Magic Cubes - Order 12

On this page I will show four order 12 cubes; a simple, a pantriagonal, a diagonal , and the complete order 12 Hendricks simple magic cube which includes 8 order 4 inlaid pandiagonal magic cubes and 48 order 4 pandiagonal magic squares.

Not shown on this page is my order 12 composition magic cube, which appears on the Composition Cubes page.

Three more inlaid magic cubes and an inlaid tesseract appear on my Inlaid Cubes page.

Poyo Simple Magic cube

Contains no magic squares. Corners of some internal cubes are O. K.

de Winkel's Pantriagonal

Contains no magic squares, but all pantriagonals are correct.

Benson & Jacoby

This diagonal type cube contains 42 simple magic squares.

Hendricks Inlaid

Contains 48 pandiagonal magic squares and 8 order 4 magic cubes.

My Composition cube

Contains 27 order 4 magic cubes arranged as an order 3 cube.

More Inlaid cubes

3 different inlaid cubes and 1 inlaid tesseract

Poyo Simple Magic cube

Poyo’s associated simple magic cube was obtained from Suzuki's original web page. [1]
There are no magic squares in this cube. However, the corners of sub-cubes 2, 3, 6, 7, and 12 starting on all odd rows are O.K. Sub-cubes 3, 4, 7, 11 starting on all even rows are also sum to 8/12 of S (which is 10374).

I will show the top horizontal plane only.

```1728   145   289  1296  1152  721  865   720   576  1297  1441   144
13  1572  1428   445   589  996  852  1021  1165   420   276  1597
25  1560  1416   457   601  984  840  1033  1177   408   264  1609
1692   181   325  1260  1116  757  901   684   540  1333  1477   108
1680   193   337  1248  1104  769  913   672   528  1345  1489    96
61  1524  1380   493   637  948  804  1069  1213   372   228  1645
73  1512  1368   505   649  936  792  1081  1225   360   216  1657
1644   229   373  1212  1068  805  949   636   492  1381  1525    60
1632   241   385  1200  1056  817  961   624   480  1393  1537    48
109  1476  1332   541   685  900  756  1117  1261   324   180  1693
121  1464  1320   553   697  888  744  1129  1273   312   168  1705
1596   277   421  1164  1020  853  997   588   444  1429  1573    12```

[1] Matsumi Suzuki's math pages are now available at MathForum

de Winkel's Pantriagonal type magic cube

This order 12 magic cube is not associated, and does not contain any magic squares. However, it is pantriagonal, which means that all broken triagonal lines (as well as the four main triagonals) sum correctly.
It is also 'complete', a term defined by Kanji Setsuda for the feature that every pantriagonal contains m/2 complement pairs, spaced m/2 apart.
Still another highlight of the cube is the fact that the eight corners of all 2x2x2, 4x4x4, 6x6x6, 7x7x7, 8x8x8, 10x10x10, and 12x12x12 sub-cubes contained in it sum to 8/12 of S. This feature holds when wrap-around is considered, so in effect, there are 123 sub-cubes of each size mentioned.

The above characteristics seem to indicate that this could be considered a Most-perfect magic cube. However, this cube is not the highest possible order cube, which is also a requirement, just as the pantriagonal magic square is the highest possible class of magic square.
Some confusion is caused by the fact that the pantriagonal magic cube has the similar feature as the pandiagonal magic square. That is; a plane may be moved from one side to the other of the cube, just as a row or column may be moved from one side to the other of a magic square, with the cube or square remaining magic.
To recap: A pandiagonal magic square has all 4 possible lines passing through each cell, summing to the magic constant. A perfect magic cube has all 13 possible lines passing through each cell, summing to the magic constant.

```Horizontal plane 1 - Top
1  1716    25  1692    49  1668   133  1608   109  1632    85  1656
1727    14  1703    38  1679    62  1595   122  1619    98  1643    74
3  1714    27  1690    51  1666   135  1606   111  1630    87  1654
1725    16  1701    40  1677    64  1593   124  1617   100  1641    76
5  1712    29  1688    53  1664   137  1604   113  1628    89  1652
1723    18  1699    42  1675    66  1591   126  1615   102  1639    78
12  1705    36  1681    60  1657   144  1597   120  1621    96  1645
1718    23  1694    47  1670    71  1586   131  1610   107  1634    83
10  1707    34  1683    58  1659   142  1599   118  1623    94  1647
1720    21  1696    45  1672    69  1588   129  1612   105  1636    81
8  1709    32  1685    56  1661   140  1601   116  1625    92  1649
1722    19  1698    43  1674    67  1590   127  1614   103  1638    79
Horizontal plane 2
1584   157  1560   181  1536   205  1452   265  1476   241  1500   217
146  1571   170  1547   194  1523   278  1463   254  1487   230  1511
1582   159  1558   183  1534   207  1450   267  1474   243  1498   219
148  1569   172  1545   196  1521   280  1461   256  1485   232  1509
1580   161  1556   185  1532   209  1448   269  1472   245  1496   221
150  1567   174  1543   198  1519   282  1459   258  1483   234  1507
1573   168  1549   192  1525   216  1441   276  1465   252  1489   228
155  1562   179  1538   203  1514   287  1454   263  1478   239  1502
1575   166  1551   190  1527   214  1443   274  1467   250  1491   226
153  1564   177  1540   201  1516   285  1456   261  1480   237  1504
1577   164  1553   188  1529   212  1445   272  1469   248  1493   224
151  1566   175  1542   199  1518   283  1458   259  1482   235  1506
Horizontal plane 3
289  1428   313  1404   337  1380   421  1320   397  1344   373  1368
1439   302  1415   326  1391   350  1307   410  1331   386  1355   362
291  1426   315  1402   339  1378   423  1318   399  1342   375  1366
1437   304  1413   328  1389   352  1305   412  1329   388  1353   364
293  1424   317  1400   341  1376   425  1316   401  1340   377  1364
1435   306  1411   330  1387   354  1303   414  1327   390  1351   366
300  1417   324  1393   348  1369   432  1309   408  1333   384  1357
1430   311  1406   335  1382   359  1298   419  1322   395  1346   371
298  1419   322  1395   346  1371   430  1311   406  1335   382  1359
1432   309  1408   333  1384   357  1300   417  1324   393  1348   369
296  1421   320  1397   344  1373   428  1313   404  1337   380  1361
1434   307  1410   331  1386   355  1302   415  1326   391  1350   367
Horizontal plane 4
1296   445  1272   469  1248   493  1164   553  1188   529  1212   505
434  1283   458  1259   482  1235   566  1175   542  1199   518  1223
1294   447  1270   471  1246   495  1162   555  1186   531  1210   507
436  1281   460  1257   484  1233   568  1173   544  1197   520  1221
1292   449  1268   473  1244   497  1160   557  1184   533  1208   509
438  1279   462  1255   486  1231   570  1171   546  1195   522  1219
1285   456  1261   480  1237   504  1153   564  1177   540  1201   516
443  1274   467  1250   491  1226   575  1166   551  1190   527  1214
1287   454  1263   478  1239   502  1155   562  1179   538  1203   514
441  1276   465  1252   489  1228   573  1168   549  1192   525  1216
1289   452  1265   476  1241   500  1157   560  1181   536  1205   512
439  1278   463  1254   487  1230   571  1170   547  1194   523  1218
Horizontal plane 5
577  1140   601  1116   625  1092   709  1032   685  1056   661  1080
1151   590  1127   614  1103   638  1019   698  1043   674  1067   650
579  1138   603  1114   627  1090   711  1030   687  1054   663  1078
1149   592  1125   616  1101   640  1017   700  1041   676  1065   652
581  1136   605  1112   629  1088   713  1028   689  1052   665  1076
1147   594  1123   618  1099   642  1015   702  1039   678  1063   654
588  1129   612  1105   636  1081   720  1021   696  1045   672  1069
1142   599  1118   623  1094   647  1010   707  1034   683  1058   659
586  1131   610  1107   634  1083   718  1023   694  1047   670  1071
1144   597  1120   621  1096   645  1012   705  1036   681  1060   657
584  1133   608  1109   632  1085   716  1025   692  1049   668  1073
1146   595  1122   619  1098   643  1014   703  1038   679  1062   655
Horizontal plane 6
1008   733   984   757   960   781   876   841   900   817   924   793
722   995   746   971   770   947   854   887   830   911   806   935
1006   735   982   759   958   783   874   843   898   819   922   795
724   993   748   969   772   945   856   885   832   909   808   933
1004   737   980   761   956   785   872   845   896   821   920   797
726   991   750   967   774   943   858   883   834   907   810   931
997   744   973   768   949   792   865   852   889   828   913   804
731   986   755   962   779   938   863   878   839   902   815   926
999   742   975   766   951   790   867   850   891   826   915   802
729   988   753   964   777   940   861   880   837   904   813   928
1001   740   977   764   953   788   869   848   893   824   917   800
727   990   751   966   775   942   859   882   835   906   811   930
Horizontal plane 7
1585   132  1609   108  1633    84  1717    24  1693    48  1669    72
143  1598   119  1622    95  1646    11  1706    35  1682    59  1658
1587   130  1611   106  1635    82  1719    22  1695    46  1671    70
141  1600   117  1624    93  1648     9  1708    33  1684    57  1660
1589   128  1613   104  1637    80  1721    20  1697    44  1673    68
139  1602   115  1626    91  1650     7  1710    31  1686    55  1662
1596   121  1620    97  1644    73  1728    13  1704    37  1680    61
134  1607   110  1631    86  1655     2  1715    26  1691    50  1667
1594   123  1618    99  1642    75  1726    15  1702    39  1678    63
136  1605   112  1629    88  1653     4  1713    28  1689    52  1665
1592   125  1616   101  1640    77  1724    17  1700    41  1676    65
138  1603   114  1627    90  1651     6  1711    30  1687    54  1663
Horizontal plane 8
288  1453   264  1477   240  1501   156  1561   180  1537   204  1513
1442   275  1466   251  1490   227  1574   167  1550   191  1526   215
286  1455   262  1479   238  1503   154  1563   178  1539   202  1515
1444   273  1468   249  1492   225  1576   165  1552   189  1528   213
284  1457   260  1481   236  1505   152  1565   176  1541   200  1517
1446   271  1470   247  1494   223  1578   163  1554   187  1530   211
277  1464   253  1488   229  1512   145  1572   169  1548   193  1524
1451   266  1475   242  1499   218  1583   158  1559   182  1535   206
279  1462   255  1486   231  1510   147  1570   171  1546   195  1522
1449   268  1473   244  1497   220  1581   160  1557   184  1533   208
281  1460   257  1484   233  1508   149  1568   173  1544   197  1520
1447   270  1471   246  1495   222  1579   162  1555   186  1531   210
Horizontal plane 9
1297   420  1321   396  1345   372  1429   312  1405   336  1381   360
431  1310   407  1334   383  1358   299  1418   323  1394   347  1370
1299   418  1323   394  1347   370  1431   310  1407   334  1383   358
429  1312   405  1336   381  1360   297  1420   321  1396   345  1372
1301   416  1325   392  1349   368  1433   308  1409   332  1385   356
427  1314   403  1338   379  1362   295  1422   319  1398   343  1374
1308   409  1332   385  1356   361  1440   301  1416   325  1392   349
422  1319   398  1343   374  1367   290  1427   314  1403   338  1379
1306   411  1330   387  1354   363  1438   303  1414   327  1390   351
424  1317   400  1341   376  1365   292  1425   316  1401   340  1377
1304   413  1328   389  1352   365  1436   305  1412   329  1388   353
426  1315   402  1339   378  1363   294  1423   318  1399   342  1375
Horizontal plane 10
576  1165   552  1189   528  1213   444  1273   468  1249   492  1225
1154   563  1178   539  1202   515  1286   455  1262   479  1238   503
574  1167   550  1191   526  1215   442  1275   466  1251   490  1227
1156   561  1180   537  1204   513  1288   453  1264   477  1240   501
572  1169   548  1193   524  1217   440  1277   464  1253   488  1229
1158   559  1182   535  1206   511  1290   451  1266   475  1242   499
565  1176   541  1200   517  1224   433  1284   457  1260   481  1236
1163   554  1187   530  1211   506  1295   446  1271   470  1247   494
567  1174   543  1198   519  1222   435  1282   459  1258   483  1234
1161   556  1185   532  1209   508  1293   448  1269   472  1245   496
569  1172   545  1196   521  1220   437  1280   461  1256   485  1232
1159   558  1183   534  1207   510  1291   450  1267   474  1243   498
Horizontal plane 11
1009   708  1033   684  1057   660  1141   600  1117   624  1093   648
719  1022   695  1046   671  1070   587  1130   611  1106   635  1082
1011   706  1035   682  1059   658  1143   598  1119   622  1095   646
717  1024   693  1048   669  1072   585  1132   609  1108   633  1084
1013   704  1037   680  1061   656  1145   596  1121   620  1097   644
715  1026   691  1050   667  1074   583  1134   607  1110   631  1086
1020   697  1044   673  1068   649  1152   589  1128   613  1104   637
710  1031   686  1055   662  1079   578  1139   602  1115   626  1091
1018   699  1042   675  1066   651  1150   591  1126   615  1102   639
712  1029   688  1053   664  1077   580  1137   604  1113   628  1089
1016   701  1040   677  1064   653  1148   593  1124   617  1100   641
714  1027   690  1051   666  1075   582  1135   606  1111   630  1087
Horizontal plane 12 - Bottom
864   877   840   901   816   925   732   985   756   961   780   937
866   851   890   827   914   803   998   743   974   767   950   791
862   879   838   903   814   927   730   987   754   963   778   939
868   849   892   825   916   801  1000   741   976   765   952   789
860   881   836   905   812   929   728   989   752   965   776   941
870   847   894   823   918   799  1002   739   978   763   954   787
853   888   829   912   805   936   721   996   745   972   769   948
875   842   899   818   923   794  1007   734   983   758   959   782
855   886   831   910   807   934   723   994   747   970   771   946
873   844   897   820   921   796  1005   736   981   760   957   784
857   884   833   908   809   932   725   992   749   968   773   944
871   846   895   822   919   798  1003   738   979   762   955   786```

Benson & Jacoby's Diagonal type magic cube

This cube appears in Magic Cubes New Recreations [1] where the authors refer to it as ‘perfect’ because all planar squares are magic. By the new Hendricks consistent magic cube definitions, it should be called a diagonal type cube because both main diagonals of all 3m planar squares are correct. C. Boyer refers to this type as perfect.(For a short time I called this type a myers cube, after R.L.Myers who constructed an order 8 cube of this type).

So this cube has 3m+6 simple magic squares. An additional feature (not part of the 'diagonal' definition) is that the 8 corners of all 123 (including wrap-around) order 7 sub-cubes sum to 6916 which is 8/12 of the constant of 10374 for this order 12 cube.

This cube is not associated. All pantriagonals in one of the four directions sum correctly to S.

[1] Benson & Jacoby, Magic Cubes: New Recreations, Dover Publ., 1981, pp.103-115 (fig. 11-4 for the top Horizontal Plane)

```Horizontal plane 1 - top
659  1538   698  1511   623  1454  1091   242  1130   215  1055   158
353  1196  1313   512   389  1280  1361   476   305  1232  1397   560
71   971    14   947    38   986  1655   827  1598   803  1622   842
1073   185  1028   221  1112   272   641  1481   596  1517   680  1568
1379   530   419  1214  1343   446   371  1250  1427   494   335  1166
1649   764  1604   797  1685   848    65   908    20   941   101   992
650  1547   707  1502   614  1463  1082   251  1139   206  1046   167
356  1193  1316   509   392  1277  1364   473   308  1229  1400   557
74   962   131   926    47   887  1658   818  1715   782  1631   743
1088   188  1133   212  1109   161   656  1484   701  1508   677  1457
1370   539   410  1223  1334   455   362  1259  1418   503   326  1175
1652   761  1601   800  1688   845    68   905    17   944   104   989
Horizontal plane 2
1017   177   633  1576  1125   232   585  1468  1065   280   688  1528
1431   550   375  1162  1323   490   423  1270  1383   442   315  1210
1588   748    88  1005  1696   777   136   897  1636   729   117   957
718  1558  1102   147   610  1491  1150   255   670  1443  1035   195
297  1180  1353   568   405  1240  1305   460   345  1288  1413   520
135   975  1671   730    27   922  1719   838    87   874  1618   778
1012   172   628  1581  1120   237   580  1473  1060   285   693  1533
1438   543   382  1155  1330   483   430  1263  1390   435   322  1203
1593   753    57  1000  1701   808     9   892  1641   856   112   952
711  1551  1059   154   603  1534  1011   262   663  1582  1042   202
292  1185  1348   573   400  1245  1300   465   340  1293  1408   525
142   982  1678   723    34   915  1726   831    94   867  1611   771
Horizontal plane 3
1075  1482  1026   223   679   270   643   186   594  1519  1111  1566
361  1260  1417   504   325  1176  1369   540   409  1224  1333   456
1651   763  1602   799  1686   846    67   907    18   943   102   990
649   241   708  1501  1056  1464  1081  1537  1140   205   624   168
1363   474   307  1230  1399   558   355  1194  1315   510   391  1278
61   972    24   937    37   996  1645   828  1608   793  1621   852
1086  1483  1135   210   678   163   654   187   703  1506  1110  1459
372  1249  1428   493   336  1165  1380   529   420  1213  1344   445
1650   906  1603   798   103   847    66   762    19   942  1687   991
660   252   697  1512  1045  1453  1092  1548  1129   216   613   157
1362   475   306  1231  1398   559   354  1195  1314   511   390  1279
84   817   121   936  1632   877  1668   961  1705   792    48   733
Horizontal plane 4
1142   263   671  1442  1034   203   710  1559  1103   146   602  1499
1304   461   344  1289  1412   521   296  1181  1352   569   404  1241
1718   830    95   866  1619   779   134   974  1679   722    35   923
584   176  1061   284  1121  1529  1016  1472   629  1580   689   233
422  1271  1382   443   314  1211  1430   551   374  1163  1322   491
140   893  1673   728   116   917  1724   749    89   872  1700   773
1019  1550   626  1583   611   230   587   254  1058   287  1043  1526
1301   464   341  1292  1409   524   293  1184  1349   572   401  1244
1727   839    86   875  1610   770   143   983  1670   731    26   914
581  1469  1064   281   692  1532  1013   173   632  1577  1124   236
431  1262  1391   434   323  1202  1439   542   383  1154  1331   482
5   896  1640   857   113   956  1589   752    56  1001  1697   812
Horizontal plane 5
652  1545  1132   213   616  1461  1084   249   700  1509  1048   165
358  1191  1318   507   394  1275  1366   471   310  1227  1402   555
81   964  1605   784    45   988  1653   820   129   940  1629   736
1071   190   591  1522  1107   274   639  1486  1023   226   675  1570
1372   537   412  1221  1336   453   364  1257  1420   501   328  1173
1654   759    22   939  1690   843    70   903  1606   795   106   987
657  1540  1137   208   621  1456  1089   244   705  1504  1053   160
351  1198  1311   514   387  1282  1359   478   303  1234  1395   562
76   969  1708   789    40   885  1660   825   124   933  1624   741
1078   183   706  1515  1114   159   658  1479  1030   207   682  1563
1377   532   417  1216  1341   448   369  1252  1425   496   333  1168
1647   766    15   946  1683   850    63   910  1599   802    99   994
Horizontal plane 6
1014   175   631  1578  1122   235   582  1471  1063   282   690  1531
1440   541   384  1164  1332   481   432  1261  1381   433   324  1201
1722   750    91   870  1699   775   138   894  1675   726   115   919
720  1560  1093   156   601  1489  1152   264   661  1452  1033   193
294  1183  1350   570   402  1243  1302   463   343  1291  1410   523
144   973  1669   732    36   913  1728   829    85   876  1620   769
1015   174   630  1579  1123   234   583  1470  1062   283   691  1530
1429   552   373  1153  1321   492   421  1272  1392   444   313  1212
1591   751    54  1003  1698   810     7   895  1638   859   114   954
577  1549  1068   277   612  1536  1009   253   636  1573  1044   240
295  1182  1351   571   403  1242  1303   462   342  1290  1411   522
133   984  1680   721    25   924  1717   840    96   865  1609   780
Horizontal plane 7
647  1478  1022   227   683  1562  1079   182   590  1523  1115   266
365  1256  1424   500   329  1172  1373   536   413  1217  1337   452
83   911  1706   791    98   878  1667   767   122   935  1682   734
1085   245   704  1505  1052   164   653  1541  1136   209   620  1460
1367   470   302  1226  1403   554   359  1190  1319   515   395  1274
1661   824   128   929  1625   740    77   968  1712   785    41   884
638  1487  1031   218   674  1571  1070   191   599  1514  1106   275
368  1253  1421   497   332  1169  1376   533   416  1220  1340   449
62   902  1607   794   107   995  1646   758    23   938  1691   851
1076   248   593  1520  1049   269   644  1544  1025   224   617  1565
1358   479   311  1235  1394   563   350  1199  1310   506   386  1283
1664   821   125   932  1628   737    80   965  1709   788    44   881
Horizontal plane 8
1149   256   669  1444  1041   196   717  1552  1101   148   609  1492
1299   466   339  1294  1407   526   291  1186  1347   574   399  1246
1720   837    52   873  1612   813     4   981  1672   861    28   921
586  1467  1066   279   694  1527  1018   171   634  1575  1126   231
429  1264  1389   436   321  1204  1437   544   381  1156  1329   484
3   898  1635   862   111   958  1587   754    51  1006  1695   814
1144   261   664  1449  1036   201   712  1557  1096   153   604  1497
1306   459   346  1287  1414   519   298  1179  1354   567   406  1239
1725   832    93   868  1617   772   141   976  1677   724    33   916
579  1474  1095   286   687  1498  1143   178   627  1450  1119   238
424  1269  1384   441   316  1209  1432   549   376  1161  1324   489
10   891  1642   855   118   951  1594   747    58   999  1702   807
Horizontal plane 9
655  1542  1134   211   619  1458  1087   246   702  1507  1051   162
349  1200  1309   505   385  1284  1357   480   312  1236  1393   564
79   967  1710   787    42   882  1663   823   126   931  1626   738
1069  1477   600  1513   684   276   637   181  1032   217  1116  1572
1375   534   415  1219  1339   450   367  1254  1422   498   331  1170
1657   768   132   925  1681   744    73   912  1716   781    97   888
642   247  1027   222  1050  1567  1074  1543   595  1518   618   271
360  1189  1320   516   396  1273  1368   469   301  1225  1404   553
78   966  1711   786    43   883  1662   822   127   930  1627   739
1080   192   589  1524  1105   265   648  1488  1021   228   673  1561
1374   535   414  1218  1338   451   366  1255  1423   499   330  1171
1656   757    13   948  1692   841    72   901  1597   804   108   985
Horizontal plane 10
578   179   635  1574  1118  1535  1010  1475  1067   278   686   239
1436   545   377  1157  1328   485   428  1265  1388   440   320  1205
2   890    59   998   119   959  1586   746  1643   854  1703   815
1148  1556  1097   152   605   197   716   260   665  1448  1037  1493
290  1187  1355   575   398  1247  1298   467   338  1286  1406   527
1592   833  1637   860  1616   809     8   977    53  1004    32   953
719   170   662  1451  1127  1490  1151  1466  1094   155   695   194
1433   548   380  1160  1325   488   425  1268  1385   437   317  1208
11   755    50  1007  1694   950  1595   899  1634   863   110   806
1145  1553  1100   149   608   200   713   257   668  1445  1040  1496
299  1178  1346   566   407  1238  1307   458   347  1295  1415   518
1721   980  1676   725    29   776   137   836    92   869  1613   920
Horizontal plane 11
640  1480  1024   225   676  1569  1072   189   592  1521  1113   273
370  1251  1426   495   334  1167  1378   531   418  1215  1342   447
69   909  1713   796   105   880  1665   760    21   928  1684   844
1083   243   699  1510  1047   166   651  1546  1131   214   622  1462
1360   477   304  1233  1396   561   352  1197  1312   513   388  1281
1666   826   130   927  1630   735    82   963  1714   783    39   879
645  1485  1029   220   681  1564  1077   184   597  1516  1108   268
363  1258  1419   502   327  1174  1371   538   411  1222  1335   454
64   904  1600   801   100   993  1648   765    16   945  1689   849
1090   250   598  1503  1054   267   646  1539  1138   219   615  1455
1365   472   309  1228  1401   556   357  1192  1317   508   393  1276
1659   819   123   934  1623   742    75   970  1707   790    46   886
Horizontal plane 12 - Bottom
1146   259  1099   150  1038   199   714  1555   667  1446   606  1495
1308   457   348  1285  1416   517   300  1177  1356   565   408  1237
1590   834  1639   858  1615   811     6   978    55  1002    31   955
588  1476   625  1584   685  1525  1020   180  1057   288  1117   229
426  1267  1386   439   318  1207  1434   547   378  1159  1326   487
12   889    49  1008   120   949  1596   745  1633   864  1704   805
1147   258  1098   151  1039   198   715  1554   666  1447   607  1494
1297   468   337  1296  1405   528   289  1188  1345   576   397  1248
1723   835  1674   727  1614   774   139   979    90   871    30   918
709  1465   672  1441   696  1500  1141   169  1104   145  1128   204
427  1266  1387   438   319  1206  1435   546   379  1158  1327   486
1   900    60   997   109   960  1585   756  1644   853  1693   816```

Hendricks Inlaid order-12 magic cube

This is an order 12 normal magic cube with 8 order 4 inlaid pantriagonal magic cubes and 48 order 4 pandiagonal magic squares placed within a single layer ‘expansion shell’.

The order 12 cube is simple magic and not associated. It consists of the consecutive numbers from 1 to 123 and contains no order 12 magic squares or other extra features (aside from the inlays).

The constant sum for the order 12 cube is 10,374 which is the required sum for a normal order 12. The sum for the order 4 cubes and squares is 3,458 (which is 4/12 of 10,374. Obviously these inlaid cubes and squares are not normal because they cannot contain consecutive numbers.

This illustration is a feeble attempt to show the parts of the cube.

Variations of this cube just by operations on the 8 inlaid cubes are;

• Each sub-cube may translocate planar faces in 64 ways.
• Each sub-cube may rotate and/or reflect in 48 ways.
• Sub-cubes may be placed in different locations in the main cube in 8! ways.

This gives a total of 123,863,040 possible variations involving the inlaid cubes only. Still available are the variations involving the 48 pandiagonal magic squares!

Each pandiagonal magic square can have 144 variations due to row, column translocation and each square has 8 aspects. However, all four squares in a ‘stack’ must have the same transformation and aspect in order to keep the integrity of the 1-agonals in the order 12 cube. The 12 stacks themselves may be relocated in the cube in any of 12! ways.
So there are 6,621,718,118,400 variations involving the squares alone (if my math is correct).

Multiplying these two numbers together gives the total variations for this one cube. Remember, the cube itself can also be shown in any of 48 aspects.

AND we haven't even started to look at the variations possible in the 'shell'. For instance, all lines of 4 numbers that border an inlaid square or cube plane also sum to 3458. The location of these lines can be rearranged, a long as care is taken to keep complete stacks together to preserve the integrity of the lines at right angles.

John Hendricks devotes about 20 pages in his book to this cube. He shows the solution set he used, as well as algebraic and numerical listings, and construction figures. [1][2]

Following is the listing for the 12 horizontal planes of the cube. I will indicate the inlaid horizontal magic squares and planes of the order 4 cubes with blue text.
Following this listing, are those for two of the eight inlaid pantriagonal cubes

```Horizontal plane 1 – Top         Top 4 pandiagonal magic squares
942  1230   355   222  1651   643  1075    67  1518  1363   510   798
966   474  1400   401  1183   619  1110   174  1700   101  1483   763
751  1267   317  1340   534  1122   607  1543    41  1616   258   978
882  1328   546  1255   329   703  1026  1628   246  1555    29   847
859   389  1195   462  1412  1014   715   113  1471   186  1688   870
775   487  1386  1495    90  1098   666  1674   199   378  1207   919
811  1242   343   234  1639  1062   630    55  1530  1351   522   955
727   306  1280   569  1303  1146   583     6  1580   269  1603  1002
990  1435   437  1172   414   595  1134  1711   161  1448   138   739
835  1160   426  1423   449  1038   691  1460   126  1723   149   894
906   557  1315   294  1292   679  1050   281  1591    18  1568   823
930   499  1374  1507    78   655  1087  1662   211   366  1219   786

Horizontal plane 2      Top plane of the top 4 pantriagonal magic cubes
940  1245   352  1371   490   640  1089  1534    63  1660   201   789
760   563  1308   422  1165   904   681   157  1718    12  1571  1113
1125   289  1274   456  1439   693   892   143  1464   266  1585   748
610  1296   311  1417   434  1042   831  1442   121  1607   288   975
963  1310   565  1163   420   819  1054  1716   155  1573    14   622
772  1209   364  1359   526  1101   664  1498    75  1648   237   921
813   508  1353   382  1215  1060   633   219  1642    93  1504   952
735  1394   457  1271   336   879   706  1632   263  1465    98  1138
1150  1188   395  1333   542   718   867  1550    37  1691   180   723
585   397  1190   540  1331  1017   856    35  1548   182  1693  1000
988   479  1416   314  1249   844  1029   241  1610   120  1487   597
933   496  1389   346  1227   657  1072   207  1678    57  1516   796

Horizontal plane 3      2nd plane of the top 4 pantriagonal magic cubes
790   483  1378   357  1240  1090   639   196  1665    70  1527   939
970  1429   446  1284   299   826  1047  1595   276  1454   133   615
603  1175   432  1298   553  1035   838  1561     2  1728   167   982
1120   410  1153   575  1320   688   897    24  1583   145  1706   753
765   444  1427   301  1286   909   676   278  1597   131  1452  1108
958   519  1366   369  1204   627  1066   232  1653    82  1491   807
915  1222   375  1348   513   670  1095  1509    88  1635   226   778
993   552  1343   385  1178   849  1024   170  1681    47  1560   592
580   326  1261   467  1404  1012   861   108  1475   253  1622  1005
1143  1259   324  1406   469   711   874  1477   110  1620   251   730
742  1321   530  1200   407   886   699  1703   192  1538    25  1131
795  1234   339  1384   501  1071   658  1521    52  1671   214   934

