patterns have been discussed thoroughly on my Arnoux Patterns page.

Suffice to say here that all 163 patterns appear in de Winkel’s cube for each of the following conditions.

  • When x and y are stepped by 1 and z is stepped by 3, 5, 7, 9, 11, and 13 (these six are the only possible steps for order 16).
  • When x and z are stepped by 1 and y is stepped by 3, 5, 7, 9, 11, and 13.
  • If x is stepped by 1 and y varied as above to test a planar array, no step will produce all correct patterns.

The top horizontal plane of Aale de Winkel's order 16 pantriagonal magic cube. 

   1  4080    33  4048    65  4016    97  3984   241  3872   209  3904   177  3936   145  3968
4095    18  4063    50  4031    82  3999   114  3855   226  3887   194  3919   162  3951   130
   3  4078    35  4046    67  4014    99  3982   243  3870   211  3902   179  3934   147  3966
4093    20  4061    52  4029    84  3997   116  3853   228  3885   196  3917   164  3949   132
   5  4076    37  4044    69  4012   101  3980   245  3868   213  3900   181  3932   149  3964
4091    22  4059    54  4027    86  3995   118  3851   230  3883   198  3915   166  3947   134
   7  4074    39  4042    71  4010   103  3978   247  3866   215  3898   183  3930   151  3962
4089    24  4057    56  4025    88  3993   120  3849   232  3881   200  3913   168  3945   136
  16  4065    48  4033    80  4001   112  3969   256  3857   224  3889   192  3921   160  3953
4082    31  4050    63  4018    95  3986   127  3842   239  3874   207  3906   175  3938   143
  14  4067    46  4035    78  4003   110  3971   254  3859   222  3891   190  3923   158  3955
4084    29  4052    61  4020    93  3988   125  3844   237  3876   205  3908   173  3940   141
  12  4069    44  4037    76  4005   108  3973   252  3861   220  3893   188  3925   156  3957
4086    27  4054    59  4022    91  3990   123  3846   235  3878   203  3910   171  3942   139
  10  4071    42  4039    74  4007   106  3975   250  3863   218  3895   186  3927   154  3959
4088    25  4056    57  4024    89  3992   121  3848   233  3880   201  3912   169  3944   137 

 Perfect Order 16

This order 16 cube was received from Abhinav Soni via email attachment on October 3, 2003.

It is a “Perfect” magic cube so contains 3m planar, 6 oblique, and 6m-6 two-segment oblique pandiagonal magic squares, a total of 144 order 16 squares.
It has a total of 3m2 orthogonal lines, 6m2 planar diagonals, and 4m2 triagonals that sum correctly to 32776

CompactPlus patterns

For all 163 starting cells, cubes of each order 2, 4, 5, 6, 8, 9, 10, and 16, the 8 corner cells sum to 16388, or 8/16 of the magic constant. Note that this is 1 less then for the pantriagonal cube above.
Of course, because of wrap-around, many of these combinations of 8 numbers will be identical (only in a different order).

Arnoux patterns

All 163 Arnoux patterns appear in Soni’s cube for each of the following conditions.

  • When x and y are stepped by 1 and z is stepped by 3, 5, 7, 9, 11, and 13.
  • When x and z are stepped by 1 and y is stepped by 3, 5, 7, 9, 11, and 13.
  • If just x is stepped by 1 and y steps varied as above to test the planar arrays, all Arnoux patterns again sum correctly.
  • Realize that the even numbers cannot be used as steps because 2 is a factor of 16, so a 16 number sequence is impossible.

The top horizontal plane of Abhinav Soni's order 16 perfect magic cube. 

   1   546  1091  1636  4085  3542  2999  2456    16   559  1102  1645  4092  3547  3002  2457
 512   991  1470  1949  3596  3115  2634  2153   497   978  1459  1940  3589  3110  2631  2152
 513  1058  1603  3940  3573  3030  2487   152   528  1071  1614  3949  3580  3035  2490   153
1024  1503  1982  3741  3084  2603  2122   361  1009  1490  1971  3732  3077  2598  2119   360
1025  1570  3907  3428  3061  2518   183   664  1040  1583  3918  3437  3068  2523   186   665
1536  2015  3774  3229  2572  2091   330   873  1521  2002  3763  3220  2565  2086   327   872
1537  3874  3395  2916  2549   214   695  1176  1552  3887  3406  2925  2556   219   698  1177
2048  3807  3262  2717  2060   299   842  1385  2033  3794  3251  2708  2053   294   839  1384
3841  3362  2883  2404   245   726  1207  1688  3856  3375  2894  2413   252   731  1210  1689
3840  3295  2750  2205   268   811  1354  1897  3825  3282  2739  2196   261   806  1351  1896
3329  2850  2371   100   757  1238  1719  3992  3344  2863  2382   109   764  1243  1722  3993
3328  2783  2238   413   780  1323  1866  3689  3313  2770  2227   404   773  1318  1863  3688
2817  2338    67   612  1269  1750  4023  3480  2832  2351    78   621  1276  1755  4026  3481
2816  2271   446   925  1292  1835  3658  3177  2801  2258   435   916  1285  1830  3655  3176
2305    34   579  1124  1781  4054  3511  2968  2320    47   590  1133  1788  4059  3514  2969
2304   479   958  1437  1804  3627  3146  2665  2289   466   947  1428  1797  3622  3143  2664

