 |
 |
 |
 |
 |
|
|
|
|
|
| Introduction | Order 15 Perfect | Order 15 Composition |
| Pantriagonal Order 16 | Perfect Order 16 | Arnoux Perfect Order 17 |
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.
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.
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. Contact me if you would like a copy of the complete cube.
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.
This 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, 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
This order 15 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.
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”.)
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
This order 16 cube, which is my file ‘Cube_16-Soni.xls’, 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
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!
Christian Boyer spotted a reference to this work in "Arithmétique Graphique - Les Espaces ArithmétiquesHypermagiques",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, two each on four different pages
Just as a point of interest, these are the eight mistakes he made.
|
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), so contains
51 orthogonal, 6 oblique,
and 96 oblique 2-segment pandiagonal magic squares.
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
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 enarge).
Arnoux’s cube is the earliest normal perfect 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.[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.
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
Please send me Feedback about my Web
site!
Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 14, 2009
Copyright © 2003 by Harvey D. Heinz