Horizontal plane 4      3rd plane of the top 4 pantriagonal magic cubes
947   358  1239   484  1377   647  1082    69  1528   195  1666   782
1119   312  1295   433  1418   687   898   122  1441   287  1608   754
766   566  1309   419  1164   910   675   156  1715    13  1574  1107
969  1307   564  1166   421   825  1048  1717   158  1572    11   616
604  1273   290  1440   455  1036   837  1463   144  1586   265   981
779   370  1203   520  1365  1094   671    81  1492   231  1654   914
806  1347   514  1221   376  1067   626  1636   225  1510    87   959
1144  1189   398  1332   539   712   873  1547    36  1694   181   729
741  1415   480  1250   313   885   700  1609   242  1488   119  1132
994   458  1393   335  1272   850  1023   264  1631    97  1466   591
579   396  1187   541  1334  1011   862    38  1549   179  1692  1006
926  1383   502  1233   340   650  1079  1672   213  1522    51   803

Horizontal plane 5      4th plane of the top 4 pantriagonal magic cubes
781  1372   489  1246   351  1081   648  1659   202  1533    64   948
609  1154   409  1319   576  1041   832  1584    23  1705   146   976
964  1428   443  1285   302   820  1053  1598   277  1451   132   621
759   445  1430   300  1283   903   682   275  1596   134  1453  1114
1126   431  1176   554  1297   694   891     1  1562   168  1727   747
949  1360   525  1210   363   636  1057  1647   238  1497    76   816
924   381  1216   507  1354   661  1104    94  1503   220  1641   769
586   323  1260   470  1405  1018   855   109  1478   252  1619   999
987   529  1322   408  1199   843  1030   191  1704    26  1537   598
736  1344   551  1177   386   880   705  1682   169  1559    48  1137
1149  1262   325  1403   468   717   868  1476   107  1621   254   724
804   345  1228   495  1390  1080   649    58  1515   208  1677   925

Horizontal plane 6       2nd set of the 4 magic squares
785   497  1376  1505    80  1088   656  1664   209   368  1217   929
761   475  1397   404  1182  1112   620   175  1697   104  1482   965
980  1266   320  1337   535   605  1121  1542    44  1613   259   752
845  1325   547  1254   332  1028   704  1625   247  1554    32   881
872   392  1194   463  1409   713  1013   116  1470   187  1685   860
956  1244   341   236  1637   629  1061    53  1532  1349   524   812
920   488  1385  1496    89   665  1097  1673   200   377  1208   776
1004   307  1277   572  1302   581  1145     7  1577   272  1602   728
737  1434   440  1169   415  1136   596  1710   164  1445   139   989
896  1157   427  1422   452   689  1037  1457   127  1722   152   836
821   560  1314   295  1289  1052   680   284  1590    19  1565   905
797  1229   356   221  1652  1076   644    68  1517  1364   509   941

Horizontal plane 7      3rd set of the 4 magic squares
788  1220   365   212  1661  1085   653    77  1508  1373   500   932
824  1253   331  1326   548  1049   677  1553    31  1626   248   908
893   464  1410   391  1193   692  1040   188  1686   115  1469   833
740   403  1181   476  1398  1133   593   103  1481   176  1698   992
1001  1338   536  1265   319   584  1148  1614   260  1541    43   725
953   521  1352  1529    56   632  1064  1640   233   344  1241   809
917  1205   380   197  1676   668  1100    92  1493  1388   485   773
869  1421   451  1158   428   716  1016  1721   151  1458   128   857
848   296  1290   559  1313  1025   701    20  1566   283  1589   884
977   571  1301   308  1278   608  1124   271  1601     8  1578   749
764  1170   416  1433   439  1109   617  1446   140  1709   163   968
800   512  1361  1520    65  1073   641  1649   224   353  1232   944

Horizontal plane 8      1st plane of the bottom 4 magic cubes
792  1391   494  1225   348  1092   637  1680   205  1514    59   937
746  1161   418  1312   567   890   695  1575    16  1714   153  1127
1115  1419   436  1294   309   683   902  1605   286  1444   123   758
624   454  1437   291  1276  1056   817   268  1587   141  1462   961
973   424  1167   561  1306   829  1044    10  1569   159  1720   612
960  1355   506  1213   384   625  1068  1644   217  1502    95   805
913   362  1211   528  1357   672  1093    73  1500   239  1646   780
721   316  1251   477  1414   865   720   118  1485   243  1612  1152
1140   538  1329   399  1192   708   877   184  1695    33  1546   733
599  1335   544  1186   393  1031   842  1689   178  1552    39   986
998  1269   334  1396   459   854  1019  1467   100  1630   261   587
793   350  1247   492  1369  1069   660    61  1536   203  1658   936