 Arnoux Perfect Order 17

On April 17th, 1887, the Frenchman Gabriel Arnoux deposited [1] 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! But see the reference in [4]. New terminology from an old source now refers to this type of hypercube as Nasik.

Christian Boyer spotted a reference to this work in "Arithmétique Graphique - Les Espaces Arithmétiques Hypermagiques", Gabriel Arnoux, 1894, page 61 and looked up the paper.

The Académie des Sciences would not allow it to be copied, but did give Christian permission to photograph the individual pages. He then kindly supplied me with the images on a CD .

I typed the numbers from these page images into a spreadsheet modeled after other spreadsheets I have been using to investigate magic cubes. I was not able to scan the images successfully because of the difficulty OCR had with the handwritten figures. I also checked that the number range 1 to 4913 contained no missing and duplicate numbers.

This work by Arnoux is monumental, considering he did all calculations by hand and wrote all figures manually. Some numbers were hard to read, due to fading or unclear handwriting. However, the magic of the spreadsheet quickly resolved what was the correct number. Nevertheless, Arnoux did make eight definite mistakes (probably typographical errors), two each on four different pages.

Just as a point of interest, these are the eight typos on his pages.

Plane (z)

Row (y)

Column (x)

His number

Correct number

1

16

3

2730

2740

1

12

4

251

260

2

5

1

3834

3934

2

5

12

4591

4491

12

15

9

427

426

12

15

10

3766

3767

13

11

2

1122

1071

13

11

16

160

211

This order 17 cube is indeed a perfect magic cube (by John Hendricks definition, now called nasik), so contains
                    51 orthogonal, 6 oblique, and 96 oblique 2-segment pandiagonal magic squares.
I now refer to this type of perfect as nasik (as per C. Planck's 1905 definition).

It consists of the numbers from 1 to 4913. Each of the 4913 numbers is part of 13 lines of 17 numbers, each of which sum to 41769 (the magic constant).
Or to put it another way, there are 3m2 orthogonal rows (1-agonals), 6m2 diagonals (2-agonals) and 4m2 triagonals (3-agonals) that sum to 41769.

Arnoux Patterns

Arnoux claimed his cube was ‘hypermagic’. By this he meant that most patterns consisting of chess knight like moves between sets of 17 numbers, would sum correctly.

I spent over a month of spare time checking his claim. While quite a few patterns do appear in his cube, perhaps not as many as he thought. The multitude of pattern variations and the 173 possible starting positions for each pattern make it impractical to check all possibilities. My investigation was limited to X and Z steps of 1, Y steps of 2, 3, 4, and 5. If Y was also stepped by 1, this would of course be a triagonal (or broken triagonal depending on the starting position of the pattern).

I checked all starting positions in the top, front, and left faces of the cube. I also considered each of these planes as a pandiagonal magic square, and tested for Y steps of 2, 3, 4, and 5 for all 289 starting positions. It seems that in the 3-D case starting in the top plane, all patterns with Y steps of 2, 4, and 5 are correct. For the 2-D case concerning the top plane, all Y steps of 2, 3, and 4 are correct. In each case, that is 172 patterns (including wraparound) that are correct!

A complete report and results of my investigation are reported on the Arnoux Patterns page.
I also included tests of other magic cubes and magic squares. This property is very general, to a lesser or greater degree, in all types of magic hypercubes. Surprisingly, I have seen no other reference to this property in the literature (up to 2003) in the 118 years since Arnoux announced it.

This is horizontal plane 1 of Arnoux's cube.
The complete listing for the cube may be obtained by downloading ArnouxCube.doc.

Horizontal plane 1 - Top 	Z = 1
4718  1060  4215   614   347  3897  2609  2397  1909  3452  1676  2138  1426  3496  3023   207  4585
  84  4507  4903  1092  4156   850   311  3877  2815  2563  1834  3377  1612  2060  1219  3613  2896
3594  3098   246  4437  4816  1021  4090   642   440  3768  2683  2485  2018  3409  1553  2294  1175
2082  1309  3473  2959   180  4620  4860   980  4315   598   420  3970  2845  2413  1923  3339  1483
3286  1701  2050  1288  3687  3139    99  4525  4789   910  4103   730   291  3832  2775  2592  1972
2509  1900  3228  1507  2173  1159  3548  3069   278  4572  4728  1129  4067   705   510  4000  2687
3942  2884  2547  1839  3446  1471  2148  1376  3708  2982   191  4496  4675   923  4183   588   370
 796   530  3854  2797  2471  1784  3232  1588  2027  1232  3655  3167   222  4447  4892   899  4176
1015  4055   652   475  4031  2829  2418  1997  3213  1569  2228  1404  3566  3092   164  4383  4678
4596  4657   988  4257   820   382  3961  2759  2347  1795  3328  1460  2102  1340  3743  3123   111
3045    41  4390  4768   884  4119   749   571  3991  2718  2578  1765  3301  1661  2270  1247  3671
1432  3697  3009   260  4353  4753  1084  4299   674   490  3913  2647  2372  1876  3195  1515  2200
1700  2113  1344  3631  2937    66  4482  4638   938  4228   859   516  3875  2858  2336  1857  3391
1735  3257  1628  2310  1388  3584  3148    29  4463  4834  1121  4133   772   446  3799  2669  2453
2631  2446  1949  3431  1533  2222  1318  3508  2957   138  4342  4696  1045  4335   804   392  4022
 334  3822  2740  2324  1811  3355  1733  2246  1265  3727  2915   136  4544  4863   962  4246   746
4270   692   553  3780  2736  2518  1979  3268  1640  2193  1195  3520  3036    13  4418  4805  1153

      This photo of the original sheet is by Christian Boyer (click to enlarge).

Arnoux’s cube is the earliest normal nasik perfect magic cube that I have been able to locate (as of 2003).[4] It preceded by one year the order 8 and two order 11 perfect cubes of F.A.P.Barnard.[2] A.H. Frost [3] had published an order 9 perfect cube (but with non-consecutive numbers) in 1878.

[1] Gabriel Arnoux, Cube Diabolique de Dix-Sept, Académie des Sciences, Paris, France, April 17, 1887.
[2]
F.A.P. Barnard, Theory of Magic Squares and Magic Cubes, Memoirs of the National Academy of Science, 4,1888,pp. 209-270.
[3]
A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123 plus plates 1 and 2.
[4] My 2010 update page contains information on a A. H. Frost normal order-9 nasik cube!

The following is quoted from an email of Sept. 22/03 from Christian Boyer.
Once again he was of tremendous help to me with locating material and assisting with translation to English.

Dear Harvey,
Some information for you about Gabriel Arnoux!
As I said earlier, his name is pronounced "R-noo", without the "x".
As you know, in French, we do not pronounce all the letters. We like the
difficulties...
 

Gabriel Arnoux lived in Les Mées, a small town located in the French Alps.
Unfortunately, I do not have his lifetime (18xx-19xx). I think something
like 1830/50-1910/30.
 

He was an officer in the French navy, and served on the frigate "Uranie"
when the commander of this warship was Edmond Jurien de la Gravière.
Edmond Jurien de la Gravière (1812-1892) became later the President of the
Académie des Sciences in Paris.
 

That's why Gabriel Arnoux thought to send his magic cube in 1887 to the
Académie des Sciences, directly through the president.
He was retired from the navy when he sent this cube of order 17.
 

After this cube, he wrote several books. I think that his 4 main books are:
        - Les espaces arithmétiques hypermagiques, Paris, 1894
        - Introduction à l'étude des fonctions arithmétiques, Paris, 1906
        - Les espaces arithmétiques, leurs transformations, Paris, 1908
        - Essai de géométrie analytique modulaire à deux dimensions, Paris, 1911
 

He wrote also some articles in "Les Tablettes du Chercheur", recreational
mathematics magazine, for example an article about the piquet, card game.
He published also some math papers in the "Comptes-Rendus de l'AFAS".
FAS = Association Française pour l'Avancement des Sciences.
After 1900, he became a friend of Gaston Tarry (1843-1913), another French
specialist of magic squares.
 

Gabriel Arnoux had a disability (clubfoot) and some diseases: malaria, and
regular headaches.
He said doing mathematics and magic squares was his way to forget his
numerous physical problems.
 

I have never found any clear explanation of his construction methods in his
books.
Mainly, he tried to found a new mathematical theory, that he called
Arithmétique Graphique = Graphical Arithmetics. Not a big success in the
history of mathematics, I think.
He spoke also about magic hypercubes, but also without any clear method.
 

With his magic cube of order 17, he sent only to the Académie the 4 pages of
explanations that you have in ArnouxExplications1.jpg & 2.jpg.
The other images (ArnouxLettres1.jpg & 2.jpg) were the letters joined to the
cube, explaining his personal situation. Nothing mathematical in them.
Kind Regards

Christian

This page was originally posted November 2003
It was last updated February 25, 2013
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz