# Magic Cubes - Order 13

 Golunski Pantriagonal Laio Perfect cube Broken oblique plane Soni Order 14 Simple

### Golunski pantriagonal order 13 magic cube

I received this cube via email from Bogdan Golunski of Germany on November 26, 2003. It is not associated, and must be classed as a pantriagonal magic cube even though it contains many magic squares.
All 13 of the horizontal planes and all 13 of the planes parallel to the front of the cube are pandiagonal magic squares. All broken diagonals and 12 of the 13 main diagonals in 1 direction of the vertical planes parallel to the sides of the cube sum incorrectly. Because of this, only 1 of these 13 planes is a simple magic square. 3 of the 6 oblique planes are simple magic squares and 3 are pandiagonal magic squares.
Total for the cube, 4 simple magic squares and 29 pandiagonal magic squares.

All pantriagonals in this cube sum correctly. So, because all 3m orthogonal planes in this cube are not magic squares (and the same type), this cube must be classed only as pantriagonal magic! A most unusual cube!

```Plane 1 - Top
456  2108   204   915   606    58  1290  1541  1869   823  1157  1857  1403
1044  1768  1454   402  2133   184   854   655   138  1345  1563  1983   764
1618  1914   787  1037  1694  1508   479  2196   209   964   594    31  1256
613    22  1188  1672  1995   835  1060  1817  1434   377  2104   272   898
2041   323   987   664    41  1309  1604  1893   747  1108  1749  1466   355
1823  1515   387  2149   260   882   584    92  1237  1634  1877   684  1163
1907   798  1100  1722  1420   436  2090   277   871   518   155  1314  1679
89  1221  1587  1952   737  1123  1711  1358   484  2165   335   888   637
241   946   563   117  1207  1533  2003   808  1179  1734  1477   422  2057
1409   450  2047   172   995   647   160  1235  1649  1949   703  1081  1790
701  1015  1835  1491   499  2075   292   926   540    75  1278  1586  1974
1336  1655  2028   726  1143  1769  1380   412  2124   229   955   523     7
1009   552   120  1268  1557  1928   780  1077  1806  1366   340  2180   304
Plane 2 - Top-1
1220  1598  1951   730  1127  1708  1360   487  2161   332   894   628    91
943   569   108  1209  1532  2014   807  1172  1738  1474   424  2060   237
447  2049   175   991   644   166  1226  1651  1948   714  1080  1783  1413
1026  1834  1484   503  2072   294   929   536    72  1284  1577  1976   700
1661  2019   728  1142  1780  1379   405  2128   226   957   526     3  1333
554   123  1264  1554  1934   771  1079  1805  1377   339  2173   308  1006
2107   197   919   603    60  1293  1537  1866   829  1148  1859  1402   467
1759  1456   401  2144   183   847   659   135  1347  1566  1979   761  1050
1917   783  1034  1700  1499   481  2195   220   963   587    35  1253  1620
15  1192  1669  1997   838  1056  1814  1440   368  2106   271   909   612
325   986   675    40  1302  1608  1890   749  1111  1745  1463   361  2032
1511   384  2155   251   884   583   103  1236  1627  1881   681  1165  1826
802  1097  1724  1423   432  2087   283   862   520   154  1325  1678  1900
Plane 3 - Top-2
1447   403  2143   194   846   652   139  1344  1568  1982   757  1047  1765
786  1030  1697  1505   472  2197   219   974   586    28  1257  1617  1919
1185  1673  1994   840  1059  1810  1437   374  2097   273   908   623    14
988   674    51  1301  1601  1894   746  1113  1748  1459   358  2038   316
380  2152   257   875   585   102  1247  1626  1874   685  1162  1828  1514
1101  1721  1425   435  2083   280   868   511   156  1324  1689  1899   795
1597  1962   729  1120  1712  1357   489  2164   328   891   634    82  1222
566   114  1200  1534  2013   818  1171  1731  1478   421  2062   240   939
2046   177   994   640   163  1232  1642  1950   713  1091  1782  1406   451
1845  1483   496  2076   291   931   539    68  1281  1583  1967   702  1025
2025   719  1144  1779  1390   404  2121   230   954   528     6  1329  1658
125  1267  1550  1931   777  1070  1807  1376   350  2172   301  1010   551
196   912   607    57  1295  1540  1862   826  1154  1850  1404   466  2118
Plane 4 - Top-3
111  1206  1525  2015   817  1182  1730  1471   425  2059   242   942   562
174   996   643   159  1229  1648  1941   715  1090  1793  1405   444  2050
1494   495  2069   295   928   541    71  1277  1580  1973   693  1027  1844
725  1135  1781  1389   415  2120   223   958   525     8  1332  1654  2022
1269  1553  1927   774  1076  1798  1378   349  2183   300  1003   555   122
911   600    61  1292  1542  1865   822  1151  1856  1395   468  2117   207
394  2145   193   857   651   132  1348  1565  1984   760  1043  1762  1453
1033  1693  1502   478  2188   221   973   597    27  1250  1621  1916   788
1666  1998   837  1061  1813  1433   371  2103   264   910   622    25  1184
676    50  1312  1600  1887   750  1110  1750  1462   354  2035   322   979
2148   254   881   576   104  1246  1637  1873   678  1166  1825  1516   383
1725  1422   437  2086   276   865   517   147  1326  1688  1910   794  1094
1961   740  1119  1705  1361   486  2166   331   887   631    88  1213  1599
Plane 5 - Top-4
1696  1498   475  2194   212   975   596    38  1249  1614  1920   785  1035
1991   841  1058  1815  1436   367  2100   270   901   624    24  1195  1665
52  1311  1611  1886   743  1114  1747  1464   357  2031   319   985   667
250   878   582    95  1248  1636  1884   677  1159  1829  1513   385  2151
1426   434  2088   279   861   514   153  1317  1690  1909   805  1093  1718
739  1130  1704  1354   490  2163   333   890   627    85  1219  1590  1963
1203  1531  2006   819  1181  1741  1470   418  2063   239   944   565   107
993   645   162  1225  1645  1947   706  1092  1792  1416   443  2043   178
506  2068   288   932   538    73  1280  1576  1970   699  1018  1846  1493
1141  1772  1391   414  2131   222   951   529     5  1334  1657  2018   722
1555  1930   770  1073  1804  1369   351  2182   311  1002   548   126  1266
599    54  1296  1539  1867   825  1147  1853  1401   459  2119   206   922
2136   195   856   662   131  1341  1569  1981   762  1046  1758  1450   400
Plane 6 - Top-5
642   164  1228  1641  1944   712  1083  1794  1415   454  2042   171   997
2079   287   925   542    70  1282  1579  1966   696  1024  1837  1495   505
1778  1382   416  2130   233   950   522     9  1331  1659  2021   718  1138
1932   773  1069  1801  1375   342  2184   310  1013   547   119  1270  1552
53  1289  1543  1864   827  1150  1849  1398   465  2110   208   921   610
186   858   661   142  1340  1562  1985   759  1048  1761  1446   397  2142
1501   471  2191   218   966   598    37  1260  1613  1913   789  1032  1698
834  1062  1812  1438   370  2096   267   907   615    26  1194  1676  1990
1313  1610  1897   742  1107  1751  1461   359  2034   315   982   673    43
874   579   101  1239  1638  1883   688  1158  1822  1517   382  2153   253
438  2085   281   864   510   150  1323  1681  1911   804  1104  1717  1419
1129  1715  1353   483  2167   330   892   630    81  1216  1596  1954   741
1528  2012   810  1183  1740  1481   417  2056   243   941   567   110  1199
Plane 7 - Middle
1055  1816  1435   372  2099   263   904   621    17  1196  1675  2001   833
1612  1896   753  1106  1744  1465   356  2036   318   978   670    49  1304
575    98  1245  1629  1885   687  1169  1821  1510   386  2150   255   877
2089   278   866   513   146  1320  1687  1902   806  1103  1728  1418   431
1714  1364   482  2160   334   889   632    84  1212  1593  1960   732  1131
2009   816  1174  1742  1480   428  2055   236   945   564   112  1202  1524
161  1230  1644  1940   709  1089  1785  1417   453  2053   170   990   646
298   924   535    74  1279  1581  1969   692  1021  1843  1486   507  2078
1388   407  2132   232   961   521     2  1335  1656  2023   721  1134  1775
775  1072  1797  1372   348  2175   312  1012   558   118  1263  1556  1929
1288  1536  1868   824  1152  1852  1394   462  2116   199   923   609    64
849   663   141  1351  1561  1978   763  1045  1763  1449   393  2139   192
474  2187   215   972   589    39  1259  1624  1912   782  1036  1695  1503
Plane 8 - Botom+5
935   534    67  1283  1578  1971   695  1017  1840  1492   498  2080   297
413  2123   234   960   532     1  1328  1660  2020   723  1137  1771  1385
1074  1800  1368   345  2181   303  1014   557   129  1262  1549  1933   772
1535  1861   828  1149  1854  1397   458  2113   205   914   611    63  1299
654   143  1350  1572  1977   756  1049  1760  1451   396  2135   189   855
2190   211   969   595    30  1261  1623  1923   781  1029  1699  1500   476
1809  1439   369  2101   266   900   618    23  1187  1677  2000   844  1054
1898   752  1117  1743  1458   360  2033   320   981   666    46  1310  1603
94  1242  1635  1876   689  1168  1832  1509   379  2154   252   879   578
282   863   515   149  1316  1684  1908   797  1105  1727  1429   430  2082
1363   493  2159   327   893   629    86  1215  1589  1957   738  1122  1716
813  1180  1733  1482   427  2066   235   938   568   109  1204  1527  2005
1227  1646  1943   705  1086  1791  1408   455  2052   181   989   639   165
Plane 9 - Botom+4
754  1116  1754  1457   353  2037   317   983   669    42  1307  1609  1889
1238  1632  1882   680  1170  1831  1520   378  2147   256   876   580    97
867   512   151  1319  1680  1905   803  1096  1729  1428   441  2081   275
492  2170   326   886   633    83  1217  1592  1953   735  1128  1707  1365
1177  1739  1473   429  2065   246   937   561   113  1201  1529  2008   809
1643  1945   708  1082  1788  1414   446  2054   180  1000   638   158  1231
545    66  1276  1582  1968   697  1020  1836  1489   504  2071   299   934
2129   225   962   531    12  1327  1653  2024   720  1139  1774  1381   410
1802  1371   341  2178   309  1005   559   128  1273  1548  1926   776  1071
1860   821  1153  1851  1399   461  2109   202   920   602    65  1298  1546
134  1352  1571  1988   755  1042  1764  1448   398  2138   185   852   660
214   965   592    36  1252  1625  1922   792  1028  1692  1504   473  2192
1432   373  2098   268   903   614    20  1193  1668  2002   843  1065  1808
Plane 10 - Bottom+3
231   953   533    11  1338  1652  2017   724  1136  1776  1384   406  2126
1373   344  2174   306  1011   550   130  1272  1559  1925   769  1075  1799
820  1146  1855  1396   463  2112   198   917   608    56  1300  1545  1871
1343  1573  1987   766  1041  1757  1452   395  2140   188   848   657   140
968   588    33  1258  1616  1924   791  1039  1691  1497   477  2189   216
366  2102   265   905   617    16  1190  1674  1993   845  1064  1819  1431
1118  1753  1468   352  2030   321   980   671    45  1303  1606  1895   745
1628  1879   686  1161  1833  1519   389  2146   249   880   577    99  1241
516   148  1321  1683  1901   800  1102  1720  1430   440  2092   274   860
2169   337   885   626    87  1214  1594  1956   731  1125  1713  1356   494
1736  1479   420  2067   245   948   560   106  1205  1526  2010   812  1173
1942   710  1085  1784  1411   452  2045   182   999   649   157  1224  1647
77  1275  1575  1972   694  1022  1839  1485   501  2077   290   936   544
Plane 11 - Bottom+2
1875   683  1167  1824  1521   388  2157   248   873   581    96  1243  1631
152  1318  1685  1904   796  1099  1726  1421   442  2091   285   859   509
336   896   625    80  1218  1591  1958   734  1121  1710  1362   485  2171
1476   426  2058   247   947   571   105  1198  1530  2007   814  1176  1732
707  1087  1787  1407   449  2051   173  1001   648   168  1223  1640  1946
1286  1574  1965   698  1019  1841  1488   497  2074   296   927   546    76
959   524    13  1337  1663  2016   717  1140  1773  1386   409  2122   228
346  2177   302  1008   556   121  1274  1558  1936   768  1068  1803  1370
1145  1848  1400   460  2114   201   913   605    62  1291  1547  1870   831
1564  1989   765  1052  1756  1445   399  2137   190   851   653   137  1349
591    29  1255  1622  1915   793  1038  1702  1496   470  2193   213   970
2095   269   902   619    19  1186  1671  1999   836  1066  1818  1442   365
1755  1467   363  2029   314   984   668    47  1306  1602  1892   751  1109
Plane 12 - Bottom+1
2179   305  1004   553   127  1265  1560  1935   779  1067  1796  1374   343
1847  1393   464  2111   203   916   601    59  1297  1538  1872   830  1156
1980   767  1051  1767  1444   392  2141   187   853   656   133  1346  1570
32  1251  1619  1921   784  1040  1701  1507   469  2186   217   967   593
262   906   616    21  1189  1667  1996   842  1057  1820  1441   376  2094
1469   362  2040   313   977   672    44  1308  1605  1888   748  1115  1746
679  1164  1830  1512   390  2156   259   872   574   100  1240  1633  1878
1322  1682  1906   799  1095  1723  1427   433  2093   284   870   508   145
895   636    79  1211  1595  1955   736  1124  1706  1359   491  2162   338
423  2064   238   949   570   116  1197  1523  2011   811  1178  1735  1472
1084  1789  1410   445  2048   179   992   650   167  1234  1639  1939   711
1585  1964   691  1023  1838  1490   500  2070   293   933   537    78  1285
530     4  1339  1662  2027   716  1133  1777  1383   411  2125   224   956
Plane 13 - Bottom
1686  1903   801  1098  1719  1424   439  2084   286   869   519   144  1315
635    90  1210  1588  1959   733  1126  1709  1355   488  2168   329   897
2061   244   940   572   115  1208  1522  2004   815  1175  1737  1475   419
1786  1412   448  2044   176   998   641   169  1233  1650  1938   704  1088
1975   690  1016  1842  1487   502  2073   289   930   543    69  1287  1584
10  1330  1664  2026   727  1132  1770  1387   408  2127   227   952   527
307  1007   549   124  1271  1551  1937   778  1078  1795  1367   347  2176
1392   457  2115   200   918   604    55  1294  1544  1863   832  1155  1858
758  1053  1766  1455   391  2134   191   850   658   136  1342  1567  1986
1254  1615  1918   790  1031  1703  1506   480  2185   210   971   590    34
899   620    18  1191  1670  1992   839  1063  1811  1443   375  2105   261
364  2039   324   976   665    48  1305  1607  1891   744  1112  1752  1460
1160  1827  1518   381  2158   258   883   573    93  1244  1630  1880   682```

### Laio Perfect order 13 magic cube

This cube is not associated. It is perfect and so contains 9m pandiagonal magic squares. That is 3 * 13 = 39 orthogonal order 13 pandiagonal magic squares, 6 oblique, and 6m-6 broken oblique order 13 pandiagonal magic squares.
The fact that there are 6m-6 broken oblique pandiagonal magic squares in a perfect cube was first mentioned by Rosser and Walker in 1938 [1]. It was mentioned again in Liao's paper [2].

This cube contains    507         1-agonals (rows, columns and pillars) [3]
1014  pan-2-agonals (pan-diagonals)
676  pan-3-agonals (pan-triagonals)
The discrepancy between the above agonals (number lines) is due to the fact that the same line of numbers appear in several different magic squares.
This cube appeared in a technical report by
F. Liao, T. Katayama and K. Takaba of Kyoto University in 1999. [2]
Following is a listing of this cube. Then I show an example broken oblique plane.

[1] B. Rosser and R. J. Walker, Magic Squares: Published papers and Supplement, 1939, a bound volume at Cornell University, catalogued as QA 165 R82+pt.1-4. All papers are very technical. There are NO diagrams.
[2]  F. Liao, T. Katayama and K. Takaba, On the Construction of Pandiagonal Magic Cubes, Kyoto Univ. Technical Report # 99021, 1999
[3]
H. D. Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0, page 165.

### Laio Order 13 Perfect Magic Cube

```Plane 1 - Top
1   184   367   550   733   916  1099  1282  1465  1648  1831  2014  2197
1746  1929  2112    98   281   464   647   830  1013  1027  1197  1380  1563
1294  1477  1660  1843  2026  2040    26   196   379   562   745   928  1111
842   856  1039  1222  1392  1575  1758  1941  2124   110   293   476   659
221   391   574   757   940  1123  1306  1489  1672  1855  1869  2052    38
1953  2136   122   305   488   671   685   868  1051  1234  1417  1587  1770
1501  1684  1698  1881  2064    50   233   416   586   769   952  1135  1318
880  1063  1246  1429  1612  1782  1965  2148   134   317   500   514   697
428   611   781   964  1147  1330  1513  1527  1710  1893  2076    62   245
2160   146   329   343   526   709   892  1075  1258  1441  1624  1807  1977
1539  1722  1905  2088    74   257   440   623   806   976  1159  1342  1356
1087  1270  1453  1636  1819  2002  2172   158   172   355   538   721   904
635   818  1001  1171  1185  1368  1551  1734  1917  2100    86   269   452
Plane 2
931  1114  1297  1480  1663  1846  2016  2030    16   199   382   565   748
479   662   845   846  1029  1212  1395  1578  1761  1944  2127   113   296
2042    28   211   394   577   760   943  1126  1309  1492  1675  1858  1872
1590  1773  1956  2139   125   308   491   674   688   871  1041  1224  1407
1138  1321  1504  1687  1701  1884  2067    40   223   406   589   772   955
517   700   883  1066  1236  1419  1602  1785  1968  2151   137   320   503
65   235   418   601   784   967  1150  1333  1516  1530  1713  1896  2079
1797  1980  2163   149   332   346   529   712   895  1078  1261  1431  1614
1345  1359  1542  1725  1908  2091    77   260   430   613   796   979  1162
724   907  1090  1273  1456  1626  1809  1992  2175   161   175   358   541
272   455   625   808   991  1174  1188  1371  1554  1737  1920  2103    89
2004  2187     4   187   370   553   736   919  1102  1285  1468  1651  1821
1383  1566  1749  1932  2115   101   284   467   650   820  1003  1017  1200
Plane 3
1848  1862  2045    31   214   397   580   763   946  1129  1312  1495  1665
1227  1410  1593  1776  1959  2142   128   311   494   664   678   861  1044
775   958  1141  1324  1507  1690  1691  1874  2057    43   226   409   592
323   506   520   690   873  1056  1239  1422  1605  1788  1971  2154   140
1886  2069    55   238   421   604   787   970  1153  1336  1519  1533  1716
1434  1617  1800  1983  2166   152   335   349   532   715   885  1068  1251
982  1165  1348  1362  1545  1728  1911  2081    67   250   433   616   799
361   544   727   910  1080  1263  1446  1629  1812  1995  2178   164   178
2106    79   262   445   628   811   994  1177  1191  1374  1557  1740  1923
1641  1824  2007  2190     7   190   373   556   739   922  1105  1275  1458
1020  1203  1386  1569  1752  1935  2118   104   274   457   640   823  1006
568   751   934  1117  1300  1470  1653  1836  2019  2033    19   202   385
116   299   469   652   835   849  1032  1215  1398  1581  1764  1947  2130
Plane 4
412   595   778   961  1144  1314  1497  1680  1694  1877  2060    46   229
2157   143   313   496   510   693   876  1059  1242  1425  1608  1791  1974
1523  1706  1889  2072    58   241   424   607   790   973  1156  1339  1509
1071  1254  1437  1620  1803  1986  2169   155   338   339   522   705   888
619   802   985  1168  1351  1365  1535  1718  1901  2084    70   253   436
167   181   364   534   717   900  1083  1266  1449  1632  1815  1998  2181
1730  1913  2096    82   265   448   631   814   997  1180  1194  1377  1560
1278  1461  1644  1827  2010  2193    10   193   376   559   729   912  1095
826  1009  1023  1206  1389  1572  1755  1925  2108    94   277   460   643
205   388   571   754   924  1107  1290  1473  1656  1839  2022  2036    22
1950  2120   106   289   472   655   838   852  1035  1218  1401  1584  1767
1485  1668  1851  1865  2048    34   217   400   583   766   949  1119  1302
864  1047  1230  1413  1596  1779  1962  2145   118   301   484   667   681
Plane 5
1329  1512  1526  1709  1892  2075    61   244   427   610   793   963  1146
708   891  1074  1257  1440  1623  1806  1989  2159   145   328   342   525
256   439   622   805   988  1158  1341  1355  1538  1721  1904  2087    73
2001  2184   157   171   354   537   720   903  1086  1269  1452  1635  1818
1367  1550  1733  1916  2099    85   268   451   634   817  1000  1183  1184
915  1098  1281  1464  1647  1830  2013  2196    13   183   366   549   732
463   646   829  1012  1026  1209  1379  1562  1745  1928  2111    97   280
2039    25   208   378   561   744   927  1110  1293  1476  1659  1842  2025
1574  1757  1940  2123   109   292   475   658   841   855  1038  1221  1404
1122  1305  1488  1671  1854  1868  2051    37   220   403   573   756   939
670   684   867  1050  1233  1416  1599  1769  1952  2135   121   304   487
49   232   415   598   768   951  1134  1317  1500  1683  1697  1880  2063
1794  1964  2147   133   316   499   513   696   879  1062  1245  1428  1611
Plane 6
2090    76   259   442   612   795   978  1161  1344  1358  1541  1724  1907
1638  1808  1991  2174   160   174   357   540   723   906  1089  1272  1455
1173  1187  1370  1553  1736  1919  2102    88   271   454   637   807   990
552   735   918  1101  1284  1467  1650  1833  2003  2186     3   186   369
100   283   466   649   832  1002  1016  1199  1382  1565  1748  1931  2114
1845  2028  2029    15   198   381   564   747   930  1113  1296  1479  1662
1211  1394  1577  1760  1943  2126   112   295   478   661   844   858  1028
759   942  1125  1308  1491  1674  1857  1871  2054    27   210   393   576
307   490   673   687   870  1053  1223  1406  1589  1772  1955  2138   124
1883  2066    52   222   405   588   771   954  1137  1320  1503  1686  1700
1418  1601  1784  1967  2150   136   319   502   516   699   882  1065  1248
966  1149  1332  1515  1529  1712  1895  2078    64   247   417   600   783
345   528   711   894  1077  1260  1443  1613  1796  1979  2162   148   331
Plane 7
810   993  1176  1190  1373  1556  1739  1922  2105    91   261   444   627
189   372   555   738   921  1104  1287  1457  1640  1823  2006  2189     6
1934  2117   103   286   456   639   822  1005  1019  1202  1385  1568  1751
1482  1652  1835  2018  2032    18   201   384   567   750   933  1116  1299
848  1031  1214  1397  1580  1763  1946  2129   115   298   481   651   834
396   579   762   945  1128  1311  1494  1677  1847  1861  2044    30   213
2141   127   310   493   676   677   860  1043  1226  1409  1592  1775  1958
1689  1703  1873  2056    42   225   408   591   774   957  1140  1323  1506
1055  1238  1421  1604  1787  1970  2153   139   322   505   519   702   872
603   786   969  1152  1335  1518  1532  1715  1898  2068    54   237   420
151   334   348   531   714   897  1067  1250  1433  1616  1799  1982  2165
1727  1910  2093    66   249   432   615   798   981  1164  1347  1361  1544
1262  1445  1628  1811  1994  2177   163   177   360   543   726   909  1092
Plane 8
1571  1754  1937  2107    93   276   459   642   825  1008  1022  1205  1388
1106  1289  1472  1655  1838  2021  2035    21   204   387   570   753   936
654   837   851  1034  1217  1400  1583  1766  1949  2132   105   288   471
33   216   399   582   765   948  1131  1301  1484  1667  1850  1864  2047
1778  1961  2144   130   300   483   666   680   863  1046  1229  1412  1595
1326  1496  1679  1693  1876  2059    45   228   411   594   777   960  1143
692   875  1058  1241  1424  1607  1790  1973  2156   142   325   495   509
240   423   606   789   972  1155  1338  1521  1522  1705  1888  2071    57
1985  2168   154   337   351   521   704   887  1070  1253  1436  1619  1802
1364  1547  1717  1900  2083    69   252   435   618   801   984  1167  1350
899  1082  1265  1448  1631  1814  1997  2180   166   180   363   546   716
447   630   813   996  1179  1193  1376  1559  1742  1912  2095    81   264
2192     9   192   375   558   741   911  1094  1277  1460  1643  1826  2009
Plane 9
291   474   657   840   854  1037  1220  1403  1586  1756  1939  2122   108
1867  2050    36   219   402   585   755   938  1121  1304  1487  1670  1853
1415  1598  1781  1951  2134   120   303   486   669   683   866  1049  1232
950  1133  1316  1499  1682  1696  1879  2062    48   231   414   597   780
498   512   695   878  1061  1244  1427  1610  1793  1976  2146   132   315
2074    60   243   426   609   792   975  1145  1328  1511  1525  1708  1891
1622  1805  1988  2171   144   327   341   524   707   890  1073  1256  1439
1170  1340  1354  1537  1720  1903  2086    72   255   438   621   804   987
536   719   902  1085  1268  1451  1634  1817  2000  2183   169   170   353
84   267   450   633   816   999  1182  1196  1366  1549  1732  1915  2098
1829  2012  2195    12   195   365   548   731   914  1097  1280  1463  1646
1208  1391  1561  1744  1927  2110    96   279   462   645   828  1011  1025
743   926  1109  1292  1475  1658  1841  2024  2038    24   207   390   560
Plane 10
1052  1235  1405  1588  1771  1954  2137   123   306   489   672   686   869
587   770   953  1136  1319  1502  1685  1699  1882  2065    51   234   404
135   318   501   515   698   881  1064  1247  1430  1600  1783  1966  2149
1711  1894  2077    63   246   429   599   782   965  1148  1331  1514  1528
1259  1442  1625  1795  1978  2161   147   330   344   527   710   893  1076
794   977  1160  1343  1357  1540  1723  1906  2089    75   258   441   624
173   356   539   722   905  1088  1271  1454  1637  1820  1990  2173   159
1918  2101    87   270   453   636   819   989  1172  1186  1369  1552  1735
1466  1649  1832  2015  2185     2   185   368   551   734   917  1100  1283
1014  1015  1198  1381  1564  1747  1930  2113    99   282   465   648   831
380   563   746   929  1112  1295  1478  1661  1844  2027  2041    14   197
2125   111   294   477   660   843   857  1040  1210  1393  1576  1759  1942
1673  1856  1870  2053    39   209   392   575   758   941  1124  1307  1490
Plane 11
1969  2152   138   321   504   518   701   884  1054  1237  1420  1603  1786
1517  1531  1714  1897  2080    53   236   419   602   785   968  1151  1334
896  1079  1249  1432  1615  1798  1981  2164   150   333   347   530   713
431   614   797   980  1163  1346  1360  1543  1726  1909  2092    78   248
2176   162   176   359   542   725   908  1091  1274  1444  1627  1810  1993
1555  1738  1921  2104    90   273   443   626   809   992  1175  1189  1372
1103  1286  1469  1639  1822  2005  2188     5   188   371   554   737   920
638   821  1004  1018  1201  1384  1567  1750  1933  2116   102   285   468
17   200   383   566   749   932  1115  1298  1481  1664  1834  2017  2031
1762  1945  2128   114   297   480   663   833   847  1030  1213  1396  1579
1310  1493  1676  1859  1860  2043    29   212   395   578   761   944  1127
689   859  1042  1225  1408  1591  1774  1957  2140   126   309   492   675
224   407   590   773   956  1139  1322  1505  1688  1702  1885  2055    41
Plane 12 - Bottom plus 1
533   703   886  1069  1252  1435  1618  1801  1984  2167   153   336   350
68   251   434   617   800   983  1166  1349  1363  1546  1729  1899  2082
1813  1996  2179   165   179   362   545   728   898  1081  1264  1447  1630
1192  1375  1558  1741  1924  2094    80   263   446   629   812   995  1178
740   923  1093  1276  1459  1642  1825  2008  2191     8   191   374   557
275   458   641   824  1007  1021  1204  1387  1570  1753  1936  2119    92
2020  2034    20   203   386   569   752   935  1118  1288  1471  1654  1837
1399  1582  1765  1948  2131   117   287   470   653   836   850  1033  1216
947  1130  1313  1483  1666  1849  1863  2046    32   215   398   581   764
482   665   679   862  1045  1228  1411  1594  1777  1960  2143   129   312
2058    44   227   410   593   776   959  1142  1325  1508  1678  1692  1875
1606  1789  1972  2155   141   324   507   508   691   874  1057  1240  1423
1154  1337  1520  1534  1704  1887  2070    56   239   422   605   788   971
Plane 13 - Bottom
1450  1633  1816  1999  2182   168   182   352   535   718   901  1084  1267
998  1181  1195  1378  1548  1731  1914  2097    83   266   449   632   815
377   547   730   913  1096  1279  1462  1645  1828  2011  2194    11   194
2109    95   278   461   644   827  1010  1024  1207  1390  1573  1743  1926
1657  1840  2023  2037    23   206   389   572   742   925  1108  1291  1474
1036  1219  1402  1585  1768  1938  2121   107   290   473   656   839   853
584   767   937  1120  1303  1486  1669  1852  1866  2049    35   218   401
119   302   485   668   682   865  1048  1231  1414  1597  1780  1963  2133
1695  1878  2061    47   230   413   596   779   962  1132  1315  1498  1681
1243  1426  1609  1792  1975  2158   131   314   497   511   694   877  1060
791   974  1157  1327  1510  1524  1707  1890  2073    59   242   425   608
326   340   523   706   889  1072  1255  1438  1621  1804  1987  2170   156
1902  2085    71   254   437   620   803   986  1169  1352  1353  1536  1719```

### An Oblique broken plane

This is one of the 6m-6 broken oblique planes from the above cube. All are pandiagonal magic squares. this is the case for all perfect magic cubes! [1][2]

[1] B. Rosser and R. J. Walker, Magic Squares: Published papers and Supplement, 1939, a bound volume at Cornell University, catalogued as QA 165 R82+pt.1-4. All papers are very technical. There are NO diagrams. The bound book contains:

• On the Transformation Group for Diabolic Magic Squares of Order Four, Reprinted from Bulletin of the American Mathematical Society, June 1938.
• The Algebraic Theory of Diabolic Magic Squares, Reprinted from Duke Mathematical Journal Vol. 5, No. 4, Dec. 1939, pp 705-728.
Section 6 (pp 727-728) has been crossed out.
• A continuation of The Algebraic Theory of Diabolic Magic Squares on typewritten pages numbered 729 – 753. (This starts with a new section 6, renamed from ‘conclusion’ to ‘Latin Squares’). Pages 736 to 753 is about diabolic (perfect) magic cubes and points out there are 3m diabolic magic squares parallel to the faces of the cubes and 6m diabolic magic squares parallel to six diagonal planes.

[2]  F. Liao, T. Katayama and K. Takaba, On the Construction of Pandiagonal Magic Cubes, Kyoto Univ. Technical Report # 99021, 1999

### Soni Order 14 Simple

This is a simple magic cube and is not associated. Only rows, columns pillars and the 4 main triagonals sum correctly to 19215.
This cube contains no magic squares. I received it from Abinhav Soni as an email attachment on  Nov. 29, 2003

I will show the top horizontal plane only.

```1100  1141  1224  1258  1292  1326  1066   592   730   826   572  2033  1779  1875
1214  1248  1282  1372  1063  1097  1131  1735  1824  1913   686   775   864   610
593   676   367   401   484   525   559   813   909   655  1773  1869  2007   381
1050  1084  1167  1201  1235  1318  1352  1956  2045   426   858   947   357  1818
1500  1534  1575  1658  1692  1383  1466   992   395  1905  2001  1747   464   903
1271  1305  1339  1030  1120  1154  1188  1792   509   941   687   489  1950  2039
1679  1419  1453  1487  1521  1604  1645   527  1995  1741  1830   547   986   781
1964  2445  2541  1944  1690    64   160  2472  2513  2596  2630  2664  2698  2438
1392   109   198  2058  2490  2579  1982  2586  2620  2654  2744  2435  2469  2503
2528  2624  2027  1430   154   292  1753   250   333    24    58   141   182   216
241   330  1798  2573  2662  1729  1475  2079  2113  2196  2230  2264  2347  2381
2707  1767  1562   286    32  1836  2618   471   505   546   629   663   354   437
77  1881  2656  2402  1861  1607   324  2300  2334  2368  2059  2149  2183  2217
1899  1652    26   115  1919  2701  2496   650   390   424   458   492   575   616```

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Harvey Heinz   harveyheinz@shaw.ca
This page last updated October 14, 2009
Copyright © 2003 by Harvey D. Heinz