Horizontal plane 9      2nd plane of the bottom 4 magic cubes
938   337  1236   503  1382   638  1091    50  1523   216  1669   791
984   303  1288   442  1425   840  1033   129  1450   280  1599   601
613   573  1318   412  1155  1045   828   147  1708    22  1581   972
1106  1300   555  1173   430   674   911  1726   165  1563     4   767
755  1282   297  1431   448   899   686  1456   135  1593   274  1118
770   373  1224   515  1346  1103   662    86  1511   228  1633   923
815  1368   517  1202   371  1058   635  1655   230  1489    84   950
1007  1198   405  1323   532   863  1010  1540    27  1701   190   578
590  1408   471  1257   322  1022   851  1618   249  1479   112   995
1129   465  1402   328  1263   697   888   255  1624   106  1473   744
732   387  1180   550  1341   876   709    45  1558   172  1683  1141
935  1380   481  1238   359   659  1070  1667   194  1525    72   794

Horizontal plane 10      3rd plane of the bottom 4 magic cubes
783   504  1381   338  1235  1083   646   215  1670    49  1524   946
1105  1438   453  1275   292   673   912  1588   267  1461   142   768
756  1168   423  1305   562   900   685  1570     9  1719   160  1117
983   417  1162   568  1311   839  1034    15  1576   154  1713   602
614   435  1420   310  1293  1046   827   285  1606   124  1443   971
951   516  1345   374  1223   634  1059   227  1634    85  1512   814
922  1201   372  1367   518   663  1102  1490    83  1656   229   771
1130   543  1336   394  1185   698   887   177  1690    40  1551   743
731   333  1270   460  1395   875   710    99  1468   262  1629  1142
1008  1252   315  1413   478   864  1009  1486   117  1611   244   577
589  1330   537  1191   400  1021   852  1696   183  1545    34   996
802  1237   360  1379   482  1078   651  1526    71  1668   193   927

Horizontal plane 11      4th plane of the bottom 4 magic cubes
945  1226   347  1392   493   645  1084  1513    60  1679   206   784
623   556  1299   429  1174  1055   818   166  1725     3  1564   962
974   298  1281   447  1432   830  1043   136  1455   273  1594   611
745  1287   304  1426   441   889   696  1449   130  1600   279  1128
1116  1317   574  1156   411   684   901  1707   148  1582    21   757
777  1214   383  1356   505  1096   669  1501    96  1643   218   916
808   527  1358   361  1212  1065   628   240  1645    74  1499   957
600  1401   466  1264   327  1032   841  1623   256  1474   105   985
997  1179   388  1342   549   853  1020  1557    46  1684   171   588
722   406  1197   531  1324   866   719    28  1539   189  1702  1151
1139   472  1407   321  1258   707   878   250  1617   111  1480   734
928   491  1370   349  1248   652  1077   204  1657    62  1535   801

Horizontal plane 12      4th set of the 4 magic squares
943   511  1362  1519    66   642  1074  1650   223   354  1231   799
907  1256   330  1327   545   678  1051  1556    30  1627   245   822
834   461  1411   390  1196  1039   690   185  1687   114  1472   895
991   402  1184   473  1399   594  1135   102  1484   173  1699   738
726  1339   533  1268   318  1147   582  1615   257  1544    42  1003
774  1206   379   198  1675  1099   667    91  1494  1387   486   918
810   523  1350  1531    54  1063   631  1638   235   342  1243   954
858  1424   450  1159   425  1015   714  1724   150  1459   125   871
883   293  1291   558  1316   702  1027    17  1567   282  1592   846
750   570  1304   305  1279  1123   606   270  1604     5  1579   979
967  1171   413  1436   438   618  1111  1447   137  1712   162   762
931  1218   367   210  1663   654  1086    79  1506  1375   498   787```
This is one of the 8 inlaid order 4 pantriagonal magic cubes. It is in the top left back position of the order 12 cube above, so it appears in planes 2, 3, 4, and 5. It is pantriagonal (as are the other 7) so all 16 pantriagonals in each of the four directions sum correctly to 3458. It is also ‘complete’, because Every pantriagonal contains m/2 complement pairs, spaced m/2 apart.
```Top                       II                        III                       IV
563  1308   422  1165    1429   446  1284   299     312  1295   433  1418    1154   409  1319   576
289  1274   456  1439    1175   432  1298   553     566  1309   419  1164    1428   443  1285   302
1296   311  1417   434     410  1153   575  1320    1307   564  1166   421     445  1430   300  1283
1310   565  1163   420     444  1427   301  1286     312  1295   433  1418     431  1176   554  1297```
This is another of the eight inlaid cubes. This one is located in the bottom right front of the order 12 cube. therefore, the top plane is located in the in the lower right quadrant of plane 8 of the listing for the mother cube.
```Top                       II                        III                       IV
118  1485   243  1612    1540    27  1701   190     177  1690    40  1551    1623   256  1474   105
184  1695    33  1546    1618   249  1479   112      99  1468   262  1629    1557    46  1684   171
1689   178  1552    39     255  1624   106  1473    1486   117  1611   244      28  1539   189  1702
1467   100  1630   261      45  1558   172  1683    1696   183  1545    34     250  1617   111  1480```

[1] John R. Hendricks, Inlaid Magic Squares and Cubes, self-published, 1999, 0-9684700-1-7, 188+ pages.
[2] John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd edition, self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated by Holger Danielsson.

 This page was originally posted March 2003 It was last updated February 15, 2010 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz