Order-9 Magic Stars

But_Sqrs.gif (1906 bytes) Home.gif (5413 bytes) But_Strs.gif (1861 bytes) But_Patt.gif (1837 bytes)

 

Introduction to Order-9

Cell designations & some of the features of Order-9 magic stars.

Characteristics - Order-9

Adjacent complement pairs, numbers at points, three triangles, solutions with each point total, etc.

Order 9A List of Basic Solutions

A partial list of the 3014 basic solutions with complement number, complement pair number, point totals, and special feature remarks.

Order 9B List of Basic Solutions

A partial list of the 1676 basic solutions for this pattern.
Same information as for pattern A.

Order 9C List of Basic Solutions

A partial list of the 1676 basic solutions for this pattern.
Same information as for pattern A.

Definitions

Review magic star definitions and general features. Use Back button to return.

Star Examples

One example of each of 16 patterns, Orders 5 to 11. Use Back button to return

Star Examples - 2

One example of each of 14 patterns, Orders 12 to 14. Use Back button to return

 

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Introduction to Order-9

Order-9 has three different patterns and is the smallest order where the number of solutions is not the same for each pattern.

Patterns A and B are continuous, pattern C consists of three superimposed triangles. (3 x 3 = 9).

Pattern A 3014 solutions 2:12:43 running time
Pattern B 1676 solutions 19:18 running time
Pattern C 1676 solutions 18:01 running time
(Program ran on a 200 Mhz Pentium Pro)

Each solution contains the consecutive series of integers from 1 to 18. The sum of this series is 171, there are 9 lines with each cell appearing in 2 lines, so the magic constant (S) is (2 x 171)/9 = 38.

Order-9B has no solutions with b = 2.

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order9-1b.gif (4596 bytes) order9-1c.gif (4581 bytes)

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Order-9 Adjacent Complement Pairs

Order-9     Adjacent complement pairs
Order-9A Order-9B Order-9C
Solution #'s Pair # Solution #'s Pair # Solution #'s Pair #
2 3 2 4 5 4 7 8 5
1071 1072 978 99 100 66 36 37 29
1595 1596 1318 261 262 202 64 65 48
2076 2077 1467 303 304 242 66 67 49
2189 2190 1483 367 368 297 91 92 67
2192 2193 1484 491 492 387 101 102 129
2249 2250 1485 609 610 472 330 331 276
2254 2255 1486 638 639 496 535 536 434
883 884 659 983 984 701
The last four pairs have sequential numbers but there are other solutions (belonging to other pairs) between them. 902 903 668 1087 1088 751
956 957 698 1279 1280 801
1011 1012 714 Numbers 48 and 49 are adjacent pairs.
1068 1069 727
1161 1162 779 Last 3 are exactly 50 pair numbers apart.
1166 1167 781
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Numbers at the points

Order-9A

Pattern 9A has different cell names and many more solutions.then patterns 9B and 9C which have the same cell names and the same number of basic solutions.

Odd numbers at the points
Order-9A has 24 solutions with the points all odd numbers. The complements of these 24 solutions have all even numbers at the points.
Order-9B has 5 solutions with all odd numbers at the points, and their complements with all even numbers at the points.
Order-9C also has 5 solutions with all odd numbers at the points, and their complements with all even numbers at the points.

Low numbers at the points
Order-9A also has 24 solutions with the numbers 1 to 9 at the points. The complements of these 24 solutions have numbers 10 to 18 at the points.
Order-9B and Order-9C both also have 5 solutions with numbers 1 to 9 at the points, and their complements with numbers 10 to 18 at the points
*** These two features always appear together, and always with the same number of solutions (for a given order and pattern).

All odd orders have these types of solutions
Singly-even (6, 10, 14) orders cannot have these types of solutions because of parity.
Doubly-even orders may have these types although order-8 does not. Order-12 does, and presumably orders 16 , 20, etc do also.

