Composition Magic Cubes

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Introduction

Intro to composition magic cubes and a composition magic square.

Order-9 cube

An order 9 composition magic cube consists of 27 order 3 magic cubes.

Order-12 cube

An order 12 composition magic cube consists of 27 order 4 magic cubes.

Order-9 cube-2

A new way of constructing composition magic squares and cubes.

Order-15 cube

A cube consisting of 27 order 3 cubes is on my Large Cubes page.

 

 Introduction

Magic cubes come in as many varieties as magic squares. However, probably because of the shear volume of numbers to be manipulated when constructing magic cubes, these variations have been seldom explored.
Some example cube variations that I show on this site are; prime number, multiply, and inlaid magic cubes.

On this page I show another variety copied from magic squares, the composition magic cube.

To illustrate the concept, I show an order 12 composition magic square, consisting of 16 order 3 magic squares arranged as an order 4 magic square.

8 1 6
3 5 7
4 9 2
This is the first of 16 order 3 magic squares that make up the order 12 square.
1 8 12 13
14 11 7 2
15 10 6 3
4 5 9 16
This pattern square is index # 112 (of 880)
but is the first order 4 associated magic square.
The 16 3x3 squares are placed according to the numbers in this square.
 

8

1

6

71

64

69

107

100

105

116

109

114

3

5

7

66

68

70

102

104

106

111

113

115

4

9

2

67

72

65

103

108

101

112

117

110

125

118

123

98

91

96

62

55

60

17

10

15

120

122

124

93

95

97

57

59

61

12

14

16

121

126

119

94

99

92

58

63

56

13

18

11

134

127

132

89

82

87

53

46

51

26

19

24

129

131

133

84

86

88

48

50

52

21

23

25

130

135

128

85

90

83

49

54

47

22

27

20

35

28

33

44

37

42

80

73

78

143

136

141

30

32

34

39

41

43

75

77

79

138

140

142

31

36

29

40

45

38

76

81

74

139

144

137


The complete order 12 composition magic square. Numbers used are 1 to 122.   S = 870.

 

15 228 360 411
438 333 201 42
465 306 174 69
96 147 279 492

This 4x4 magic square is composed of the magic sums of the 16 order 3 squares within the order 12 magic square.

My first example of a composition magic cube is an order 9, constructed by assembling 27 order 3 magic cubes, arranged as an order 3 cube.

My second example is an order 12 magic cube. It consists of 27 order 4 magic cubes assembled in an order 3 arrangement.. In each case, the magic sums of the individual cubes are assembled in the same arrangement, to form a new order 3 cube with the same constant as the order 9 (or order 12) composition cube.
Notice that for the order 12 cube, I am using the order 3 as the assembly pattern and filling the cube with order 4 sub-cubes. In the composition magic square I used the opposite arrangement!

Order-9 cube

This order-9 composition magic cube is constructed from 27 order 3 magic cubes

Order 3 index #1 is the pattern for the placement of the 27 sub-cubes, and also for each of the other 26 order 3 cubes.

Plane 1_Top    Plane 2        Plane 3_Bottom
 2  13  27     16  21   5     24   8  10
22   9  11      3  14  25     17  19   6
18  20   4     23   7  12      1  15  26

The other 26 cubes are constructed to this same pattern, by using the second group of 27 numbers for the second cube, the third group of 27 numbers for the third cube, etc.
The cubes indicated in the top plane of the pattern order 3 are placed appropriately in the top 3 planes of the order 9. The order 3 cubes indicated in the middle and bottom layers are similarly placed in the middle 3 and bottom 3 planes of the order 9 cube.
I have indicated the location of the first and second 3x3 cubes with green and red type.

The order 9 composition magic cube.

Plane 1_Top                                    Plane 2_Top-1
 29   40   54  326  337  351  704  715  729     43   48   32  340  345  329  718  723  707
 49   36   38  346  333  335  724  711  713     30   41   52  327  338  349  705  716  727
 45   47   31  342  344  328  720  722  706     50   34   39  347  331  336  725  709  714
569  580  594  218  229  243  272  283  297    583  588  572  232  237  221  286  291  275
589  576  578  238  225  227  292  279  281    570  581  592  219  230  241  273  284  295
585  587  571  234  236  220  288  290  274    590  574  579  239  223  228  293  277  282
461  472  486  515  526  540   83   94  108    475  480  464  529  534  518   97  102   86
481  468  470  535  522  524  103   90   92    462  473  484  516  527  538   84   95  106
477  479  463  531  533  517   99  101   85    482  466  471  536  520  525  104   88   93
Plane 3_Top - 2                                Plane 4 Top - 3
 51   35   37  348  332  334  726  710  712    407  418  432  542  553  567  110  121  135
 44   46   33  341  343  330  719  721  708    427  414  416  562  549  551  130  117  119
 28   42   53  325  339  350  703  717  728    423  425  409  558  560  544  126  128  112
591  575  577  240  224  226  294  278  280     56   67   81  353  364  378  650  661  675
584  586  573  233  235  222  287  289  276     76   63   65  373  360  362  670  657  659
568  582  593  217  231  242  271  285  296     72   74   58  369  371  355  666  668  652
483  467  469  537  521  523  105   89   91    596  607  621  164  175  189  299  310  324
476  478  465  530  532  519   98  100   87    616  603  605  184  171  173  319  306  308
460  474  485  514  528  539   82   96  107    612  614  598  180  182  166  315  317  301
Plane 5_Middle                                 Plane 6_Bottom + 3
421  426  410  556  561  545  124  129  113    429  413  415  564  548  550  132  116  118
408  419  430  543  554  565  111  122  133    422  424  411  557  559  546  125  127  114
428  412  417  563  547  552  131  115  120    406  420  431  541  555  566  109  123  134
 70   75   59  367  372  356  664  669  653     78   62   64  375  359  361  672  656  658
 57   68   79  354  365  376  651  662  673     71   73   60  368  370  357  665  667  654
 77   61   66  374  358  363  671  655  660     55   69   80  352  366  377  649  663  674
610  615  599  178  183  167  313  318  302    618  602  604  186  170  172  321  305  307
597  608  619  165  176  187  300  311  322    611  613  600  179  181  168  314  316  303
617  601  606  185  169  174  320  304  309    595  609  620  163  177  188  298  312  323
Plane 7_Bottom + 2                             Plane 8_Bottom + 1
623  634  648  191  202  216  245  256  270    637  642  626  205  210  194  259  264  248
643  630  632  211  198  200  265  252  254    624  635  646  192  203  214  246  257  268
639  641  625  207  209  193  261  263  247    644  628  633  212  196  201  266  250  255
434  445  459  488  499  513  137  148  162    448  453  437  502  507  491  151  156  140
454  441  443  508  495  497  157  144  146    435  446  457  489  500  511  138  149  160
450  452  436  504  506  490  153  155  139    455  439  444  509  493  498  158  142  147
  2   13   27  380  391  405  677  688  702     16   21    5  394  399  383  691  696  680
 22    9   11  400  387  389  697  684  686      3   14   25  381  392  403  678  689  700
 18   20    4  396  398  382  693  695  679     23    7   12  401  385  390  698  682  687
Plane 9_Bottom
645  629  631  213  197  199  267  251  253 
638  640  627  206  208  195  260  262  249 
622  636  647  190  204  215  244  258  269 
456  440  442  510  494  496  159  143  145 
449  451  438  503  505  492  152  154  141 
433  447  458  487  501  512  136  150  161 
 24    8   10  402  386  388  699  683  685 
 17   19    6  395  397  384  692  694  681 
  1   15   26  379  393  404  676  690  701 

The numbers in the following order 3 magic cube are the magic sums of the 27 cubes in the order 9 composition cube. S = 3285.

Plane 1_Top              Plane 2                  Plane 3_Bottom
 123   1014   2148       1257   1662    366       1905    609    771
1743    690    852        204   1095   1986       1338   1500    447
1419   1581    285       1824    528    933         42   1176   2067

All order 3 cubes are associated, as we expected. The order 9 cube is simple magic with no extra features except that it is associated. Because all the cubes are odd order associated, the middle plane in each of the 3 orthogonal orientations is a magic square. 

 Order-12 cube

This order 3 magic cube is the pattern for the placement of the 27 order 4 cubes in the order 12 magic composition cube shown below. It is index # 4 of 4.

All the order 3 cubes In the first order 9 and the order 12 cube have exactly the same characteristics. That is: they all are associated, have 3 magic squares in the 3 center orthogonal planes, and have all pantriagonals in one direction correct. Also in all cases there are 2 oblique squares with correct rows, 4 oblique squares with correct columns, and 3 of the 6 oblique squares have all pandiagonals in one direction correct. These are the features common to the four basic order 3 cubes.

I_Top             II                III_Bottom
 8   12   22      15   25    2      19    5   18
24    7   11       1   14   27      17   21    4
10   23    9      26    3   13       6   16   20

This is the first of the 27 order 4 cubes.
64 is added to each number in this cube to obtain cube 2, 64 is added to each number in cube 2 to obtain cube 3, etc. Numbers used are 1 to 1728. 1728 is 12 cubed and also 27 times 64.

I_Top                  II                     III                    IV_Bottom
 1   63   62    4      48   18   19   45      32   34   35   29      49   15   14   52
60    6    7   57      21   43   42   24      37   27   26   40      12   54   55    9
56   10   11   53      25   39   38   28      41   23   22   44       8   58   59    5
13   51   50   16      36   30   31   33      20   46   47   17      61    3    2   64

I have again indicated the location of the first and second  4x4 cubes with green and red type.

The order 12 composition magic cube

Plane 1 _ Top           order 4 cubes represented by the top plane of the order 3 
                      pattern cube are in these top 4 planes.
 449    511    510    452    705    767    766    708   1345   1407   1406   1348
 508    454    455    505    764    710    711    761   1404   1350   1351   1401
 504    458    459    501    760    714    715    757   1400   1354   1355   1397
 461    499    498    464    717    755    754    720   1357   1395   1394   1360
1473   1535   1534   1476    385    447    446    388    641    703    702    644
1532   1478   1479   1529    444    390    391    441    700    646    647    697
1528   1482   1483   1525    440    394    395    437    696    650    651    693
1485   1523   1522   1488    397    435    434    400    653    691    690    656
 577    639    638    580   1409   1471   1470   1412    513    575    574    516
 636    582    583    633   1468   1414   1415   1465    572    518    519    569
 632    586    587    629   1464   1418   1419   1461    568    522    523    565
 589    627    626    592   1421   1459   1458   1424    525    563    562    528
 Plane 2 _ Top - 1
 496    466    467    493    752    722    723    749   1392   1362   1363   1389
 469    491    490    472    725    747    746    728   1365   1387   1386   1368
 473    487    486    476    729    743    742    732   1369   1383   1382   1372
 484    478    479    481    740    734    735    737   1380   1374   1375   1377
1520   1490   1491   1517    432    402    403    429    688    658    659    685
1493   1515   1514   1496    405    427    426    408    661    683    682    664
1497   1511   1510   1500    409    423    422    412    665    679    678    668
1508   1502   1503   1505    420    414    415    417    676    670    671    673
 624    594    595    621   1456   1426   1427   1453    560    530    531    557
 597    619    618    600   1429   1451   1450   1432    533    555    554    536
 601    615    614    604   1433   1447   1446   1436    537    551    550    540
 612    606    607    609   1444   1438   1439   1441    548    542    543    545
Plane 3 _ Top - 2
 480    482    483    477    736    738    739    733   1376   1378   1379   1373
 485    475    474    488    741    731    730    744   1381   1371   1370   1384
 489    471    470    492    745    727    726    748   1385   1367   1366   1388
 468    494    495    465    724    750    751    721   1364   1390   1391   1361
1504   1506   1507   1501    416    418    419    413    672    674    675    669
1509   1499   1498   1512    421    411    410    424    677    667    666    680
1513   1495   1494   1516    425    407    406    428    681    663    662    684
1492   1518   1519   1489    404    430    431    401    660    686    687    657
 608    610    611    605   1440   1442   1443   1437    544    546    547    541
 613    603    602    616   1445   1435   1434   1448    549    539    538    552
 617    599    598    620   1449   1431   1430   1452    553    535    534    556
 596    622    623    593   1428   1454   1455   1425    532    558    559    529
Plane 4 _ Top - 3
 497    463    462    500    753    719    718    756   1393   1359   1358   1396
 460    502    503    457    716    758    759    713   1356   1398   1399   1353
 456    506    507    453    712    762    763    709   1352   1402   1403   1349
 509    451    450    512    765    707    706    768   1405   1347   1346   1408
1521   1487   1486   1524    433    399    398    436    689    655    654    692
1484   1526   1527   1481    396    438    439    393    652    694    695    649
1480   1530   1531   1477    392    442    443    389    648    698    699    645
1533   1475   1474   1536    445    387    386    448    701    643    642    704
 625    591    590    628   1457   1423   1422   1460    561    527    526    564
 588    630    631    585   1420   1462   1463   1417    524    566    567    521
 584    634    635    581   1416   1466   1467   1413    520    570    571    517
 637    579    578    640   1469   1411   1410   1472    573    515    514    576
Plane 5 _ Top - 4   order 4 cubes represented by the middle plane of the order 3
                       pattern cube are in these middle 4 planes. 
 897    959    958    900   1537   1599   1598   1540     65    127    126     68
 956    902    903    953   1596   1542   1543   1593    124     70     71    121
 952    906    907    949   1592   1546   1547   1589    120     74     75    117
 909    947    946    912   1549   1587   1586   1552     77    115    114     80
   1     63     62      4    833    895    894    836   1665   1727   1726   1668
  60      6      7     57    892    838    839    889   1724   1670   1671   1721
  56     10     11     53    888    842    843    885   1720   1674   1675   1717
  13     51     50     16    845    883    882    848   1677   1715   1714   1680
1601   1663   1662   1604    129    191    190    132    769    831    830    772
1660   1606   1607   1657    188    134    135    185    828    774    775    825
1656   1610   1611   1653    184    138    139    181    824    778    779    821
1613   1651   1650   1616    141    179    178    144    781    819    818    784
Plane 6 _ Top - 5
 944    914    915    941   1584   1554   1555   1581    112     82     83    109
 917    939    938    920   1557   1579   1578   1560     85    107    106     88
 921    935    934    924   1561   1575   1574   1564     89    103    102     92
 932    926    927    929   1572   1566   1567   1569    100     94    95      97
  48     18     19     45    880    850    851    877   1712   1682   1683   1709
  21     43     42     24    853    875    874    856   1685   1707   1706   1688
  25     39     38     28    857    871    870    860   1689   1703   1702   1692
  36     30     31     33    868    862    863    865   1700   1694   1695   1697
1648   1618   1619   1645    176    146    147    173    816    786    787    813
1621   1643   1642   1624    149    171    170    152    789    811    810    792
1625   1639   1638   1628    153    167    166    156    793    807    806    796
1636   1630   1631   1633    164    158    159    161    804    798    799    801
Plane 7 _ Bottom + 5
 928    930    931    925   1568   1570   1571   1565     96     98     99     93
 933    923    922    936   1573   1563   1562   1576    101     91     90    104
 937    919    918    940   1577   1559   1558   1580    105     87     86    108
 916    942    943    913   1556   1582   1583   1553     84    110    111     81
  32     34     35     29    864    866    867    861   1696   1698   1699   1693
  37     27     26     40    869    859    858    872   1701   1691   1690   1704
  41     23     22     44    873    855    854    876   1705   1687   1686   1708
  20     46     47     17    852    878    879    849   1684   1710   1711   1681
1632   1634   1635   1629    160    162    163    157    800    802    803    797
1637   1627   1626   1640    165    155    154    168    805    795    794    808
1641   1623   1622   1644    169    151    150    172    809    791    790    812
1620   1646   1647   1617    148    174    175    145    788    814    815    785
Plane 8 _ Bottom + 4
 945    911    910    948   1585   1551   1550   1588    113     79     78    116
 908    950    951    905   1548   1590   1591   1545     76    118    119     73
 904    954    955    901   1544   1594   1595   1541     72    122    123     69
 957    899    898    960   1597   1539   1538   1600    125     67     66    128
  49     15     14     52    881    847    846    884   1713   1679   1678   1716
  12     54     55      9    844    886    887    841   1676   1718   1719   1673
   8     58     59      5    840    890    891    837   1672   1722   1723   1669
  61      3      2     64    893    835    834    896   1725   1667   1666   1728
1649   1615   1614   1652    177    143    142    180    817    783    782    820
1612   1654   1655   1609    140    182    183    137    780    822    823    777
1608   1658   1659   1605    136    186    187    133    776    826    827    773
1661   1603   1602   1664    189    131    130    192    829    771    770    832

Plane 9 _ Bottom + 3  order 4 cubes represented by the bottom plane of the order 3
                        pattern cube are in these bottom 4 planes. 
1153   1215   1214   1156    257    319    318    260   1089   1151   1150   1092
1212   1158   1159   1209    316    262    263    313   1148   1094   1095   1145
1208   1162   1163   1205    312    266    267    309   1144   1098   1099   1141
1165   1203   1202   1168    269    307    306    272   1101   1139   1138   1104
1025   1087   1086   1028   1281   1343   1342   1284    193    255    254    196
1084   1030   1031   1081   1340   1286   1287   1337    252    198    199    249
1080   1034   1035   1077   1336   1290   1291   1333    248    202    203    245
1037   1075   1074   1040   1293   1331   1330   1296    205    243    242    208
 321    383    382    324    961   1023   1022    964   1217   1279   1278   1220
 380    326    327    377   1020    966    967   1017   1276   1222   1223   1273
 376    330    331    373   1016    970    971   1013   1272   1226   1227   1269
 333    371    370    336    973   1011   1010    976   1229   1267   1266   1232
Plane 10 _ Bottom + 2
1200   1170   1171   1197    304    274    275    301   1136   1106   1107   1133
1173   1195   1194   1176    277    299    298    280   1109   1131   1130   1112
1177   1191   1190   1180    281    295    294    284   1113   1127   1126   1116
1188   1182   1183   1185    292    286    287    289   1124   1118   1119   1121
1072   1042   1043   1069   1328   1298   1299   1325    240    210    211    237
1045   1067   1066   1048   1301   1323   1322   1304    213    235    234    216
1049   1063   1062   1052   1305   1319   1318   1308    217    231    230    220
1060   1054   1055   1057   1316   1310   1311   1313    228    222    223    225
 368    338    339    365   1008    978    979   1005   1264   1234   1235   1261
 341    363    362    344    981   1003   1002    984   1237   1259   1258   1240
 345    359    358    348    985    999    998    988   1241   1255   1254   1244
 356    350    351    353    996    990    991    993   1252   1246   1247   1249
Plane 11 _ Bottom + 1
1184   1186   1187   1181    288    290    291    285   1120   1122   1123   1117
1189   1179   1178   1192    293    283    282    296   1125   1115   1114   1128
1193   1175   1174   1196    297    279    278    300   1129   1111   1110   1132
1172   1198   1199   1169    276    302    303    273   1108   1134   1135   1105
1056   1058   1059   1053   1312   1314   1315   1309    224    226    227    221
1061   1051   1050   1064   1317   1307   1306   1320    229    219    218    232
1065   1047   1046   1068   1321   1303   1302   1324    233    215    214    236
1044   1070   1071   1041   1300   1326   1327   1297    212    238    239    209
 352    354    355    349    992    994    995    989   1248   1250   1251   1245
 357    347    346    360    997    987    986   1000   1253   1243   1242   1256
 361    343    342    364   1001    983    982   1004   1257   1239   1238   1260
 340    366    367    337    980   1006   1007    977   1236   1262   1263   1233
Plane 12 _ Bottom
1201   1167   1166   1204    305    271    270    308   1137   1103   1102   1140
1164   1206   1207   1161    268    310    311    265   1100   1142   1143   1097
1160   1210   1211   1157    264    314    315    261   1096   1146   1147   1093
1213   1155   1154   1216    317    259    258    320   1149   1091   1090   1152
1073   1039   1038   1076   1329   1295   1294   1332    241    207    206    244
1036   1078   1079   1033   1292   1334   1335   1289    204    246    247    201
1032   1082   1083   1029   1288   1338   1339   1285    200    250    251    197
1085   1027   1026   1088   1341   1283   1282   1344    253    195    194    256
 369    335    334    372   1009    975    974   1012   1265   1231   1230   1268
 332    374    375    329    972   1014   1015    969   1228   1270   1271   1225
 328    378    379    325    968   1018   1019    965   1224   1274   1275   1221
 381    323    322    384   1021    963    962   1024   1277   1219   1218   1280

This order 3 magic cube is constructed from the magic sums of the 27 cubes in the order 12 magic cube.
The magic constant for this cube (and the order 12 cube) is 10374.

I_Top                   II                      III_Bottom
1922   2946   5506      3714   6274    386      4738   1154   4482
6018   1666   2690       130   3458   6786      4226   5250    898
2434   5762   2178      6530    642   3202      1410   3970   4994

The order 12 cube is simple magic and has no special features except that it is associated. The order 4 cubes and of course the order 3 cubes are also associated.

 Order-9 cube-2

This order 9 cube also consists of 27 order 3 associated magic cubes.

A second method of constructing a composition cube is by forming the sub-cubes by multiplication instead of addition. As expected, the numbers used will be much larger and there will be gaps in the series of numbers used. So the resulting composition will not be a normal cube. A more serious problem results if the numbers in the generating cubes are consecutive. That is, there will be many duplicate numbers.

Using unique prime numbers in the generating cubes ensures that all 729 numbers generated are different. Unfortunately, the two Suzuki cubes I am using as generators have 3 prime numbers in common. They are 683, 839, and 1259.This causes 3 duplicate numbers to appear in the order 9 composition cube. They are 683*839, 683*1259 and 839*1259.

The numbers of cube B are multiplied by the number in cube A to form the order 3 cube that goes in that position in the order 9 cube.

Plane 1_Top             Plane 2                 Plane 3_Bottom
 263   2309   2087      1439   1487   1733      2957   863     839
2129    107   2423      1847   1553   1259       683   2999    977
2267   2243    149      1373   1619   1667      1019    797   2843

A. Above layout and multiplier pattern is the Suzuki-prime-1 magic cube.

Plane 1_Top             Plane 2                 Plane 3_Bottom
2153    929    227       509   1607   1193       647    773   1889
 839    947   1523      1787   1103    419       683   1259   1367
 317   1433   1559      1013    599   1697      1979   1277     53

B. Above numbers are multiplied by numbers in pattern A. This cube is the Suzuki-prime-2.

The sums of these 27 sub-cubes generated as above, form the cube below. Magic constant for that cube and the order 9 cube which follows, is 15,416,631

Plane 1_Top                     Plane 2                        Plane 3_Bottom
 870267   7640481  6905883      4761651  4920483  5734497      9784713  2855667  2776251
7044861    354063  8017707      6111723  5138877  4166031      2260047  9923691  3232893
7501503   7422087   493041      4543257  5357271  5516103      3371871  2637273  9407487

C. This cube is formed from the constants of the 27 order 3 cubes in the order 9 composition magic cube.

Features of the 30 order 3 cubes and the order 9 cube are identical. Each cube is simple and associated, but has no other features. So it follows that each cube has 3 magic squares in the central three orthogonal planes. It also follows that the order 9 cube has 3 x 27 = 81 order 3 magic squares.

Notice that the order 9 and 12 cubes shown previously and their attendant order 3 cubes have all pantriagonals in one direction correct. Also 3 of the 6 oblique squares have all pandiagonals in one direction correct. All four basic order 3 cubes have these features. Presumably, the difference with this construction is that none of the cubes use consecutive numbers, so none of these cubes are normal.

It is an interesting, but understandable observation, that all 729 numbers in the composition cube are composite (they have at least two factors. The magic constant is also a composite number.

The order 9 composition-2 cube.

Plane 1_Top
 566239    244327     59701   4971277   2145061    524143   4493311   1938823    473749
 220657    249061    400549   1937251   2186623   3516607   1750993   1976389   3178501
  83371    376879    410017    731953   3308797   3599731    661579   2990671   3253633
4583737   1977841    483283    230371     99403     24289   5216719   2250967    550021
1786231   2016163   3242467     89773    101329    162961   2032897   2294581   3690229
 674893   3050857   3319111     33919    153331    166813    768091   3472159   3777457
4880851   2106043    514609   4829179   2083747    509161    320797    138421     33823
1902013   2146849   3452641   1881877   2124121   3416089    125011    141103    226927
 718639   3248611   3534253    711031   3214219   3496837     47233    213517    232291
Plane 2_Top-1
 133867    422641    313759   1175281   3710563   2754637   1062283   3353809   2489791
 469981    290089    110197   4126183   2546827    967471   3729469   2301961    874453
 266419    157537    446311   2339017   1383091   3918373   2114131   1250113   3541639
1083661   3421303   2539897     54463    171949    127651   1233307   3893761   2890639
3804523   2348287    892051    191209    118021     44833   4329901   2672569   1015237
2156677   1275271   3612913    108391     64093    181579   2454499   1451377   4111831
1153903   3643069   2704531   1141687   3604501   2675899     75841    239443    177757
4051129   2500501    949873   4008241   2474029    939817    266263    164347     62431
2296471   1357933   3847099   2272159   1343557   3806371    150937     89251    252853
Plane 3_Top-2
 170161    203299    496807   1493923   1784857   4361701   1350289   1613251   3942343
 179629    331117    359521   1577047   2907031   3156403   1425421   2627533   2852929
 520477    335851     13939   4569511   2948593    122377   4130173   2665099    110611
1377463   1645717   4021681     69229     82711    202123   1567681   1872979   4577047
1454107   2680411   2910343     73081    134713    146269   1654909   3050557   3312241
4213291   2718733    112837    211753    136639      5671   4795117   3094171    128419
1466749   1752391   4282363   1451221   1733839   4237027     96403    115177    281461
1548361   2854153   3098989   1531969   2823937   3066181    101767    187591    203683
4486393   2894959    120151   4438897   2864311    118879    294871    190273      7897
Plane 4_Top-3
3098167   1336831    326653   3201511   1381423    337549   3731149   1609957    393391
1207321   1362733   2191597   1247593   1408189   2264701   1453987   1641151   2639359
 456163   2062087   2243401    471379   2130871   2318233    549361   2483389   2701747
3976591   1715863    419269   3343609   1442737    352531   2710627   1169611    285793
1549633   1749109   2812981   1302967   1470691   2365219   1056301   1192273   1917457
 585499   2646751   2879473    492301   2225449   2421127    399103   1804147   1962781
2956069   1275517    311671   3485707   1504051    367513   3589051   1548643    378409
1151947   1300231   2091079   1358341   1533193   2465737   1398613   1578649   2538841
 435241   1967509   2140507    513223   2320027   2524021    528439   2388811   2598853
Plane 5_Middle
 732451   2312473   1716727    756883   2389609   1773991    882097   2784931   2067469
2571493   1587217    602941   2657269   1640161    623053   3096871   1911499    726127
1457707    861961   2441983   1506331    890713   2523439   1755529   1038067   2940901
 940123   2968129   2203471    790477   2495671   1852729    640831   2023213   1501987
3300589   2037241    773893   2775211   1712959    650707   2249833   1388677    527521
1871011   1106353   3134359   1573189    930247   2635441   1275367    754141   2136523
 698857   2206411   1637989    824071   2601733   1931467    848503   2678869   1988731
2453551   1514419    575287   2893153   1785757    678361   2978929   1838701    698473
1390849    822427   2329981   1640047    969781   2747443   1688671    998533   2828899
Plane 6_Bottom+3
 931033   1112347   2718271    962089   1149451   2808943   1121251   1339609   3273637
 982837   1811701   1967113   1015621   1872133   2032729   1183639   2181847   2369011
2847781   1837603     76267   2942773   1898899     78811   3429607   2213041     91849
1195009   1427731   3488983   1004791   1200469   2933617    814573    973207   2378251
1261501   2325373   2524849   1060699   1955227   2122951    859897   1585081   1721053
3655213   2358619     97891   3073387   1983181     82309   2491561   1607743     66727
 888331   1061329   2593597   1047493   1251487   3058291   1078549   1288591   3148963
 937759   1728607   1876891   1105777   2038321   2213173   1138561   2098753   2278789
2717167   1753321     72769   3204001   2067463     85807   3298993   2128759     88351
Plane 7_Bottom+2
6366421   2747053    671239   1858039    801727    195901   1806367    779431    190453
2480923   2800279   4503511    724057    817261   1314349    703921    794533   1277797
 937369   4237381   4609963    273571   1236679   1345417    265963   1202287   1308001
1470499    634507    155041   6456847   2786071    680773   2103481    907633    221779
 573037    646801   1040209   2516161   2840053   4567477    819703    925219   1487971
 216511    978739   1064797    950683   4297567   4675441    309709   1400041   1523143
2193907    946651    231313   1715941    740413    180919   6120979   2641147    645361
 854941    964993   1551937    668683    754759   1213831   2385277   2692321   4329889
 323023   1460227   1588621    252649   1142101   1242523    901231   4074019   4432237
Plane 8_Bottom+1
1505113   4751899   3527701    439267   1386841   1029559    427051   1348273   1000927
5284159   3261571   1238983   1542181    951889    361597   1499293    925417    351541
2995441   1771243   5018029    874219    516937   1464511    849907    502561   1423783
 347647   1097581    814819   1526491   4819393   3577807    497293   1570039   1165561
1220521    753349    286177   5359213   3307897   1256581   1745899   1077631    409363
 691879    409117   1159051   3037987   1796401   5089303    989701    585223   1657969
 518671   1637533   1215667    405673   1280779    950821   1447087   4568701   3391699
1820953   1123957    426961   1424239    879091    333943   5080441   3135829   1191217
1032247    610381   1729243    807361    477403   1352509   2879959   1702957   4824571
Plane 9_Bottom
1913179   2285761   5585773    558361    667099   1630207    542833    648547   1584871
2019631   3722863   4042219    589429   1086517   1179721    573037   1056301   1146913
5851903   3776089    156721   1707877   1102051     45739   1660381   1071403     44467
 441901    527959   1290187   1940353   2318227   5665111    632119    755221   1845553
 466489    859897    933661   2048317   3775741   4099633    667291   1230043   1335559
1351657    872191     36199   5935021   3829723    158947   1933483   1247629     51781
 659293    787687   1924891    515659    616081   1505533   1839421   2197639   5370427
 695977   1282921   1392973    544351   1003423   1089499   1941769   3579337   3886381
2016601   1301263     54007   1577263   1017769     42241   5626297   3630511    150679

This page was originally posted January 2003
It was last updated March 03, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2009 by Harvey D. Heinz