Cube_Update-5

Introduction

April 20, 2004. Material about magic cubes continues to appear, but at a slower rate. I am including some material here on other magic objects.

 3x3x11 Magic Cuboid Order-10 Pantriagonal Cube Perfect Mod-5 Order-5 cube Magic Knight Tour - 1 Magic Knight Tour - 2 Order-12 Magic Knight Tour Cube Edward's Multiply Square Symmetric Magic Diamonds Durupt Bent-diagonal Order-8 Cube Fillion Bent-diagonal Order-16 Cube Recent Publications and Postings Acknowledgements New Tesseract Pages Nasik (Perfect) Hypercube Generators

3x3x11 Magic Cuboid

Magic rectangles are similar to magic squares except (as the name suggests) they are bigger in one dimension then the other. All rows sum to a magic constant, and all columns sum to a different magic constant. Just as with magic squares, the concept can be carried over to higher dimensions. Obviously, diagonals are not required to be magic, because they cannot extend from corner to corner in an object with different dimensions.

The term Magic Rectangle has been in use at least since 1908. [1]
The term cuboid is used by Abhinav Soni, but is defined in a popular mathematics dictionary[2]
Aale de Winkel uses the term Magic Beam and Magic Hyperbeam for 3 and higher dimension objects of this type. [3]

(Even, even) orders are relatively simple to construct. (Odd, odd) are more difficult. (Even, odd) normal magic rectangles are impossible.

 Here are two simple examples of magic rectangles from Marián Trenkler. [4] 1  7  6  4     and     01 10 14 09 06   8  2  3  5             15 02 07 11 05                          08 12 03 04 13

If the even, even order rectangle desired has 1 dimension an even multiple of the other, there is a very simple method available. Using a magic square of order equal to the smaller dimension for a pattern, distribute the numbers from 1 to m among the m cells of the rectangle.

Example:

 01 08 10 15 12 13 03 06 07 02 16 09 14 11 05 04
 01 08 58 63 60 61 03 06 07 02 64 57 62 59 05 04 09 16 50 55 52 53 11 14 05 10 56 49 54 51 13 12 17 24 42 47 44 45 19 22 23 18 48 41 46 43 21 20 25 32 34 39 36 37 27 30 31 26 40 33 38 35 29 28

As this figure consists of 4 order-4 magic squares, they may be arranged to form an order-8 magic square. (I used this same idea to construct a pyramid of 16 order-4 squares for the frontispiece of the Magic Square Lexicon. [5]

Here is an Order-8 magic square with inlaid 4x6 magic rectangle.
This was constructed by then 90 year old E. W. Shineman.,Jr.  © 2005 (used by permission).
The 8x8 is a pandiagonal magic square  S=260
The 4x6 is a magic rectangle  S=195 and 130; (195/6=32.5*4=130).
All sets of 8 numbers of like color sum to 260
 Order-8 magic square with inlaid 4x6 magic rectangle 10 7 58 31 34 47 18 55 56 57 8 33 32 17 48 9 16 1 64 25 40 41 24 49 50 63 2 39 26 23 42 15 13 4 61 28 37 44 21 52 51 62 3 38 27 22 43 14 11 6 59 30 35 46 19 54 53 60 5 36 29 20 45 12

As I mentioned above, magic rectangles can be extended to the third (or higher) dimension.
Here is a 3x3x11 constructed by Abhinav Soni [6]

A 3x3x11 magic cuboid.   Rows sum to 550, columns and pillars sum to 150.

The text listing:
Front face                         Middle plane                       Back face
88 95 82 84 69 08 48 19 26 14 17   34 43 66 57 54 79 04 40 53 64 56   28 12 02 09 27 63 98 91 71 72 77
61 45 35 42 60 96 32 58 38 39 44   22 29 16 18 03 41 81 85 92 80 83   67 76 99 90 87 13 37 07 20 31 23
01 10 33 24 21 46 70 73 86 97 89   94 78 68 75 93 30 65 25 05 06 11   55 62 49 51 36 74 15 52 59 47 50

[1] Andrews, W. S., Magic Squares & Cubes, Open Court, 1908, page 170. Also the same page in Edition 2, 1960
[2] Harper-Collins Mathematical Dictionary
[3] Aale de Winkel, The Magic Encyclopedia, at http://www.magichypercubes.com/Encyclopedia/index.html
[4] Marián Trenkler, The Mathematical Gazette, March,1999
[5] The Lexicon is available at http://www.magic-squares.net/BookSale.htm
[6] Soni no longer has a web page (Nov./09).
Mitsutoshi Nakamura also has a good site on magic rectangles, at http://homepage2.nifty.com/googol/magcube/en/rectangles.htm

Order-10 Pantriagonal Cube

Recently I was reminded that in November, 2005, Walter Trump advised me of a transformation from single even to pantriagonal magic cubes. He reported that he first saw it (as applied to squares) on Mutsumi Suzuki’s now defunct Web site.

This is the procedure for an order 10 associated simple magic cube
Starting with the X planes, exchange plane x=6 with x=10 and plane x=7 with plane x=9.
With this new cube, exchange plane y=6 with y=10 and plane y=7 with plane y=9.
Finally, with this new cube, exchange plane z=6 with z=10 and plane z=7 with plane z=9.
The resulting cube is a non-associated pantriagonal magic cube.

I tested this procedure out on the on the Planck order-10 associated cube. The following is the resulting pantriagonal cube. Being pantriagonal, this cube can be changed to another pantriagonal  magic cube simply by moving any outside plane to the opposite side of the cube.  However, it is no longer associated!

```Horizontal plane 10 – top                             Horizontal plane 9 _ top-1
1000  999  903   94    6  991  992    8    7    5     191  109  898  897  805  110  102  893  894  106
990  912   83   17  986  981   19   18   14  985     120  889  888  814  185  111  882  883  117  116
921   72   28  977  976   30   29   23  974  975     880  879  823  174  126  871  872  128  127  125
61   39  968  967  935   40   32  963  964   36     870  832  163  137  866  861  139  138  134  865
50  959  958  944   55   41  952  953   47   46     841  152  148  857  856  150  149  143  854  855
910  909   93    4   95  901  902  998   97   96     101  192  808  807  195  200  199  803  804  896
920   82   13   84  916  911  989   88   87  915     181  819  818  184  115  190  812  813  887  186
71   22   73  927  926  980   79   78  924  925     830  829  173  124  175  821  822  878  177  176
31   62  938  937   65   70   69  933  934  966     840  162  133  164  836  831  869  168  167  835
51  949  948   54   45   60  942  943  957   56     151  142  153  847  846  860  159  158  844  845

Horizontal plane 8_  top-2                           Horizontal plane 7 _ top-3
800  702  293  207  796  791  209  208  204  795     310  699  698  604  395  301  692  693  307  306
711  282  218  787  786  220  219  213  784  785     690  689  613  384  316  681  682  318  317  315
271  229  778  777  725  230  222  773  774  226     680  622  373  327  676  671  329  328  324  675
240  769  768  734  265  231  762  763  237  236     631  362  338  667  666  340  339  333  664  665
760  759  743  254  246  751  752  248  247  245     351  349  658  657  645  350  342  653  654  346
710  292  203  294  706  701  799  298  297  705     391  609  608  394  305  400  602  603  697  396
281  212  283  717  716  790  289  288  714  715     620  619  383  314  385  611  612  688  387  386
221  272  728  727  275  280  279  723  724  776     630  372  323  374  626  621  679  378  377  625
261  739  738  264  235  270  732  733  767  266     361  332  363  637  636  670  369  368  634  635
750  749  253  244  255  741  742  758  257  256     341  352  648  647  355  360  359  643  644  656

Horizontal plane 6 _ top-4                           Horizontal plane 5 _ bottom+4
501  492  408  597  596  410  409  403  594  595     100   99    3  904  905   91   92  908  997  906
481  419  588  587  515  420  412  583  584  416      90   12  913  914   86   81  919  988  917   85
430  579  578  524  475  421  572  573  427  426      21  922  923   77   76  930  979  928   74   75
570  569  533  464  436  561  562  438  437  435     931  932   68   67   35  970  939   63   64  936
560  542  453  447  556  551  449  448  444  555     941   59   58   44  945  950   52   53  947  956
491  402  493  507  506  600  499  498  504  505      10    9  993  994  996    1    2   98  907  995
411  482  518  517  485  490  489  513  514  586      20  982  983  987   16   11   89  918  984   15
471  529  528  474  425  480  522  523  577  476     971  972  978   27   26   80  929  973   24   25
540  539  463  434  465  531  532  568  467  466     961  969   38   37  965  940  962   33   34   66
550  452  443  454  546  541  559  458  457  545     960   49   48  954  955  951   42   43   57  946

Horizontal plane 4 _ bottom+3                        Horizontal plane 3 _ bottom+2
801  802  198  197  105  900  809  193  194  806     300  202  703  704  296  291  709  798  707  295
811  189  188  114  815  820  182  183  817  886     211  712  713  287  286  720  789  718  284  285
180  179  123  824  825  171  172  828  877  826     721  722  278  277  225  780  729  273  274  726
170  132  833  834  166  161  839  868  837  165     731  269  268  234  735  740  262  263  737  766
141  842  843  157  156  850  859  848  154  155     260  259  243  744  745  251  252  748  757  746
891  899  108  107  895  810  892  103  104  196     210  792  793  797  206  201  299  708  794  205
890  119  118  884  885  881  112  113  187  816     781  782  788  217  216  290  719  783  214  215
130  129  873  874  876  121  122  178  827  875     771  779  228  227  775  730  772  223  224  276
140  862  863  867  136  131  169  838  864  135     770  239  238  764  765  761  232  233  267  736
851  852  858  147  146  160  849  853  144  145     250  249  753  754  756  241  242  258  747  755

Horizontal plane 2 _ bottom+1                        Horizontal plane 1 _ bottom
601  399  398  304  605  610  392  393  607  696     401  502  503  497  496  510  599  508  494  495
390  389  313  614  615  381  382  618  687  616     511  512  488  487  415  590  519  483  484  516
380  322  623  624  376  371  629  678  627  375     521  479  478  424  525  530  472  473  527  576
331  632  633  367  366  640  669  638  364  365     470  469  433  534  535  461  462  538  567  536
641  642  358  357  345  660  649  353  354  646     460  442  543  544  456  451  549  558  547  455
700  309  308  694  695  691  302  303  397  606     591  592  598  407  406  500  509  593  404  405
320  319  683  684  686  311  312  388  617  685     581  589  418  417  585  520  582  413  414  486
330  672  673  677  326  321  379  628  674  325     580  429  428  574  575  571  422  423  477  526
661  662  668  337  336  370  639  663  334  335     440  439  563  564  566  431  432  468  537  565
651  659  348  347  655  650  652  343  344  356     450  552  553  557  446  441  459  548  554  445```

Perfect Mod-5

At first glance this is an ordinary simple magic associated order-5 magic cube. Because it is associated, the center plane in each of the 3 orientations is an associated magic square. The cube contains no other magic squares.

However, on closer inspection, all possible lines of 5 numbers sum to a multiple of 5. This includes the straight (1 segment) lines, and also the 2-segment and 3-segment lines.

So this is a Perfect magic cube (Modulo 5). 75 1-agonals, 150 2-agonals, and 100 3-agonals all sum to 0 mod 5.
Because it is perfect mod 5, this cube also contains 45 order-5 pandiagonal magic squares (mod 5).

Walter Trump sent me this cube via email on Sept. 3, 2005. He easily constructed it using a magic cube generating program written by Peter Bartsch.

```Horizontal plane 5 – top    Horizontal plane 4          Horizontal plane 3
94   23   52  106   40     111   45   99    3   57       8   62  116   50   79
55  109   38   92   21      97    1   60  114   43     119   48   77    6   65
36   95   24   53  107      58  112   41  100    4      80    9   63  117   46
22   51  110   39   93      44   98    2   56  115      61  120   49   78    7
108   37   91   25   54       5   59  113   42   96      47   76   10   64  118

Horizontal plane 2          Horizontal plane 1 - bottom
30   84   13   67  121      72  101   35   89   18
11   70  124   28   82      33   87   16   75  104
122   26   85   14   68      19   73  102   31   90
83   12   66  125   29     105   34   88   17   71
69  123   27   81   15      86   20   74  103   32```

Magic Knight Tour - 1  .....  on order-8 cube surfaces.

 On my unusual cubes page I show an illustration of an 8x8x8 cube with a chess knight tour on each of the six surfaces. It was constructed by H. E. Dudeney and published in 1917. If each move of the knight is numbered, rows and columns sum to various values. However, he does mention that knight tours of 8x8 (chess)boards are possible with all rows and columns summing to the same constant value. However, at that time no KT had been discovered where the diagonals also summed correctly. It was proven in August of 2003 that it is impossible for a knight tour on an 8x8 board to have row and columns, and also the 2 main diagonals all sum to the same value. This is the probable reason that in the Knight Tour community, a Knight Tour is considered magic if only the rows and columns sum correctly. Unfortunately, this causes much confusion among magic square fans, where the main diagonals must also sum to the constant. Awani Kumar is an active investigator of Knight Tours. In May 2007, he improved on Dudeney's cube faces tour. Each of the six faces of the 8x8x8 cube shown here consists of a Magic Knight Tour (rows and columns sum to 1540, but not diagonals. I have shown the first 32 moves of each face, and the last 32 moves using different colors. More on Knight Tours and links to other sites are on my Knight Tours page. Awani Kumar's MKT on the six faces of an order-8 cube. (email of  May 4, 2007)

Listing of Knight moves for the above cube surface tours.

```Rows & columns = 1540, diagonals = 1494 & 1592             diagonals = 1504 & 1496
203  182  207  186  209  164  223  166       55   38  347  330   35   58  351  326
206  185  204  183  222  167  162  211      346  331   56   37  350  327   34   59
181  202  187  208  163  210  165  224       39   54  329  348   57   36  325  352
188  205  184  201  168  221  212  161      332  345   40   53  328  349   60   33
175  180  189  220  193  200  169  214       41   52  333  344   45   62  339  324
190  219  174  177  172  213  196  199      334  343   44   49  340  323   46   61
179  176  217  192  197  194  215  170       51   42  341  336   63   48  321  338
218  191  178  173  216  171  198  195      342  335   50   43  322  337   64   47

Rows & columns = 1540, diagonals = 1866 & 1870             diagonals = 1504 & 1496
359   10   25  376  361    8   23  378      246  239  146  139  226  241  160  143
26  375  360    9   24  377    6  363      147  138  245  240  159  144  225  242
373  358   11   28    7  362  379   22      238  247  140  145  244  227  142  157
12   27  374  357  380   21  364    5      137  148  237  248  141  158  243  228
353  372   29   16  365    4  381   20      236  249  136  149  232  253  156  129
32   13  356  369  384   17  366    3      135  150  233  252  153  132  229  256
371  354   15   30    1  368   19  382      250  235  152  133  254  231  130  155
14   31  370  355   18  383    2  367      151  134  251  234  131  154  255  230

Rows & columns = 1540, diagonals = 1504 & 1496             diagonals = 1504 & 1496
294  319   90   67  298  315   70   87      262  123  288   97  260  125  274  111
91   66  295  318   69   88  299  314      287   98  261  124  275  110  257  128
320  293   68   89  316  297   86   71      122  263  100  285  126  259  112  273
65   92  317  296   85   72  313  300       99  286  121  264  109  276  127  258
292  307   94   77  312  301   84   73      266  101  284  117  280  113  272  107
93   78  291  308   81   76  311  302      283  120  265  104  269  108  277  114
306  289   80   95  304  309   74   83      102  267  118  281  116  279  106  271
79   96  305  290   75   82  303  310      119  282  103  268  105  270  115  278```

Magic Knight Tour - 2 ..... in an Order-8 cube

On April 28, 2007, Awani Kumar announced via email, the first solution to the magic Knight Tour of an order-8 cube.

When the steps a chess knight takes while traveling from cell to cell through the cube are numbered

This cube consists of the numbered steps of a knight as it travels through the 512 cells of the cube. All rows, columns and pillars sum correctly to 2052. Only two of the four triagonals sum to this value. The other two sum to 2500 and 2020.  Features of this cube were confirmed by Dan Thomasson email of April 30,2007.

However, Knight Tour fans do not require that the n-agonals sum correctly for a tour to be considered magic. Magic square and cube fans, of course, would consider this cube to be only semi-magic.
There are no magic squares in this cube, because none of the planar diagonals sum correctly (not a requirement).

The tour is re-entrant, meaning that the last cell visited is exactly one knight move away from the first cell. Therefore the tour may be started on any cell, and will successfully visit all 512 cells in the cube. Of course, because the numbers are now in different positions, the tour will no longer be magic.
Kumar reports however, that the tour will be magic if started at position 257 instead of position 1.

More on Knight Tours, and links to other sites are on my Knight Tours page.
An 0rder-4 cube Magic Knight Tour is shown on my Unusual cubes pages.

Addendum:  Listed here is the cube mentioned above. However, it is NOT magic because only 2 of the four triagonals sum correctly!
On May 8, 2007, Kumar announced a new magic knight tour order-8 cube that had all 4 triagonals correct. This cube is simple magic, not associated, and includes no order 8 magic squares, but consists of  a reentry knight tour! I list it on my Knight Tours page.

Listing for the order-8 Knight Tour cube.

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

Order-12 Magic Knight Tour Cube

On June 19 2007, Awani Kumar announced via email, the first solution to the magic Knight Tour of an order-12 cube.

This is a simple magic cube with all  rows, columns, pillars, and the 4 main triagonals summing correctly to 10374. It is classed as simple magic, not associated, with no included order 12 magic squares. The knight tour is not quite re-entrant.

The Knight Tours of both this cube and the magic order-8 cube  (May 8) mentioned above were confirmed correct by  Guenter Stertenbrink.

```Cube_12-Kumar-KT.xls
1
1136   601   578  1143   626  1117   624  1091  1088   649   658  1063      583  1138  1129   608  1119   628  1089   622   659  1062  1085   652
579  1142  1133   604  1099   616  1109   634   663  1058  1081   656     1132   605   582  1139   614  1097   636  1111  1084   653   662  1059
602   591  1144  1121  1118   625  1092   623   650   671  1064  1073     1137  1128   607   586   627  1120   621  1090  1061  1076   651   670
1141  1124   603   590   615  1100   633  1110  1057  1080   655   666      606   587  1140  1125  1098   613  1112   635   654   667  1060  1077
758   987   970   743   996  1015   730   717  1052   685   694  1027      975   738   755   990   735   716   997  1010   695  1026  1049   688
985   744   757   972  1017  1006   707   728   691  1030  1053   684      756   973   992   737   710   721  1024  1003  1056   681   690  1031
748   965   984   761   734   713  1000  1011   686   699  1028  1045      977   768   749   964   993  1014   731   720  1025  1048   687   698
967   762   747   982   711   724  1021  1002  1029  1044   683   702      750   979   962   767  1020  1007   706   725   682   703  1032  1041
942   773   796   947   908   919   818   813   876   887   850   845      769   938   951   800   831   804   901   922   863   836   869   890
795   948   781   934   817   814   907   920   849   846   875   888      952   799   930   777   902   921   832   803   870   889   864   835
772   939   950   797   918   905   816   819   886   873   848   851      943   776   793   946   801   830   923   904   833   862   891   872
949   798   931   780   815   820   917   906   847   852   885   874      794   945   784   935   924   903   802   829   892   871   834   861
2
1146  1135   600   577  1102   609  1108   639  1066  1087   648   657      593   584  1151  1130   611  1104   637  1106   645   660  1067  1086
597   580  1147  1134   631  1116   617  1094   641   664  1071  1082     1150  1131   596   581  1114   629  1096   619  1070  1083   644   661
592  1145  1122   599   610  1101   640  1107   672  1065  1074   647     1127   594   585  1152  1103   612  1105   638  1075   646   669  1068
1123   598   589  1148  1115   632  1093   618  1079   642   665  1072      588  1149  1126   595   630  1113   620  1095   668  1069  1078   643
986   759   742   971  1018  1005   708   727  1038  1051   676   693      739   974   991   754   709   722  1023  1004   673   696  1039  1050
741   988   969   760   995  1016   729   718   677   692  1035  1054      976   753   740   989   736   715   998  1009  1034  1055   680   689
968   745   764   981   712   723  1022  1001   700  1037  1046   675      765   980   961   752  1019  1008   705   726  1047   674   697  1040
763   966   983   746   733   714   999  1012  1043   678   701  1036      978   751   766   963   994  1013   732   719   704  1033  1042   679
956   787   782   933   914   909   812   823   882   877   844   855      791   960   929   778   805   826   927   900   837   858   895   868
941   774   955   788   811   824   913   910   843   856   881   878      770   937   792   959   928   899   806   825   896   867   838   857
790   957   932   779   912   915   822   809   880   883   854   841      953   786   783   936   827   808   897   926   859   840   865   894
771   940   789   958   821   810   911   916   853   842   879   884      944   775   954   785   898   925   828   807   866   893   860   839
3
1164   573   550  1171  1190   537  1212   519  1242   503  1222   491      547  1174  1165   572   539  1208   517  1194  1219   494  1247   498
551  1170  1161   576   527  1188   529  1214  1227   486  1239   506     1168   569   546  1175  1202   525  1200   531  1234   511  1230   483
574   555  1172  1157   538  1205   520  1195   504  1241   492  1221     1173  1156   571   558  1191   540  1209   518   493  1220   497  1248
1169  1160   575   554  1203   528  1197   530   485  1228   505  1240      570   559  1176  1153   526  1185   532  1215   512  1233   484  1229
1342   401  1316   399  1308  1287   418   445   468  1277   464  1249      391  1340   409  1318   431   436  1301  1290  1253   460   469  1276
403  1328   397  1330   417   446  1307  1288   449  1264   477  1268     1322   389  1336   411  1302  1289   432   435   476  1269  1260   453
386  1341   416  1315  1286  1305   448   419  1278   467  1250   463     1339   408  1317   394   433   430  1291  1304   459  1254  1275   470
1327   388  1329   414   447   420  1285  1306  1263   450  1267   478      406  1321   396  1335  1292  1303   434   429  1270   475   454  1259
362  1345  1376   375   348   327  1378  1405   316   301  1430  1411     1349   366   371  1372  1391  1396   341   330  1427  1414   317   300
383  1368   361  1346  1377  1406   347   328  1431  1410   313   304     1364   379  1350   365   342   329  1392  1395   320   297  1426  1415
1352   367   370  1369   326   345  1408  1379   302  1435  1412   309      363  1348  1373   374  1393  1390   331   344  1413   308   299  1438
1361   378  1351   368  1407  1380   325   346  1409   312   303  1434      382  1365   364  1347   332   343  1394  1389   298  1439  1416   305
4
1182  1163   564   549   522  1189   536  1211   502  1243   490  1223      565   548  1179   166  1207   524  1193   534   495  1218   499  1246
561   552  1183  1162  1187   544  1213   514   487  1226   507  1238     1178  1167   568   545   542  1201   516  1199   510  1235   482  1231
556  1181  1158   563  1206   521  1196   535  1244   501  1224   489     1155   566   557  1180   523  1192   533  1210  1217   496  1245   500
1159   562   553  1184   543  1204   513  1198  1225   488  1237   508      560  1177  1154   567  1186   541  1216   515  1236   509  1232   481
402  1325   400  1331  1282  1309   444   423  1280   465  1252   461     1323   392  1333   410   437   426  1295  1300   457  1256  1273   472
1343   404  1313   398   443   424  1281  1310  1261   452  1265   480      390  1337   412  1319  1296  1299   438   425  1272   473   456  1257
1326   385  1332   415  1312  1283   422   441   466  1279   462  1251      407  1324   393   334   427   440  1297  1294  1255   458   471  1274
387  1344   413  1314   421   442  1311  1284   451  1262   479  1266     1338   405  1320   395  1298  1293   428   439   474  1271  1258   455
384  1367  1354   353   322   349  1404  1383  1422   315   292  1429     1363   380   357  1358  1397  1386   335   340   293  1428  1419   318
1353   354  1375   376  1403  1384   321   350   289  1432  1423   314      358  1357   372  1371   336   339  1398  1385  1418   319   296  1425
1362   377   360  1359   352   323  1382  1401  1436  1421   310   291      381  1366  1355   356  1387  1400   337   334   307   294  1437  1420
359  1360   369  1370  1381  1402   351   324   311   290  1433  1424     1356   355  1374   373   338   333  1388  1399  1440  1417   306   295
5
18  1725    16  1699  1666  1693    60    39  1664  1641    82    71     1727     4  1697    30    59    40  1665  1694    83    70  1661  1644
1707    24  1717    10    53    42  1679  1684    87    66  1657  1648        6  1705    28  1719  1680  1683    54    41  1660  1645    86    67
1726     1  1700    31  1692  1671    34    61    74  1663  1640    81       19  1728    13  1698    33    62  1691  1672  1637    84    75  1662
7  1708    25  1718    47    52  1685  1674  1633    88    79  1658     1706    21  1720    11  1686  1673    48    51    78  1659  1636    85
182  1563  1546   167  1572  1591   154   141  1628   109   118  1603     1551   162   179  1566   159   140  1573  1586   119  1602  1625   112
1561   168   181  1548  1593  1582   131   152   115  1606  1629   108      180  1549  1568   161   134   145  1600  1579  1632   105   114  1607
172  1541  1560   185   158   137  1576  1587   110   123  1604  1621     1553   192   173  1540  1569  1590   155   144  1601  1624   111   122
1543   186   171  1558   135   148  1597  1578  1605  1620   107   126      174  1555  1538   191  1596  1583   130   149   106   127  1608  1617
1526  1511   220   201   250   237  1476  1495   288  1449  1458   263     1519  1534   193   212  1477  1490   255   236  1463   258   281  1456
219   202  1525  1512   227   248  1497  1486  1459   262   285  1452      194   211  1520  1533  1504  1483   230   241   284  1453  1462   259
1514  1531   200   213  1480  1491   254   233   266   287  1448  1457     1523  1506   221   208   251   240  1473  1494  1441  1464   271   282
199   214  1513  1532  1501  1482   231   244  1445  1460   267   286      222   207  1524  1505   226   245  1500  1487   270   283  1444  1461
6
1710    17  1716    15  1696  1667    38    57  1642    95    72  1649        3  1712    29  1714    37    58  1695  1668    69  1652  1643    94
23  1724     9  1702    43    56  1681  1678    65  1656  1647    90     1722     5  1704    27  1682  1677    44    55  1646    91    68  1653
2  1709    32  1715  1670  1689    64    35    96    73  1650  1639     1711    20  1713    14    63    36  1669  1690  1651  1638    93    76
1723     8  1701    26    49    46  1675  1688  1655  1634    89    80       22  1721    12  1703  1676  1687    50    45    92    77  1654  1635
1562   183   166  1547  1594  1581   132   151  1614  1627   100   117      163  1550  1567   178   133   146  1599  1580    97   120  1615  1626
165  1564  1545   184  1571  1592   153   142   101   116  1611  1630     1552   177   164  1565   160   139  1574  1585  1610  1631   104   113
1544   169   188  1557   136   147  1598  1577   124  1613  1622    99      189  1556  1537   176  1595  1584   129   150  1623    98   121  1616
187  1542  1559   170   157   138  1575  1588  1619   102   125  1612     1554   175   190  1539  1570  1589   156   143   128  1609  1618   103
218   203  1528  1509   228   247  1498  1485  1450  1471   264   273      195   210  1517  1536  1503  1484   229   242   257   280  1455  1466
1527  1510   217   204   249   238  1475  1496   261   276  1451  1470     1518  1535   196   209  1478  1489   256   235  1454  1467   260   277
198   215  1516  1529  1502  1481   232   243  1472   265   274  1447      223   206  1521  1508   225   246  1499  1488   279  1442  1465   272
1515  1530   197   216  1479  1492   253   234   275  1446  1469   268     1522  1507   224   205   252   239  1474  1493  1468   269   278  1443```

Edward's Multiply Square

Reversible numbers multiplicative magic square This magic square pair was published in Scripta Mathematica (Vol XXII, p.202) in 1957 by Ronald B Edwards. Each square consists of the numbers of the other square with the digits reversed. The first square may be bordered to create an additive magic square with S = 438.

46

39

231

11

121

21

26

69

13

253

33

42

63

22

23

143

P = 4558554

64

93

132

11

121

12

62

96

31

352

33

24

36

22

32

341

P = 8642304
 98 102 51 64 79 44 58 46 39 231 11 53 95 121 21 26 69 106 50 13 253 33 42 47 96 63 22 23 143 91 41 93 52 61 94 97

See more on this and other material  on Christian Boyer’s April 2007 Update page at multimagie/English/Multiplicative.htm

Symmetric Magic Diamonds

 Ted Harper sent me these patterns in July, 2005. This is an order-8 simple magic square , numbered from 0 to 63 so S = 252. The diamond is the result of rotating the square 45 degrees and converting all the numbers to the octal number system. Considering just the digits, notice the symmetry across the vertical center line. This is another simple magic square. This time, the integers have been converted to the binary number system in the diamond. Again consider just the digits when checking out the symmetry.

Durupt Bent-diagonal Order-8 Cube

In 2005, Arséne Durupt (France) constructed an order 8 pantriagonal magic cube with some nice additional features. At that time he was 83 years old and did this without the help of a computer.

Features:

·        Sum of the two planar diagonals in each plane is 4104 ( which = 2S)
(No planes have correctly summing diagonals, so there are no magic squares.)

·        Sums of the corners of all orders 2, 4, 6 and 8 cubes (within this main cube) is equal to S
This is with the corner of the sub-cube on any cell and includes wrap-around.

·        Because corners of all 2x2 squares sum to ½ S, this cube is classed as compact.

·        Horizontal Planar Bent Diagonals (V shaped):
on all horizontal planes, and vertical planes parallel to the front of the cube starting on columns 1 and 5,
on all vertical planes parallel to the sides, starting on columns 3 and 7.
There are no planes that have all vertical bent diagonals starting on any particular row or column.

I show a cube with similar (but slightly improved features) here. It was constructed by Abhinav Soni in 2005 (using a computer).
I show a cube by John Hendricks here that contains 28 order-4 magic cubes, so contains many bent triagonals!

Durupt’s cube.

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

Fillion Bent-diagonal Order-16 Cube

On June 10, 2007 I received an email from Jacques Fillion mentioning that he had constructed an order 16 cube with bent diagonals.

This cube has the following features:

• It is magic because all row, columns, and the 4 triagonals sum correctly.
• It is pantriagonal magic because all broken triagonals sum correctly.
• It contains no order 16 magic squares because none of the 48 pairs of main diagonals sum correctly. (In fact none of the 96 diagonals do.)
However, all diagonal pairs sum to 2S.
• Corners of all even order sub cubes (2,4,6,8,10,12,14, and 16) starting on ANY cell, sum correctly!

Horizontal Bent Diagonals (i.e. 2 sections of 8 cells at 90 degrees).

• ALL bent diagonals starting on any cell of columns 1 and 9 of all horizontal planes and all vertical planes parallel with the front of the cubes.
• ALL bent diagonals starting on any cell of columns 5 and 13 of all planes parallel to the sides of the cubes.
• No other horizontal bent diagonals.

Vertical Bent diagonals

• ALL bent diagonals starting on any cell of rows 5 and 13 of all horizontal planes
• ALL bent diagonals starting on any cell of rows1 and 9 of all vertical planes parallel (parallel to front and parallel to sides).
• No other bent diagonals.

Good work Jacques!

Recent Publications and Postings

My JRM articles

Recent publishings by this editor, in The Journal of Recreational Mathematics

 Problem 2617: Magic Cube of Primes Harvey D. Heinz JRM:31:4:2002-2003: 298 Problem 2584: Prime (Magical) Square    (Solution) Harvey D. Heinz JRM:32:1:2003-2004: 30-36 Problem 2617: Magic Cube of Primes      (Solution) Allen Wm. Johnson, Jr. JRM:32:4:2003-2004: 338-9 A Unified Classification System for Magic Hypercubes H. Heinz and J. Hendricks JRM:32:1:2003-2004: 30-36 The First  (?) Magic Cube Harvey D. Heinz JRM:33:2:2004-2005: 111-115 The First  (?) Perfect Magic Cubes Harvey D. Heinz JRM:33:2:2004-2005: 116-119 Hypercube Classes - An Update Harvey D. Heinz JRM:35:1:2006: 5-10 Magic Tesseract Classes Harvey D. Heinz JRM:35:1:2006: 11-14

Three new pages

 Magic 9x5 Hexagrams April 4, 2007 This new page shows 1 solution for each S of 46 to 54 of the 5 numbers in each of 9 lines magic hexagram. April 4, 2007 This new page starts with a discussion of "the ‘most magical magic square in 5,000 years’. (It's not!) Then is shown order-12 and order-16 Most-perfect Bent diagonal magic squares. April 14, 2007 . A little bit about another old recreation, touring the chessboard with a knight, then arranging the tour so the numbered steps form a magic square. Compact Magic Squares May 18, 2007 A proof and demonstration that all sub-arrays with even dimensions are pan-magic in a compact magic square. Addendum to Order 6 stars May 17,2007 A short article on the first solvers of the magic hexagon (6-pointed star). Magic Square Update Sept. 29/2009 3 new types. Normal squares just a subset. Postage stamp m.s., etc.

Acknowledgements

For their contributions to these five latest pages, I wish to thank the following (in no particular order).
Christian Boyer, Ted Harper, Awani Kumar, Donald Morris, Abhinav Soni, Walter Trump, and Aale de Winkel.

And thanks to the following for recent correspondence pointing out typos (or other errors) on my pages.
Jp Gesukens, Frans Lelieveld, and Yu Jianbin.

New Tesseract Pages

On November 22, 2007, I posted 11 pages devoted to magic Tesseracts. Because of space restraints, I again had to open another site.
However, all 3 sites on this subject; My Geocities Magic-squares.net site, My Shaw cubes site, and now my Shaw Tesseract site are linked by buttons in the button bars of most pages. For those of you with high speed Internet (most I hope) the delay in switching sites should be almost non-existent.
I apologize for any inconvenience. My hope is to eventually have all these sites consolidated into one big site.

The new site is http://members.shaw.ca/tesseracts/

Nasik (Perfect) Hypercube Generators

Recently, Dwane Campbell posted a new site that discusses Perfect magic hypercubes.

He uses Hendricks definition of perfect, where all possible lines must sum correctly. To make certain that he s understood correctly, he also uses Nakamura's definitions of pan-2,3-agonal for cubes, pan-2,3,4-agonal for tesseracts, pan-2,3,4,5-agonal for dimension 5 perfect hypercubes, etc. He finalizes the definition with my new suggestion of the term Nasik.

Campbell discusses a basic method of constructing this type of magic hypercube. He supplies references, examples, and some definitions.

He concludes with 3 perfect magic hypercube generators and an order-8 magic cube tester spreadsheet that may be downloaded. The tester is quite comprehensive, although applicable only to order-8 cubes.

Dwane Campbell's website is at http://home.earthlink.net/~dwanecampbel/