Order-9B Solutions Order-9C Solutions
Odd Even 1 - 9 10 - 18 Odd Even 1 - 9 10 - 18
159 818 542 1673 53 1169 609 1675
167 813 547 1672 236 1036 732 1674
285 941 607 1675 241 1033 735 1673
393 888 634 1674 627 965 894 1672
515 1019 671 1676 838 1255 945 1676
Odd Even 1 - 9 10 - 18
30 1548 1062 2993
31 1529 1063 2992
46 1524 1065 2991
52 1559 1067 2994
156 1592 1152 2998
170 1591 1153 2997
175 1587 1156 2996
177 1576 1157 2995
215 1949 1180 3012
231 1730 1183 3008
239 1623 1185 3001
387 1715 1259 3005
398 1714 1260 3004
399 1713 1261 3003
400 1706 1264 3002
408 1723 1267 3007
414 1720 1268 3006
439 1621 1273 3000
687 1858 1355 3010
689 1855 1356 3009
800 1889 1381 3011
995 1615 1435 2999
1011 7056 1441 3014
1253 1963 1500 3013

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Three Triangles

Order-9B consists of 3 super-imposed triangles. The points of these triangles all have the same value in 12 solutions (six complement pairs).

Solution     Triangle    Complement     Triangle
number      point total    number      point total 
 103           31            110          26
 106           28            108          29
 187           23           1423          34
 260           32            263          25
 289           26           1098          31
 638           30            639          27
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Point Totals – Number of basic solutions for each

Point      Pattern         Point      Pattern         Point      Pattern  
Total    A    B    C       Total    A    B    C       Total    A    B    C 
 45     24    5    5        59     10   16   16        73     44   33   33
 46     10    3    3        60     37    9    9        74     51   24   24
 47      7    9    9        61     30    9    9        75     62   37   37
 48     14    6    6        62     28   15   15        76     42   22   22
 49      9    6    6        63     30   16   16        77     61   38   38
 50     16    1    1        64     31   11   11        78     90   47   47
 51     17    3    3        65     25   18   18        79     60   43   43
 52     10    1    1        66     52   17   17        80     51   35   35
 53      7   12   12        67     23   19   19        81    114   36   36
 54     18    6    6        68     39   38   38        82     48   41   41
 55      8    5    5        69     35   30   30        83     47   56   56
 56      5    0    0        70     49   23   23        84     95   36   36
 57     11    6    6        71     42   34   34        85     55   37   37
 58     11    5    5        72     89   33   33
Total solutions for the low half of the count is            1507  838  838
The high totals are the same as these, but in reverse order
 i.e. 86 matches with 85, 87 with 84,. . ., 126 with 45.

Top.gif (1442 bytes)rule-w.gif (2726 bytes)

Order 9A Partial List of Basic Solutions

Sol..  | -------- cell value ----- Upper case indicates the points --------| Compl  Pair  Point Remarks
 #  .  A   b   c   D   e   f   G   h   i   J   k   l   M   n   O   P   Q   R  ment    #   Total
   1   1   2  17  18   3   6  11  12   5  10   8  13   7  15  14   4  16   9     4     1   90
   2   1   2  17  18   3  10   7  11   5  15   8   9   6  16  14   4  13  12     3     2   90
   3   1   2  17  18   3  10   7  11  14   6   8   9  15  16   5  13   4  12     2     2   81
   4   1   2  17  18   4   6  10  11  14   3   7  13  15  16   5  12   9   8     1     1   81
   5   1   2  18  17   3   4  14   9   8   7   6  10  15  16   5  12  13  11  1521     3   95
   6   1   2  18  17   3   4  14  12   5   7   8  13  10  15  11   6  16   9  1533     4   91
   7   1   2  18  17   3   5  13   7  10   8   4  12  14  16   6  11  15   9  1518     5   94
   8   1   2  18  17   3   5  13  10  11   4   7  15  12  16   8   9  14   6  1514     6   84
   9   1   2  18  17   3   7  11   9  14   4   6  13  15  16   5  12  10   8  1516     7   83
  10   1   2  18  17   3   8  10  14   9   5  11  15   7  16  13   4  12   6  1513     8   75
  11   1   2  18  17   3   8  10  15   6   7   5  12  14   9  13   4  11  16  1564     9   93
  12   1   2  18  17   3  11   7   9   6  16   4   8  10  14  12   5  13  15  1549    10   96
  13   1   2  18  17   3  11   7  13   4  14  10   6   8  16  12   5   9  15  1527    11   88
  14   1   2  18  17   3  11   7  14   4  13  10   6   9  15  12   5   8  16  1541    12   88
  15   1   2  18  17   3  12   6   8   9  15   5  11   7  16  13   4  14  10  1520    13   87
  16   1   2  18  17   3  13   5   9  16   8   4  11  15  14   7  10   6  12  1546    14   81
  17   1   2  18  17   3  13   5  10  15   8   7   9  14  16   6  11   4  12  1522    15   78
  18   1   2  18  17   3  13   5  15   8  10  12   7   9  16  11   6   4  14  1525    16   77
  19   1   2  18  17   3  14   4   8  15  11   5   9  13  16   7  10   6  12  1523    17   81
  20   1   2  18  17   3  14   4  10  11  13   7   6  12  16   8   9   5  15  1530    18   84

  56   1   2  18  17  10   8   3  15  13   7   5  12  14  16   6   4  11   9  1517    54   72
  57   1   3  16  18   2   5  13  15   4   6   7  11  14   9  12   8  10  17    74    55   99

 142   1   3  18  16  11   9   2  15  14   7   6  12  13  17   5   4  10   8  2152   131   66
 143   1   4  15  18   2   8  10   5   6  17   3   7  11  14   9  12  13  16   150   132  107

 245   1   4  18  15  11  10   2  13  16   7   5  12  14  17   3   6   9   8  2513   228   65
 246   1   5  14  18   2   7  11   6   4  17   3  10   8  12  13   9  16  15   263   229  108

 377   1   5  18  14  11  10   3  17  12   6   9   8  15  16   2   7   4  13  2595   346   65
 378   1   6  13  18   2   8  10  15   4   9  17   7   5  16  11  12   3  14   380   347   83

 496   1   6  18  13  11   9   5  12  17   4   3  15  16  14   2   7  10   8  2289   460   66
 497   1   7  12  18   2   3  15  14   4   5  10  17   6   9  16   8  13  11   513   461   93

 659   1   7  18  12  14   9   3  17   5  13   8   6  11  16   4   2  10  15  2721   603   71
 660   1   8  11  18   3   5  12   6   4  16   2  13   7   9  14  10  17  15   669   604  110

 820   1   8  18  11  13  12   2  16  15   5  10  14   9  17   4   3   7   6  2949   748   48
 821   1   9  10  18  12   5   3  13   7  15   8   4  11  16   2  14   6  17   554   503   87

 969   1   9  18  10  15   5   8  17   7   6   4  12  16  11   2   3  13  14  2397   885   73
 970   1  10   9  18  13   4   3  12   7  16   6   5  11  15   2  14   8  17   428   394   90

1070   1  10  18   9  15   6   8  14  11   5   4  17  12  13   3   2  16   7  2315   977   63
1071   1  11   8  18   5   9   6   2  16  14   3  17   4  13  10  15  12   7  1072   978   87

1186   1  11  18   8  15  10   5  14  13   6   9  16   7  17   3   2  12   4  2942  1080   48
1187   1  12   7  18  15   3   2  14   5  17   9   8   4  16   6  10  11  13   164   147   82

1293   1  12  18   7  15  11   5  16   4  13   9   6  10  14   2   3   8  17  2015  1164   66
1294   1  13   6  18   7   3  10  15   5   8  17  11   2  14   9  16   4  12  1252  1126   80

1384   1  13  18   6  15  10   7  12  11   8   9  16   5  17   3   2  14   4  2813  1237   50
1385   1  14   6  17  16   2   3  18   8   9  13  12   4  15   5  11   7  10  1609  1238   67

1453   1  14  18   5  12  17   4  11   7  16  10   9   3  15   6   2   8  13  2437  1286   58
1454   1  15   6  16  12   7   3   8  14  13  10  11   4  17   2  18   5   9   522   473   71

1503   1  15  18   4  10  13  11  14   6   7  17   9   5  16   2   8   3  12  2907  1310   53
1504   1  16   7  14  11   9   4   6  15  13  10  12   3  17   2  18   5   8   674   612   68

1512   1  16  15   6  11   9  12  14   8   4  18  13   3  17   2  10   5   7  2943  1312   50
1513   2   1  17  18   3   4  13   8  10   7   5  11  15  16   6  12  14   9    10     8   96

2148   2  15  11  10   6  14   8   1  17  12   9  13   4  16   3  18   5   7   334   303   69
2149   3   1  16  18   2   5  13  10   8   7   4  15  12  11  14   6  17   9   136   125   99

2511   3  13  10  12   8   1  17  14   2   5  18   9   6  15   4  16   7  11  1751  1368   81
2512   4   1  15  18   2   5  13  10   3  12  11   9   6  17  14   7  16   8   208   191   98

2721   4  11  14   9   2  10  17   5   1  15  12   3   8  13   6  16   7  18   659   603  100
2722   5   1  14  18   2   3  15  12   4   7   8   6  17  11   9  13  10  16   368   337  110

2829   5  11  15   7   3  10  18   2   1  17   9   4   8  13   6  14  12  16   906   823  103
2830   6   1  13  18   3  10   7   5  11  15   2   4  17  12   8  14   9  16   490   454  110

2887   6   7  14  11   1   9  17   2   3  16   5   4  13  10   8  15  12  18  1443  1276  116
2888   7   1  12  18   2   3  15   5   4  14   8   6  10  16  11  13  17   9   626   570  114

2948   7   6  10  15   2   3  18   5   1  14   8   4  12  11   9  17  13  16  1452  1285  121
2949   8   1  11  18   2   5  13   9   4  12   3   7  16   6  15  10  14  17   820   748  123

2972   8   6  15   9   1  11  17   3   2  16   4   5  13   7  12  10  14  18  1415  1254  117
2973   9   1  10  18   2   7  11   6   5  16   3   4  15   8  14  12  13  17   962   878  125

2990   9   6  11  12   1   7  18   2   3  15   5   4  14   8  10  16  13  17  1284  1155  124
2991  10   1   9  18   2   6  12   8   3  15   5   4  14   7  16  11  13  17  1065   972  126  Points are #'s 
                                                                                                 10 to 18
3005  10   3   7  18   4   5  11   2   8  17   1   6  14   9  12  15  16  13  1259  1131  126
3006  10   3   7  18   5   2  13   8   6  11   1   9  17   4  14  12  16  15  1268  1140  126  to the end
3007  10   3   7  18   6   1  13   9   4  12   2   8  16   5  14  11  17  15  1267  1139  126
3008  10   3   8  17   1   7  13   4   6  15   2   9  12   5  18  11  16  14  1183  1077  126
3009  10   4   6  18   2   7  11   3   8  16   1   9  12   5  17  13  15  14  1356  1211  126
3010  10   4   6  18   5   1  14   9   2  13   7   3  15   8  11  16  12  17  1355  1210  126
3011  10   4   9  15   2   7  14   3   8  13   1   6  18   5  11  16  12  17  1381  1234  126
3012  10   5   6  17   4   2  15   9   1  13   8   3  14   7  12  16  11  18  1180  1074  126
3013  10   5   7  16   1   4  17   2   8  11   3   9  15   6  12  18  14  13  1500  1308  126
3014  10   6   8  14   2   7  15   1   4  18   5   3  12   9  11  17  13  16  1441  1274  126
Top.gif (1442 bytes)
rule-w.gif (2726 bytes)


Order 9B Partial List of Basic Solutions

Sol..  | -------- cell value ----- Upper case indicates the points --------| Compl  Pair  Point  Remarks
 #  .  A   b   c   D   e   f   G   h   i   J   k   L   m   N   o   P   Q   R  ment    #   Total
   1   1   2  17  18   3   4  13  14  10  15   9  11   8   5  16   7   6  12     8    1   88
   2   1   2  17  18   3   7  10  14  13  15  12   8  11   5  16   4   6   9     7    2   76
   3   1   2  17  18   3   8   9  15  13  12  16   7   6  10  14   4  11   5     6    3   77
   4   1   2  17  18   3   9   8  16  13  15  14   6   5  11  10   4  12   7     5    4   82
   5   1   2  17  18   6   3  11  10  16  15   5  12   9   7  14   4   8  13     4    4   89
   6   1   2  17  18   6   4  10  11  16  15   3  14   5   8  13   7   9  12     3    3   94
   7   1   2  17  18   6   5   9  12  16  15   7  10   3  13   8   4  14  11     2    2   95
   8   1   2  17  18   9   5   6  15  16  12  10   7   3  13  11   4  14   8     1    1   83
   9   1   3  16  18   2   4  14  12  11  13   8  15   6   5  17   9   7  10    25    5   92
  10   1   3  16  18   2   4  14  15   8  13  11  12   6   5  17   9   7  10    30    6   89
  11   1   3  16  18   2   6  12   8  17  14   7  15   4  11  10   5  13   9    16    7   98
  12   1   3  16  18   2   6  12   8  17  15   7  14   5  11   9   4  13  10    17    8   98
  13   1   3  16  18   2   6  12  15  10  13   9  14   4   5  17  11   7   8    26    9   89
  14   1   3  16  18   2   8  10  15  12  13   9  14   4   5  17  11   7   6    24   10   85
  15   1   3  16  18   2  10   8  15  14  13  11  12   6   5  17   9   7   4    21   11   77
  16   1   3  16  18   2  11   7  13  17  14  12  10   9   6  15   5   8   4    11    7   73
  17   1   3  16  18   2  11   7  13  17  15  12   9  10   6  14   4   8   5    12    8   73
  18   1   3  16  18   2  11   7  17  13  14  12  10   5   6  15   9   8   4    22   12   77
  19   1   3  16  18   4   5  11  17   9  14   7  13   2   6  15  12   8  10    27   13   93
  20   1   3  16  18   5   2  13  15   9  12  14   7  10   6  17   4   8  11    29   14   80
  21   1   3  16  18   5   4  11   9  17  10   8  15   2  12  13   6  14   7    15   11   94
  22   1   3  16  18   6   2  12   8  17  10   7  15   4  11  14   5  13   9    18   12   94
  23   1   3  16  18   7   2  11  17   9  14  12   5  10   6  15   4   8  13    28   15   80
  24   1   3  16  18   7   4   9  11  17   8  10  13   2  12  15   6  14   5    14   10   86
  25   1   3  16  18   8   7   5  15  17  10  11   9   2  12  13   6  14   4     9    5   79
  26   1   3  16  18   9   4   7  13  17   8  10  11   2  12  15   6  14   5    13    9   82
  27   1   3  16  18  10   2   8  14  15   7  12   9   4  11  17   5  13   6    19   13   78
  28   1   3  16  18  10   2   8  17  12  15   7   6   4  11   9   5  13  14    23   15   91
  29   1   3  16  18  10   4   6  17  14  15   5   8   2  11   9   7  13  12    20   14   91
  30   1   3  16  18  11   4   5  15  17  10   8   9   2  12  13   6  14   7    10    6   82
  31   1   4  15  18   2   5  13   7  17  10  12  14   9   8  16   3  11   6    37   16   84

  75   1   4  17  16  14   2   6  13  18  11   3  10   7   8  15   5   9  12  1152   49   78
  76   1   5  14  18   2   3  15  10  12  13   6  17   7   4  16   9   8  11   107   50   96

 147   1   5  17  15  14   6   3  16  18  13   7   4  10   8  12   2   9  11  1388  100   66
 148   1   6  13  18   2  10   8  14  15  16  11   9   3  12   4   5  17   7   152  101   93

 201   1   6  17  14   9   7   8  11  18   4  15  10   5  12  16   3  13   2  1534  146   67
 202   1   7  12  18   4   2  14  10  13   6  17  11   8   9  16   3  15   5   204  147   82

 259   1   7  17  13  14   5   6  15  16  10   2  12   8   3  18   9   4  11  1593  200   69
 260   1   8  11  18   2   5  13  15   9  16  14   6   7  10   4   3  17  12   263  201   96  3 triangle point

 300   1   8  17  12  10   3  13   6  18   7   5  16   2  14   9   4  15  11  1306  239   93    totals same
 301   1   9  10  18   2   4  14  12  11   6  17  13   5   8  15   7  16   3   310  240   86

 365   1   9  17  11  16   3   8  15  14  13   4   5  12   6  10   2   7  18   118   71   71
 366   1  10   9  18   5   7   8  16  13  12   4  17   3   2  14  15  11   6   369  296   90

 423   1  10  16  11  18   4   5  17  15   9   8   3   6  12   7   2  14  13  1580  338   70
 424   1  11   8  18   3   2  15  10  12  14   4  17   5   6   7   9  16  13   425  339  109

 490   1  11  17   9  18   3   8  15  14   5   2  13   4   6  16  10   7  12  1589  386   71
 491   1  12   7  18   2   8  10  14  13  16   5  15   3   6   4  11  17   9   492  387  103

