F. A. P. Barnard's Magic Cubes

Like the papers by Rev. Frost [1], Dr. Barnard's paper [2] published in 1888, contained cubes that were years ahead of their time! And also like Frost, he seemed uninterested in simple magic cubes.

His paper considers magic squares (some quite unusual), circles and cyclovolutes, as well as magic cubes. The cubes he shows are an order 4, which is not magic by our present standards, and 3 perfect magic cubes (new definition), an order 8 and two order 11.

As far as I can determine, the first perfect magic cube was constructed by Frost. However, it was not normal. It used non-consecutive numbers from 1 to 889. the next one to be published (after Barnard's) was by Ian Howard in a JRM paper [3] in 1976 (he gave instructions on building a normal order 11).
ADDENDUM: In early 2003, Christian Boyer located an order 17 perfect normal magic cube constructed by G. Arnoux in 1887. [4]

Other early publication of cubes (that I am aware of) that were magic by present standards (rows, columns, pillars, and triagonals all correct) was an order 3 by Hugel in 1876 [5], and orders 4 and 5 cubes published in 1899 by Hermann Schubert [6].
The first published definition for this simple magic cube was by Andrews in 1908 [7].

As this page is dedicated to Dr. Barnard, I will show several unusual squares and other magic objects, as well as two of his perfect cubes.

[1] A. H. Frost, On the General Properties of Nasik Cubes, Quarterly Journal of Mathematics 15, 1878 pp 110-116
[2] F. A. P. Barnard, Theory of Magic Squares and Cubes, Nat. Academy of Sciences, Vol. IV, Sixth Memoir, 1888, pp 207-270
[3] Ian P. Howard, Pan-diagonal Associative Magic Cubes (Letter to the Editor), JRM 9:4, 1976, pp276-278.
[4] Gabriel Arnoux, Cube Diabolique de Dix-Sept, Académie des Sciences, Paris, France, April 17, 1887.
[5] Theodore Hugel, Das Problem der magishen Systeme, 1876, Verlag von A. H. Gottschick, 70pp. (German).
[6] H. Schubert, Mathematical Essays and Recreations. Translated from German to English by Thomas J. McCormack (1899, Open Court, 1903.
[7] W. S. Andrews, Magic Squares & Cubes, Open Court, 1908, 1 (page 65).

Inlaid Squares	Two squares Barnard refers to as inlaid.
Cylinder and sphere	Two unusual magic objects
Order 8 perfect cube	Order 11 perfect cube

Inlaid Squares

										505
1	2	98	97	25	76	96	95	7	8	505
99	100	3	4	75	26	5	6	93	94	505									404
17	83	33	36	66	64	62	42	82	20	505	1	2	98	97	96	95	7	8	404
84	18	67	43	50	55	54	34	19	81	505	99	100	3	4	5	6	93	94	404
29	71	63	57	52	45	48	38	70	32	505	17	83	43	50	55	54	82	20	404
72	30	40	46	47	58	51	61	31	69	505	84	18	57	52	45	48	19	81	404
21	79	41	56	53	44	49	60	78	24	505	21	79	46	47	58	51	78	24	404
80	22	59	65	35	37	39	68	23	77	505	80	22	56	53	44	49	23	77	404
86	85	14	13	74	27	12	11	92	91	505	86	85	14	13	12	11	92	91	404
16	15	87	88	28	73	89	90	10	9	505	16	15	87	88	89	90	10	9	404
505	505	505	505	505	505	505	505	505	505	505	404	404	404	404	404	404	404	404	404

Reassemble the magic square on the left, without the colored cells, and you get the magic square on the right.

Here we have an order 14 magic square that reduces to an order 10. Then reduces to orders 8, 6, and finally order 4.
Barnard calls these inlaid magic squares. However, they are quite different then John Hendricks Inlaid squares, where each inlay is also a magic square.

Cylinder and sphere

The cylinder uses numbers 1 to 105
19 vertical, 5 diagonal, and 5 circumference lines sum to 265.
The two central circles each sum to 2 x 265.
2 vertical lines are incorrect. Can you find them?

The sphere uses the numbers 1 to 55
8 circles with 8 numbers sum to 216. 5 lines with 5 numbers sum to 135. 5 lines with 3 numbers sum to 81.

Order 8 perfect cube

Barnard's paper contained 3 magic cubes of the type we now call perfect. An order 8 which is not associated, and two order 11 cubes which are associated. (Order 9 is the smallest perfect cube that can be associated.)
They precede the modern publication of normal perfect cubes by 88 years (when Howard published an order 11. The only perfect cube that I am aware of being published at an earlier date is Frost's order 9, but it was not normal.
ADDENDUM: Gabriel Arnoux of France constructed an order 17 Perfect normal magic cube in 1887. However, he did not publish it so it was not available to the general public. I have a page about the Arnoux cube.

The following is quoted from page 252 and page 265 of Barnard’s paper [1] and is in reference to the order 8 perfect cube shown here (fig. 48 in his paper).

In order to verify the properties ascribed to this cube we select for addition the terms which, in this arrangement, are brought in any direction into line. Observing that the value of S must in general be equal to the sum of an arithmetical series of which the first term is 1, the last term n³, and the number of terms n, we have

S=1/2(n³+l)=1/2(n⁴+n) (68)

And for the cube of 8 S=1/2(4104+8)=2052 (he obviously meant S=1/2(4096+8)=2052)

The row parallel to z on the right upper edge is then

469+298+148+431+21+234+340+111=2052=S

The direct diagonal of the solid is

1+93+174+207+512+420+339+306=2052=S

The transverse diagonal —yzx is

469+400+322+294+44+113+191+219=2052=S

The broken diagonal parallel to —xzy, beginning at 141 and ending at 385, is

141+278+479+128+372+235+34+385=2052=S

And so of others.

It is impossible to exhibit magic cubes to the eye (except those of small numbers, (which are necessarily imperfect) otherwise than by presenting, as here, their component squares separately. In Fig. 50 is shown the cube of 4 arranged in solid form. It is magical, except in the rows parallel to z, and in the diagonals of the faces xz, yz, and those of the solid. In this figure every cubic tessera of eight terms, however taken, will be found to give the same sum; and the cubes of the higher powers of 2 may be made to possess the same property.

It will be found to be true of the cube of 8, Fig. 48, and this cube possesses other still more remarkable properties. The eight numbers, for example, which mark the solid angles of any cube less than the cube of 8, which can be made within this magic cube, will give invariably the same sum, viz, 2052. And in any right parallelopipedon, whose terminal planes are squares of the even numbers 2 or 4, and in whose lateral edges the number of terms is even, the sum of the numbers marking the solid angles will still be the same, 2052. If the terminal planes be squares of 6, the same will be true whatever the number of terms on the lateral edges; if the terminal planes be squares of the odd numbers 3, 5, or 7, and the lateral edges contain either three or seven terms, the proposition will be true of these parallelopipedons also.

This enumeration does not exhaust all the peculiar properties of this remarkable cube.

He also says

Perfectly magic cubes may be formed on all orders from 8 upward, except the unevenly even.

Horizontal plane 1 - Top                   plane 2
  1  490   59  468    8  495   62  469     251  276  200  303  254  277  193  298
144  359  182  349  137  354  179  348     438   93  393   98  435   92  400  103
465    2  491   60  472    7  494   61     299  252  280  199  302  253  273  194
352  143  358  181  345  138  355  180     102  437   89  394   99  436   96  399
 57  466    3  492   64  471    6  493     195  300  256  279  198  301  249  274
184  351  142  357  177  346  139  356     398  101  433   90  395  100  440   95
489   58  467    4  496   63  470    5     275  196  304  255  278  197  297  250
360  183  350  141  353  178  347  140      94  397   97  434   91  396  104  439
Plane 3                                    Plane 4
328  175  382  149  321  170  379  148     126  405   65  426  123  404   72  431
  9  482   51  476   16  487   54  477     243  284  208  295  246  285  201  290
152  327  174  381  145  322  171  380     430  125  401   66  427  124  408   71
473   10  483   52  480   15  486   53     291  244  288  207  294  245  281  202
384  151  326  173  377  146  323  172      70  429  121  402   67  428  128  407
 49  474   11  484   56  479   14  485     203  292  248  287  206  293  241  282
176  383  150  325  169  378  147  324     406   69  425  122  403   68  432  127
481   50  475   12  488   55  478   13     283  204  296  247  286  205  289  242
Plane 5                                    Plane 6
449   42  507   20  456   47  510   21     315  212  264  239  318  213  257  234
336  167  374  157  329  162  371  156     118  413   73  418  115  412   80  423
 17  450   43  508   24  455   46  509     235  316  216  263  238  317  209  258
160  335  166  373  153  330  163  372     422  117  409   74  419  116  416   79
505   18  451   44  512   23  454   45     259  236  320  215  262  237  313  210
376  159  334  165  369  154  331  164      78  421  113  410   75  420  120  415
 41  506   19  452   48  511   22  453     211  260  240  319  214  261  233  314
168  375  158  333  161  370  155  332     414   77  417  114  411   76  424  119
Plane 7                                    Plane 8 - Bottom
136  367  190  341  129  362  187  340     446   85  385  106  443   84  392  111
457   34  499   28  464   39  502   29     307  220  272  231  310  221  265  226
344  135  366  189  337  130  363  188     110  445   81  386  107  444   88  391
 25  458   35  500   32  463   38  501     227  308  224  271  230  309  217  266
192  343  134  365  185  338  131  364     390  109  441   82  387  108  448   87
497   26  459   36  504   31  462   37     267  228  312  223  270  229  305  218
368  191  342  133  361  186  339  132      86  389  105  442   83  388  112  447
 33  498   27  460   40  503   30  461     219  268  232  311  222  269  225  306

All 24 planar squares are pandiagonal magic as are also the 6 oblique squares and the seven broken squares parallel to each of these[2][3] for a total of 72 pandiagonal magic squares. The 256 pantriagonals also all sum correctly.
Corners of all orders 2, 3, 4, 5, 6, 7 and 8 (including wraparound) sub-cubes also sum correctly to 2052. Also, many other shapes of parallelopipeds.

A perfect magic cube is a combination pantriagonal and pandiagonal magic cube. However, all 6 oblique must be pandiagonal magic as well. This last condition is a natural consequence of “…all lower dimension hypercubes are perfect”.
In a perfect magic cube there are 9m pandiagonal magic squares. That is, all 3m orthogonal planes, the 6 oblique planes, and the 6m-1 broken planes parallel to the oblique planes [2].
There are 13m² lines that sum correctly. Order-8 is the smallest possible perfect magic cube.

[1] F. A. P. Barnard, Theory of Magic Squares and Cubes, Nat. Academy of Sciences, Vol. IV, Sixth Memoir, 1888, pp 207-270
[2] B. Rosser and R. J. Walker, A continuation of The Algebraic Theory of Diabolic Magic Squares on typewritten pages numbered 729 – 753, (diabolic cubes pp 736-753).
[3] F. Liao, T. Katayama, K. Takaba, Technical Report 99021, School of Informatics, Kyoto University, 1999. Available on the Internet at http://www.amp.i.kyoto-u.ac.jp/tecrep/TR1999.html

Order 11 perfect cube

I decided not to put either of the two order 11 associated perfect cubes of Barnard on this page.

For anyone interested in seeing them, Barnard-11.doc is available for downloading.

I do show other order 11 perfect cubes on this site.

Howard	1976	not associated
Seimiya	1977	associated
Collison	1991	associated

F. A. P. Barnard's Magic Cubes