 564   1  12  17   8  18   2  10  14  13   6   3  11   9   4  16   7   5  15  1442  439   67
 565   1  13   6  18  11   2   7  14  16   5  10  12   9   3  17   8  15   4   295  234   73

 608   1  13  17   7  16  11   4  15  18   2  12   8   6   9  14   5  10   3  1670  471   49
 609   1  14   5  18   3   4  13   8  16  12   6  17  11   2  10   7  15   9   610  472   94

 637   1  14  17   6  12   5  15   4  18   3   7  16  10   8  13   2   9  11  1250  495   71
 638   1  15   4  18   2   7  11  10  16  14   9  13  12   3   6   5  17   8   639  496   90  3 triangle point

 680   1  15  17   5  16  10   7  12  18   2  11   9   4  13   8   3  14   6  1576  533   60    totals same
 681   1  16   7  14  13   9   2  17  18  10   3  12   5   4   8  11  15   6  1486  534   75

 700   1  16  17   4  12  13   9  10  18   6   5  15  11   2  14   7   3   8  1465  553   55
 701   1  17   6  14   3  12   9  10  18  15   7  13  11   4   2   5  16   8  1261  554   85

 731   1  17  16   4  10  13  11   8  18   7   6  15   3  12   2   5  14   9  1480  584   78
 732   2   1  18  17   3   5  13  15   8  14   9  12   4   7  16  10   6  11   749  585   92

1135   2  17  16   3  14  15   6  12  18  10   1  13   9   4   7   8   5  11  1622  769   62
1136   3   1  18  16   2   6  14  12   9  13   8  15   4   7  17  11   5  10  1149  770   94

1371   3  16  15   4  10  18   6  12  17   8   9  11   2  13   1   7  14   5  1527  829   71
1372   4   1  16  17  13   3   5  14  15   7   8  10   2  12  18   9  11   6   771  600   81

1512   4  16  11   7   2  15  14   3  17  13   5  18   9   8   1   6  12  10   421  336   92
1513   5   1  15  17   2   3  16  13   4  12  14  10   8   7  18   9   6  11   812  621   93

1585   5  15   6  12   2  11  13   3  17  14   4  18   9   8   1   7  16  10   422  337  103
1586   6   1  14  17   7   3  11   5  16   9   4  18   2  13  15  10  12   8   533  414  104

1636   6  16   5  11   2  13  12   3  17  14   4  18  10   7   1   8  15   9   350  281  100
1637   7   1  13  17   5   6  10   9  12  18   4  11   2  16   3   8  15  14   672  525  116

1645   7  13   4  14   2  12  10   5  16  15   3  18   6   9   1  11  17   8   582  449  109
1646   8   2  12  16   4   1  17   7   6  11   5  18   3  10  15  14   9  13   669  522  116

1660   8   7   9  14   3   6  15   4  11  13   5  17   1  16   2  10  18  12   604  467  123
1661   9   1  15  13   6   5  14   8   7  17   3  12   2  16   4  10  11  18   536  417  120

1671   9   8   5  16   1   7  14   4  11  17   2  18   6  10   3  12  15  13   594  459  124
1672  10   3   7  18   1   2  17   5   6  13   9  15   4  14   8  11  16  12   547  424  126  Points are #'s
1673  10   3   7  18   4   5  11   8   9  17   1  16   2  12   6  15  14  13   542  421  126    10 to 18
1674  10   4   8  16   2   5  15   7   6  14   9  13   1  17   3  11  18  12   634  492  126   here to end
1675  10   5   7  16   4   1  17   2   9  13   3  18   6  12   8  11  14  15   607  470  126
1676  10   6   5  17   8   1  12   7   9  15   2  13   4  14   3  11  18  16   671  524  126

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Order 9C Partial List of Basic Solutions

Sol..  | -------- cell value ----- Upper case indicates the points --------| Compl  Pair  Point  Remarks
 #  .  A   b   c   D   e   f   G   h   i   J   k   L   m   N   o   P   Q   R  ment    #   Total
   1   1   3  16  18   2   6  12  17   4   5   7  10  14   8  11  15  13   9     6    1   91
   2   1   3  16  18   2   6  12  17   4   5   9   8  14  10  13  11  15   7     4    2   87
   3   1   3  16  18   2   6  12  17   5   4  10   8   9  15   7  11  14  13    10    3   96
   4   1   3  16  18   2   6  12  17   5   4  10   8  15   9  13  11  14   7     2    2   84
   5   1   3  16  18   2   7  11  17   6   4  10   8   9  14   5  13  12  15    13    4   96
   6   1   3  16  18   2   8  10  17   5   6  12   4  15  11  13   9  14   7     1    1   80
   7   1   3  16  18   2   8  10  17   6   5  13   4  12  14  11   7  15   9     8    5   83
   8   1   3  16  18   2   8  10  17   7   4   6  12  13   5  11  15  14   9     7    5   88
   9   1   3  16  18   2  11   7  17   6   8  10   4   9  14   5  13  12  15    12    6   92
  10   1   3  16  18   2  12   6  17  10   5   9   8  14   4  13  11  15   7     3    3   75
  11   1   3  16  18   2  12   6  17  11   4   8  10   7   9   5  13  14  15    14    7   90
  12   1   3  16  18   2  14   4  17  10   7   9   6  13   5   8  15  11  12     9    6   79
  13   1   3  16  18   2  14   4  17  10   7   9   6  13   5  12  11  15   8     5    4   75
  14   1   3  16  18   2  14   4  17  12   5  11   6   8  10   7   9  15  13    11    7   81
  15   1   3  18  16   2   8  12  17   4   5   9   6  14  10  13  11  15   7  1310    8   83
  16   1   3  18  16   2   8  12  17   5   4   6  10   9  11   7  15  14  13  1318    9   96
  17   1   3  18  16   2   8  12  17   5   4   6  10  13   7  11  15  14   9  1312   10   88
  18   1   3  18  16   2   8  12  17   5   4  10   6   9  15   7  11  14  13  1317   11   92
  19   1   3  18  16   2   8  12  17   5   4  10   6  15   9  13  11  14   7  1309   12   80
  20   1   3  18  16   2  12   8  17   9   4  10   6   7  13   5  11  14  15  1320   13   88
  21   1   3  18  16   2  12   8  17   9   4  10   6  15   5  13  11  14   7  1308   14   72
  22   1   3  18  16   2  12   8  17   9   4  11   5  15   6  13  10  14   7  1307   15   71
  23   1   3  18  16   2  14   6  17  10   5   7   8  12   4   9  15  13  11  1315   16   79
  24   1   3  18  16   2  14   6  17  10   5   7   8  12   4  11  13  15   9  1313   17   77
  25   1   3  18  16   2  14   6  17  10   5   8   7  13   4   9  15  12  11  1314   18   77
  26   1   3  18  16   2  14   6  17  10   5  11   4   8  12   7   9  15  13  1319   19   81
  27   1   3  18  16   2  14   6  17  10   5  11   4  13   7  12   9  15   8  1311   20   71
  28   1   3  18  16   2  15   5  17  12   4  10   6   9   8   7  11  14  13  1316   21   78
  29   1   4  15  18   2   5  13  16   6   3   9  11  12  10  14   8  17   7    40   22   88

  55   1   4  18  15   3  11   9  17  10   2  13   5  16   6   8  14   7  12  1481   40   71
  56   1   5  14  18   2   7  11  15   9   3   4  17   8   6  10  13  16  12    68   41   97

  84   1   5  18  14   4  13   7  17   8   6   2  12  10   3  11  16  15   9  1404   60   83
  85   1   6  13  18   2   3  15  14   5   4  11  10   8  17   7   9  12  16    97   61  102

 153   1   6  18  13   5  16   4  17  14   3  15   2  12   8   9   7  10  11  1595  121   59
 154   1   7  12  18   2  14   4  13  10  11   9   6   3  15   8   5  17  16   169  122   93

 216   1   7  18  12   6  11   9  17   4   8   2  10  14   3  15  16  13   5  1348  176   77
 217   1   8  11  18   2   3  15  12   5   6   7  14  17   4  16  13  10   9   233  177   90

 322   1   8  18  11   4  13  10  14   9   5  12   3  15   7   6  16   2  17  1231  268   72
 323   1   9  10  18   2   3  15  11   7   5  17   6  16  13  14   4   8  12   337  269   82

 428   1   9  18  10   8  16   4  17  15   2  11   7   3  12   6   5  13  14  1588  357   68
 429   1  10   9  18   3   2  15  11   5   7  16   6  13  17  12   4   8  14   437  358   90

 531   1  10  18   9   8  15   6  16  13   3  12   5  11   7   4  14   2  17  1164  430   64
 532   1  11   8  18   5   2  13  12   3  10   6  14   7  15  16   4  17   9   537  431  101
                                                                 
 609   1  11  18   8  10  16   4  17  14   3  12   5  15   2  13   9   6   7  1675  484   45  # 1 - 9 at 
 610   1  12   7  18   8   2  10  14   3  11   4  16  15   5  17  13   9   6   576  458   89    the points

 678   1  12  18   7  11  14   6  17  10   5   2  13   8   3  16   9  15   4  1500  533   63
 679   1  13   6  18  10   2   8  15   3  12   4  16   9  11  17   7  14   5   416  345   92

 770   1  13  18   6  12  16   4  17  14   3   8   9  11   2  15   7  10   5  1667  579   47
 771   1  14   5  18  11   3   6  15   8   9   7  17   2  16  10   4  13  12   311  258   96

 842   1  14  18   5  12  17   4  16  15   3   7  10   2   9   8   6  11  13  1627  624   62
 843   1  15   4  18  13   5   2  16   3  17   6  11  10  12  14   9   8   7   103   72   85

 908   1  15  18   4  14  13   7  17   8   6   2  12  10   3  11  16   5   9  1403  664   63
 909   1  16   7  14  15   4   5  17   3  13  12   6  10  18   9   8   2  11   400  329   78

 945   1  16  18   3  15  14   6  17  10   5  11   4  13   7  12   9   2   8  1676  691   45  # 1 - 9 at 
 946   2   3  16  17   1   7  13  15   4   6  11   5  18   8  12  14  10   9   956  692   84    the points

1306   2  16  15   5  17  13   3  18   9   8   1  14   4   7  12  10  11   6  1567  824   66
1307   3   1  16  18   2   6  12  17   4   5   8   9  10  13   7  14  15  11    22   15  100

1479   3  15  16   4  17  12   5  18   9   6   2  14   1  11  10   8  13   7  1571  836   71
1480   4   1  15  18   2   7  11  16   3   8  10   5  17   9  12  14  13   6    54   39   88

1539   4  16  13   5  18   7   8  17   3  10   1  14   2  15  11   9  12   6  1536  837   83
1540   5   1  14  18   3   6  11  16   2   9   8   7  10  15   4  17  12  13    83   59  107

1594   5  13   9  11  17   4   6  18   2  12   1  16   3  15   7  14  10   8  1477  834   97
1595   6   1  13  18   2  10   8  14   7   9   4  12   5  11   3  17  16  15   153  121  112

1631   6   9  10  13  14   4   7  18   5   8   3  17   1  16   2  15  11  12  1289  807  105
1632   7   2  14  15   6   9   8  16   3  11   1  12   4  13   5  17  18  10   881  646  111

1649   7  11   6  14  15   1   8  16   4  10   5  17   2  18   3  13   9  12   897  656  108
1650   8   2  12  16   6   7   9  15   3  11   1  14   4  13   5  17  18  10   883  648  116

1660   8   7  11  12   5   6  15   9   4  10   1  16   2  14   3  17  13  18   829  611  123
1661   9   3  11  15   1   4  18   8   2  10   5  12   6  16   7  13  17  14   591  466  124

1671   9   7  10  12   5   6  15   8   4  11   1  16   2  14   3  17  13  18   830  612  125
1672  10   2   9  17   1   7  13   8   5  12   3  14   6  11   4  18  15  16   894  655  126   Points are #'s
1673  10   4   6  18   2   1  17   7   3  11   5  16   9  12   8  15  14  13   735  557  126      10 to 18
1674  10   4   6  18   3   2  15   8   1  14   5  13   7  16   9  12  17  11   732  555  126     to the end
1675  10   6   5  17   4   3  14   7   1  16   2  15   9  11   8  18  12  13   609  484  126
1676  10   7   9  12   6   5  15   8   1  14   2  13   4  16   3  18  11  17   945  691  126

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Harvey Heinz   harveyheinz@shaw.ca
This page last updated September 09, 2003
Copyright 1998, 1999 by Harvey D. Heinz