Order-10 Magic Stars

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Introduction to Order-10 Cell designations & some of the features of Order-10 magic stars.
Characteristics of Order-10 Adjacent complement pairs, numbers at points, solutions with each point total, etc.
Order-10A List A partial list of the 10,882 basic solutions with complement number, complement pair number, and point totals.
I show the first and last 10 solutions. In addition, while A = 1, I show the first solution for each value of B. Finally, I show the the first solution for each value of a greater then 1.
Order-10B List A partial list of the 115,552 basic solutions for this pattern.
Same information as for pattern A.
Order-10C List A partial list of the 10,882 basic solutions for this pattern.
Same information as for pattern A.
Related pages on this site Use the browser 'back' button to return here if you wish.
Definitions Review magic star definitions and general features.
Examples One example of each of 16 patterns, Orders 5 to 11.
Examples-2 One example of each of 14 patterns, Orders 12,13,14.

Introduction to Order-10

Pattern A consists of two super-imposed pentagons, pattern B is continuous, and pattern C is two super-imposed pentagrams. (2 and 5 are the factors of 10).

As expected, patterns A and C have the same number of basic solutions because point names are the same. However, it seems odd that B has over ten times as many as A and C. Are there equivalents in the list? I used the same algorithm for all three patterns (in fact, for all orders). If so, it is very coincidental that all matched up into complement pairs. (These numbers were later independently confirmed.)

Pattern A        10882 solutions             approx. 24 hrs. to run
Pattern B      115552 solutions             approx. 6.5 hrs. to run
Pattern C        10882 solutions             approx. 24 hrs. to run
Running times are on a 200 Mhz Pentium Pro with 32 Megs. of  RAM.

Each solution contains the consecutive series of integers from 1 to 20. The sum of this series is 210, there are 6 lines with each cell appearing in 2 lines, so the magic constant (S) is  (2 x 210)/10 = 42.

Characteristics of Order-10

Some adjacent complement pairs:

10A   1 & 26               5304 & 186               10882 & 5128
10B   1 & 254             54777 & 2432            115552 & 40180
10C   1 & 6                 5304 & 27                 10882 & 4823
In each case, the first solution shown here is the first, the first for a = 2, and the last solution in the list.

Number of complement pairs for 10B is 57,776 (115,552/2).
Number of solutions with A equal to 1 is 57,776 !
Order 10B has exactly 27,000 solutions with a = 2. They start at index number 57,777. It also has more solutions then order-11 just as order-6 has more solutions then order-7.

The first solutions with A = 3

Point Features

Feature Pattern 10A Pattern 10B Pattern 10C Comments
Number of basic solutions 10,882 115,552 10,882 B over 10 times as many as A & C
Adjacent Complement Pairs 27 26 50 B has a small percentage of A & C
Solutions with numbers 1 to 10 at the points
(See note below)
0 0 0 No, same as Order-6 & Order-8. It follows that no solutions exist with numbers 11 to 20 (these would be the complement solutions).
Solutions with all odd numbers at the points 0 0 0 No, same as Order-6 & Order-8. It follows that no solutions exist with even numbers at the points (these would be the complement solutions).
Point total of pentagon or pentagram equal Pentagon
Yes, on all
Not applicable Pentagon
Yes, on all
NOTE: Order 8A has 2 squares and the corners of these always sum the same.
Order-6 has two triangles, and the corners of these always sum the same.

Distribution of Point Totals

Two features that may be present in any order, always appear together.

  • Some solutions have the points consisting of the n lowest numbers.
  • Some solutions have the points consisting of all even numbers.

Of course, in each case, there will be the same number of solutions with the opposite condition.
There will always be the same number of solutions of each feature (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.

Summary of Point Totals  - Low Range
Point |------ Pattern ------|   Point |------ Pattern ------|   Point |------ Pattern ------|
Total A B C   Total A B C   Total A B C
56 10 54 10   74 118 1141 118   90 387 4218 387
58 17 354 17   76 166 1376 166   92 387 3956 387
60 30 530 30   78 176 1726 176   94 351 4177 351
62 16 260 16   80 174 1783 174   96 448 4478 448
64 36 388 36   82 185 1757 185   98 458 4123 458
66 33 468 33   84 254 1881 254   100 449 4311 449
68 62 672 62   86 223 3887 223   102 513 5415 513
70 71 736 71   88 306 4020 306   104 454 4939 454
72 117 1153 117                    

Above is the low range of point totals. The arrangement of point totals from 106 to 154 is a mirror image of this i.e. 106 is the same as 104, 108 is the same as 102, , 154 is the same as 56. Solutions in this high bank are the complements of the solutions in the low bank. Note that the point total distribution for patterns A and C are exactly the same. This is because the point names are exactly the same.

                     

Order-10A Partial List of Basic Solutions

Solut.  |---- Order-10A ----- Cell values ------ upper case signifies points -------|  Comple  Pair  Point
   #    A   b   c   D   e   f   G   h   i   J   k   l   M   n   o   P   Q   R   S   T   ment     #   Total
    1   1   2  19  20   3   4  15  10   5  12   6   8  16   7  18   9  11  17  14  13     26     1  128
    2   1   2  19  20   3   4  15  12   5  10   6   8  18   7  16  11   9  17  14  13     76     2  128
    3   1   2  19  20   3   4  15  13   5   9   6  11  16   7  18  12   8  17  14  10     25     3  122
    4   1   2  19  20   3   5  14  15   7   6  18  10   8  16  17  11   9  13   4  12     35     4   98
    5   1   2  19  20   3   5  14  18   4   6  11   9  16   8  17  13   7  12  15  10     54     5  114
    6   1   2  19  20   3   5  14  18   4   6  13   7  16   8  17  11   9  10  15  12     55     6  114
    7   1   2  19  20   3   6  13  14   5  10  11  12   9  15  17  16   4  18   8   7     37     7  106
    8   1   2  19  20   3   6  13  17   7   5   8  11  18   9  14  16   4  15  12  10     91     8  114
    9   1   2  19  20   3   7  12   4  11  15   9   8  10  13  18   6  14  17   5  16     14     9  116
   10   1   2  19  20   3   7  12   6   8  16   9   4  13  10  18   5  15  14  11  17     23    10  124
  201   1   2  20  19  13   7   3  16  12  11   9   8  14  10  17   5   4  15   6  18   5341   148   96
  202   1   3  18  20   2   4  16   8   6  12   5  10  15   7  19   9  13  17  14  11    222   149  128
  434   1   3  20  18  13   5   6  16   8  12  11   9  10  14  17   7   2  19   4  15   6002   324   94
  435   1   4  17  20   2   5  15   3  10  14   8  11   9  13  19   7  16  18   6  12    445   325  118
  778   1   4  20  17  12   2  11  15  10   6   9  13  14   8  19   3   7  18   5  16   8687   629   98
  779   1   5  16  20   2   7  13  10  11   8   3  17  14  12  15  18   6  19   9   4    822   630  112
 1214   1   5  20  16  13   9   4  17  15   6  10   8  18  12  11   7   2  14   3  19   7079  1014   90
 1215   1   6  15  20   2   3  17  16   4   5  19   8  10  18  13  14  11  12   7   9   1254  1015  106
 1645   1   6  20  15  13  11   3  10  17  12   2   9  19   8  14   4   5  16   7  18   7652  1391  100
 1646   1   7  14  20   2   3  17   9  10   6   5  18  13  12  16  15  11  19   8   4   1682  1392  114
 2126   1   7  20  14  15   8   5  12  16   9   6  10  17  11  13   3   4  18   2  19   6809  1787   92
 2127   1   8  13  20   2   3  17   5   9  11  10  14   7  18  16  12  15  19   4   6   2139  1788  112
 2601   1   8  20  13  17   5   7  19   6  10  12   4  16  11  14   2   3  15   9  18   8738  2187   94
 2602   1   9  12  20   2   4  16   7  11   8  15  14   5  19  17  10  18  13   3   6   2624  2188  100
 3132   1   9  20  12  17   7   6  18  10   8  13   5  16  14  11   3   2  15   4  19   6578  2609   86
 3133   1  10  11  20   2   4  16  14   3   9  19   6   8  18  15  12  17   7  13   5   3139  2610  108
 3518   1  10  20  11  17   8   6  14  15   7   4  12  19   9  13   3   2  18   5  16   7154  2917   88
 3519   1  11  10  20   2   7  13   8   4  17  14   5   6  19  16  12  18   9  15   3   3525  2918  114
 3862   1  11  20  10  17   6   9  16  12   5   4  14  19   7  15   3   2  18   8  13   6618  3196   88
 3863   1  12   9  20   3   5  14   8   4  16  13   7   6  18  17  11  19  10  15   2   3864  3197  114
 4149   1  12  20   9  17   6  10  15   4  13  14   8   7  18  16   3   2  19   5  11   6224  3411   80
 4150   1  13   8  20   3   5  14   7   4  17   9   6  10  16  15  12  19  11  18   2   4152  3412  124
 4483   1  13  20   8  17   6  11  15   4  12  14   9   7  18  16   3   2  19   5  10   6223  3675   78
 4484   1  14   7  20   9   2  11  12  13   6  15  16   5  19  17   8  18  10   4   3   4103  3370   86
 4698   1  14  20   7  15  12   8  11   6  17   2  10  13   9  19   4   3  16  18   5   9186  3838   92
 4699   1  15   6  20  12   2   8  13  18   3   9  19  11  14  16   7  17  10   5   4   3034  2518   86
 4929   1  15  20   6  17  11   8  16   4  14   7   9  12  10  19   3   2  13  18   5   9252  4021   82
 4930   1  16   6  19  11   2  10  13  15   4  12  17   9  14  18   5  20   7   8   3   3285  2731   86
 5131   1  16  20   5  17  14   6  18   7  11  12  10   9  13  19   3   2   8  15   4  10670  4164   64
 5132   1  17   8  16   9   3  14   6   7  15  12  11   4  19  18   5  20  13  10   2   3964  3264  100
 5258   1  17  20   4  14  16   8  13   6  15   7   9  11  12  18   5   3  10  19   2   7573  4253   78
 5259   1  18   7  16  12   8   6  10  11  15  13   9   5  17  19   3  20   4  14   2   2738  2270   86
 5303   1  18  17   6  15  16   5  14  11  12   7  10  13   9  19   2   8   4  20   3   3555  2925   74
 5304   2   1  19  20   3   5  14  15   7   6   8  11  17  10  13  16   4  18   9  12    186   133  118
 7975   2  17  13  10  20   1  11  19   5   7  12  15   8  14  18   3   6  16   9   4  10265  5217   76
 7976   3   1  18  20   2   4  16  11   6   9   5  13  15   7  17  14   8  19  12  10    363   253  126
 9376   3  16  13  10  19   2  11  14   9   8   1  18  15   7  17   4   6  20  12   5   1999  1662   94
 9377   4   1  17  20   2   5  15  18   3   6  13  11  12   7  19  14   9  10  16   8    722   573  114
10121   4  15  11  12  19   2   9  13   3  17   1  10  14   8  16   5   7  20  18   6   2486  2077  112
10122   5   1  16  20   2   3  17   6   4  15   7   9  11   8  18  10  14  19  12  13   1135   935  136
10511   5  13   4  20  11   2   9   3  16  14   1  17  10  12  15   8  19  18   7   6   3470  2869  116
10512   6   1  15  20   2   4  16   5   3  18   8   7   9  10  17  11  14  19  12  13   1610  1356  138
10677   6  10  18   8   3  16  15   1   9  17   2   4  19   5  12   7  14  11  20  13   5217  4215  130
10678   7   1  14  20   2   3  17   9   6  10   4  16  12   5  18  15  11  19  13   8   2083  1744  132
10794   7  10  16   9   4  12  17   1   5  19   2   3  18   6  11   8  14  15  20  13   5284  4265  140
10795   8   1  13  20   2   3  17   9   4  12   5   7  18   6  10  16  11  19  14  15   2584  2170  150
10847   8   7  17  10   2  14  16   3   5  18   4   1  19   6   9  11  12  13  20  15   5226  4223  142
10848   9   1  12  20   3   5  14   2   7  19   4   8  11   6  16  10  17  18  13  15   3095  2572  146
10866   9   6  13  14   2   8  18   4   1  19   5   3  15  11   7  17  10  20  16  12   5184  4190  150
10867  10   1  11  20   2   4  16   7   6  13   3   8  18   5   9  17  12  19  14  15   3498  2897  154
10873  10   3  13  16   4   7  15   1   8  18   2   5  17   6   9  11  14  20  12  19   4434  3626  152
10874  10   4   8  20   3   7  12   2   9  19   1   6  16   5  11  13  18  15  17  14   4352  3563  154
10875  10   4   9  19   1   5  17   7   2  16   8   3  15   6  11  14  18  12  20  13   4429  3622  154
10876  10   4  13  15   3   8  16   1   7  18   5   2  17   9   6  12  14  19  11  20   4112  3374  152
10877  10   5   7  20   8   1  13   9   4  16   2   6  18   3  11  12  15  17  19  14   4476  3668  154
10878  10   5  11  16   4   2  20   9   1  12   8   3  19   7   6  14  13  18  15  17   4142  3404  154
10879  10   5  13  14   2   6  20   1   4  17   7   3  15   9   8  11  16  19  12  18   4889  3982  152
10880  10   5  13  14   2   8  18   6   3  15   7   1  19   4   9  11  16  12  20  17   4647  3795  152
10881  10   6   8  18   3   5  16   4   7  15   1   9  17   2  13  12  19  14  20  11   4102  3369  152
10882  10   6   9  17   2   3  20   5   1  16   8   4  14   7  11  12  19  15  18  13   5128  4161  154

                     

Order-10B Partial List of Basic Solutions

Solution |-----Order-10B----- Cell values ------ upper case signifies points --------|  Comple. Pair  Point
   #     A   b   c   D   e   f   G   h   i   J   k   l   M   n   O   p   Q   R   S   T   ment     #   Total 
     1   1   2  19  20   3   4  15   5  10  12  17   7   6   8   9  13  16  14  18  11     254      1  122
     2   1   2  19  20   3   4  15   6   7  14  11   9   8   5  10  12  16  17  13  18     975      2  132
     3   1   2  19  20   3   4  15   7   6  14  11   9   8  10   5  17  16  18  12  13     906      3  122
     4   1   2  19  20   3   4  15   8   6  13  10  12   7  11   5  17  16  18   9  14    1044      4  118
     5   1   2  19  20   3   4  15   9   6  12  10  13   7  11   5  17  16  18   8  14    1042      5  116
     6   1   2  19  20   3   4  15   9   6  12  13   7  10   8   5  17  16  18  14  11     752      6  122
     7   1   2  19  20   3   4  15   9   7  11  18   8   5   6   1  10  16  17  14  13     107      7  124
     8   1   2  19  20   3   4  15  11   7   9  13  14   6   5   1  10  16  17   8  18     826      8  122
     9   1   2  19  20   3   4  15  12   6   9  11  14   8  10   5  17  16  18   7  13     904      9  112
    10   1   2  19  20   3   4  15  12   6   9  14   8  11   7   5  17  16  18  13  10     594     10  118
  2960   1   2  20  19  16   4   3  18  13   8  14  11   9   7   6  15  17  10   5  12   58639   2320   90
  2961   1   3  18  20   2   4  16   5   7  14   9  11   8   6  10  13  15  17  12  19    3679   2321  132
  5476   1   3  20  18  17   5   2  19  13   8  14  11   9   7   6  15  16  10   4  12   85272   4404   86
  5477   1   4  17  20   2   5  15   9   7  11  12  16   3   8  14  10  13  18   6  19    5765   4405  120
  8904   1   4  20  17  18   5   2  13  19   8  15  10   9   6   7  14  16   3  11  12   99243   7622   86
  8905   1   5  16  20   2   3  17   4  12   9   8  19   6  13   7  18  14  11  10  15    9510   7623  110
 12194   1   5  20  16  19   4   3  13  18   8  15  10   9   6   7  14  17   2  11  12  101055  10544   86
 12195   1   6  15  20   2   3  17   5   4  16   9  10   7   8  12  14  13  19  11  18   12405  10545  134
 16086   1   6  20  15  19   5   3  14  18   7  13  12  10   8   4  17  16   2   9  11  107378  14223   78
 16087   1   7  14  20   2   3  17   6  10   9  18  11   4   5  19   8  12  13  16  15   16151  14224  126
 20186   1   7  20  14  19   6   3  18  11  10  17   2  13   4   5  16  15   9  12   8  111963  17979   90
 20187   1   8  13  20   2   9  11   3  10  18  14   4   6   7  16  12   5  19  17  15   20230  17980  128
 24364   1   8  20  13  19   7   3  17  18   4  15  12  11   6   5  16  14   2   9  10  113247  21857   72
 24365   1   9  12  20   2   3  17   4   8  13  16   6   7   5  18  11  10  15  19  14   24500  21858  134
 28544   1   9  20  12  19   8   3  18   5  16  17   2   7   4  11  10  13  15   6  14  108356  25515   98
 28545   1  10  11  20   2   3  17   4   8  13  18   6   5  12  14  16   9  15  19   7   28590  25516  120
 32882   1  10  20  11  19   9   3  18  14   7  16  15   4   5  13   8  12   6   2  17  100986  29381   76
 32883   1  11  10  20   2   3  17   4   5  16  13   7   6  14  12  19   8  18  15   9   33122  29382  122
 36396   1  11  20  10  19   9   4  18  13   7  17  15   3  14   5  16  12   6   2   8  115171  32202   58
 36397   1  12   9  20   2   4  16  14   7   5  18   8  11   3  19  13   6  17  15  10   36408  32203  120
 40402   1  12  20   9  19  11   3  18  14   7  16  15   4   5  13   8  10   6   2  17  100985  35622   72
 40403   1  13   8  20   2   3  17   9   5  11  15  12   4  16  14  19   6  18  10   7   40440  35623  108
 43972   1  13  20   8  19  12   3  18   5  16  17   2   7   4  11  10   9  15   6  14  108353  38412   90
 43973   1  14   7  20   2   3  17   6  10   9  18  11   4  12  19  15   5  13  16   8   43982  38413  112
 47360   1  14  20   7  19  13   3  17  18   4  15  12  11   6   5  16   8   2   9  10  115347  41075   60
 47361   1  15   6  20   7   2  13   8  10  11  19   9   3  16  17  18   5  12  14   4   46423  40173  100
 50558   1  15  20   6  19  14   3  18  11  10  17   2  13   4   5  16   7   9  12   8  114366  43295   74
 50559   1  16   5  20   7   4  11   8  14   9  18  12   3  15  19  17   2  10  13   6   45640  39598   94
 53392   1  16  20   5  19  14   4  18   8  12  17  10   3  13   6  15   7  11   2   9  115134  45333   60
 53393   1  17   5  19   6   2  15   8   9  10  18  11   3  14  20  16   4  12  13   7   17190  15089  104
 55488   1  17  20   4  19  14   5  15  16   6  18   8  10   3   9  12   7   2  13  11  114403  46915   68
 55489   1  18   6  17   7   4  14   9  11   8  20  12   2  15  19  16   3  10  13   5   66529  46916   92
 57082   1  18  20   3  17  16   6  15   9  12  19   4   7   2  13   8   5  10  11  14  114078  48138   82
 57083   1  19   6  16   7   5  14  11   8   9  20  10   3  15  18  17   2  13  12   4   90254  48139   92
 57776   1  19  18   4  17  16   5  15  14   8  20  12   2   9  13  10   3   6   7  11  114738  48832   60
 57777   2   1  19  20   3   4  15   7   6  14  11   9   8  10   5  16  17  18  12  13    2432   1792  124
 84776   2  19  18   3  17   9  13  15   4  10  20   5   7   1  16   6  11   8  12  14  109994  56858   96
 84777   3   1  18  20   2   4  16   7   5  14  11   9   8  10   6  15  17  19  12  13    5033   3961  128
 99124   3  19  16   4  18  14   6  17  11   8  20   2  12   1  13  10   5   7  15   9  111360  57715   82
 99125   4   1  17  20   2   6  14   3   7  18   9  10   5  12   8  13  15  19  11  16    8533   7251  130
106020   4  19  14   5  16  12   9  18   2  13  20   1   8   3  17   7   6  15  10  11  103673  57734   98
106021   5   1  16  20   2   3  17   4   8  13  14   6   9   7  10  11  18  15  19  12   11383   9733  138
110486   5  19  12   6  18   4  14  15   3  10  20   1  11   2  17   8  13   7  16   9  105974  57773  108
110487   6   1  15  20   2   3  17   4   7  14   5  13  10   8   9  12  18  16  11  19   15949  14086  140
112758   6  18  11   7  17   3  15  16   2   9  19   4  10   1  20   5  14   8  13  12   56660  47735  114
112759   7   1  14  20   2   3  17   5   4  16  12   6   8   9  11  10  18  19  15  13   19645  17438  144
114302   7  15  11   9  10   5  18   6   2  16  17   1   8   3  20   4  13  12  19  14   52303  44401  136
114303   8   1  13  20   2   3  17  11   4  10  16   7   9   5  15   6  18  19  14  12   23556  21049  142
114944   8  13   5  16   4   3  19   6   2  15  14   1  12   7  18  11  10  17  20   9   44943  39162  144
114945   9   1  12  20   2   3  17   5   7  13   8  10  11   4  15   6  18  16  14  19   28168  25139  152
115276   9  11   8  14   4   7  17   6   3  16  13   1  12   2  20   5  10  18  19  15   45408  39479  150
115277  10   1  11  20   2   5  15   3   7  17   4   8  13   6  12   9  16  18  14  19   32785  29284  154
115543  10   8   7  17   4   2  19   9   1  13  11   6  12   3  20   5  15  18  14  16   56309  47500  154
115544  10   8   7  17   6   1  18   9   4  11  15   3  13   2  20   5  16  14  19  12   57394  48450  150
115545  10   8   7  17   9   2  14   4   6  18  11   1  12   3  20   5  15  13  19  16   56358  47549  154
115546  10   8   7  17  11   1  13   6   4  19   9   2  12   3  20   5  16  14  15  18   56346  47537  154
115547  10   8   9  15   3   5  19   4   2  17  13   1  11   6  16   7  14  18  20  12   49153  42059  152
115548  10   8  13  11   6   5  20   4   1  17   9   2  14   3  12   7  18  15  19  16   51975  44250  152
115549  10   9   5  18   6   3  15   8   2  17  11   1  13   4  20   7  12  19  16  14   54824  46308  154
115550  10   9   6  17   5   1  19   7   4  12  11   3  16   2  18   8  15  14  20  13   52479  44518  154
115551  10   9   8  15   3   4  20   5   1  16  13   2  11   6  17   7  14  18  19  12   52988  44929  152
115552  10   9   8  15   3   6  18   7   1  16  13   2  11   4  19   5  12  20  17  14   40180  35400  152

                     

Order-10C Partial List of Basic Solutions

Solut.  |---- Order-10C ----- Cell values ------ upper case signifies points -------|  Comple. Pair  Point
   #    A   b   c   D   e   f   G   h   i   J   k   l   M   n   o   P   Q   R   S   T   ment     #   Total 
    1   1   2  19  20   3  11   8  10   9  15   6   5  16   7  18  17  13  14   4  12      6     1  120
    2   1   2  19  20   3  12   7  10   9  16   6   5  15   8  18  17  13  14   4  11      5     2  118
    3   1   2  19  20   3  12   7  14  16   5  10   9  18   6  17  15   8  11   4  13     12     3  102
    4   1   2  19  20   3  12   7  14  16   5  10   9  18   8  15  17   6  13   4  12     18     4  102
    5   1   2  19  20   3  13   6  16  15   5  12  11  14   9  18  17   7   8   4  10      2     2   92
    6   1   2  19  20   3  14   5  16  15   6  12  11  13  10  18  17   7   8   4   9      1     1   90
    7   1   2  19  20   3  15   4  14  16   8   9   7  18   6  17  13  10  11   5  12     13     5  102
    8   1   2  19  20   4  12   6  15   7  14   9   3  16   8  17  18  13  11   5  10     11     6  114
    9   1   2  19  20   4  13   5  10  12  15   6   7  14  11  16  18   8  17   3   9     15     7  110
   10   1   2  19  20   4  13   5  12  15  10   8   6  18   9  14  16   7  17   3  11     20     8  108
   31   1   2  20  19   7  11   5  15  10  12   8   4  18   6  17  16   9  14   3  13   5311    21  110
   32   1   3  18  20   2   8  12   9  10  11   7   5  19   6  16  17  13  15   4  14     43    22  126
   55   1   3  20  18   8  10   6  13  11  12   7   4  19   5  17  14   9  16   2  15   7470    38  112
   56   1   4  17  20   2   9  11  10  18   3   8  12  19   6  16  15   7  13   5  14     66    39  108
  128   1   4  20  17  11  12   2  15   7  18   6   5  13   9  19  16   8  14   3  10   8999   100  102
  129   1   5  16  20   2   9  11   6  17   8   3  12  19   4  18  13  10  14   7  15    144   101  118
  229   1   5  20  16  11   8   7  10  12  13   3   9  17   6  18  15   4  19   2  14   5339   185  108
  230   1   6  15  20   2   9  11   7   5  19   8   3  12  13  16  17  18  14   4  10    248   186  126
  414   1   6  20  15  12  10   5  16   3  18   9   4  11  13  17  19   8  14   2   7   7314   351  100
  415   1   7  14  20   2   4  16   6   5  15   9   8  10  12  19  18  17  11   3  13    417   352  124
  649   1   7  20  14  13  11   4  10  12  16   3   6  17   5  19   9   8  18   2  15   8000   567  104
  650   1   8  13  20   2   5  15   7   4  16  11   6   9  14  18  17  19  10   3  12    662   568  122
  993   1   8  20  13  14   9   6  16   2  18   4   3  17   5  19  15  11  12   7  10   9284   886  110
  994   1   9  12  20   2   5  15   3  11  13   7   6  16   8  17  10  19  14   4  18   1022   887  130
 1316   1   9  20  12  17   5   8  19   4  11  10   3  18   7  16  15   6  14   2  13   8990  1167  100
 1317   1  10  11  20   2   3  17   4   9  12   5   6  19   7  15  13  18  14   8  16   1359  1168  138
 1788   1  10  20  11  17  12   2  16   6  18   7   9   8  14  19  15   4  13   3   5   9027  1571   80
 1789   1  11  10  20   2   5  15   6  17   4   3  19  16   7  18  14   9   8  13  12   1795  1572  112
 2091   1  11  20  10  17  13   2  16   6  18   8   9   7  15  19  14   5  12   3   4   8930  1815   76
 2092   1  12   9  20   2   3  17   6   5  14  13   7   8  18  15  16  19  10   4  11   2135  1816  120
 2626   1  12  20   9  18   7   8  16   3  15   6  10  11  13  17  19   2  14   5   4   6059  2256   88
 2627   1  13   8  20   3   2  17   9   5  11  14  10   7  16  18  15  19   4   6  12   2628  2257  112
 3047   1  13  20   8  19  12   3  14  15  10   5   9  18   7  16   6   2  17   4  11   8824  2575   80
 3048   1  14   7  20   2   9  11   8   6  17   3  10  12  13  16  15  19   5  18   4   3060  2576  122
 3550   1  14  20   7  19  13   3  17  16   6   9  12  15   8  18   5   2  11   4  10  10624  2933   64
 3551   1  15   6  20   2  12   8   7  11  16   4  13   9  14  18  10  19   3  17   5   3555  2934  108
 3973   1  15  20   6  19  14   3  17  13   9  10   5  18   7  16   2   8  12   4  11   8987  3226   74
 3974   1  16   5  20   2   7  13   3  17   9   8  14  11  12  18   4  19   6  10  15   3979  3227  108
 4435   1  16  20   5  19  12   6  14   4  18   2   7  15   9  17  11   8  13  10   3   9223  3560   90
 4436   1  17   4  20   2   6  14   5  11  12   8  15   7  16  18  10  19   3  13   9   4437  3561  108
 4831   1  17  20   4  19  16   3  15  11  13   8  12   9  14  18   7   5  10   6   2  10843  3858   60
 4832   1  18   3  20   9   6   7   5  17  13   4  15  10  12  19   2  14   8  11  16   2417  2067  102
 5135   1  18  20   3  17  12  10  14   2  16  11   6   9  13  19   8  15   7   5   4  10371  4104   78
 5136   1  19   4  18   7   2  15   6   9  12   5  17   8  13  20  10  16   3  14  11   9055  4105  108
 5303   1  19  18   4  17  15   6  11   9  16   2  10  14   7  20   3  13   8  12   5  10415  4272   82
 5304   2   1  19  20   3  11   8  14  15   5  10   9  18   6  16  17   7  12   4  13     27    17  106
 7975   2  19  18   3  16  11  12   9   4  17   1  10  14   6  20   7  15   8  13   5  10058  5217   96
 7976   3   1  18  20   2   8  12  15  10   5  13   7  17   6  16  19  11   9   4  14     53    36  114
 9376   3  19  16   4  17   7  14   2   8  18   1  13  10   9  20   5  12  15   6  11   8974  5323   98
 9377   4   1  17  20   2   9  11  10  15   6   5  12  19   3  16  18   7  13   8  14    118    90  120
10121   4  19  14   5  16   3  18   6   1  17   2  15   8  10  20  13  12   9  11   7   9142  5345  104
10122   5   1  16  20   2   9  11   6  13  12   3   8  19   4  14  17  10  18   7  15    225   181  134
10511   5  18  13   6  16   1  19   4   2  17   3  12  10   7  20   9  15  11   8  14   7758  5061  114
10512   6   1  16  19   3   9  11   5   8  18   2  10  12   4  20  17  14  13   7  15   5491  4329  132
10677   6  17  10   9  14   3  16   4   2  20   1   8  13   5  18   7  19  11  12  15   5279  4248  128
10678   7   1  14  20   2   9  11   8   6  17   4   3  18   5  12  19  15  16  10  13    641   559  146
10794   7  14   3  18   5   4  15   2   6  19   1  12  10   9  16  11  20   8  17  13   1266  1118  138
10795   8   1  16  17   2   9  14   6   7  15   3   5  19   4  11  20  13  18  10  12   3491  2877  146
10847   8  12   4  18   9   2  13   6   3  20   1   7  14   5  15  11  19  10  16  17   5026  4015  146
10848   9   1  15  17   3   8  14   5   7  16   2   6  18   4  11  20  12  19  10  13   3438  2838  148
10866   9  11   7  15   5   3  19   4   1  18   2   8  14   6  13  16  20  10  17  12   2875  2428  150
10867  10   2  13  17   1   9  15   4   7  16   3   5  18   6   8  20  14  19  12  11   3537  2920  152
10873  10   5  11  16   1   6  19   2   9  12   3   7  20   4   8  17  15  18  13  14   3430  2834  154
10874  10   6   9  17   2   5  18   1   8  15   4   7  16   3  13  12  20  14  11  19   3512  2895  152
10875  10   6  11  15   8   3  16   7   2  17   1   5  19   4   9  20  12  18  14  13   3859  3134  154
10876  10   7   6  19   8   3  12   9   1  20   4   2  16   5  11  15  18  13  14  17   4797  3824  154
10877  10   7  11  14   4   6  18   1   8  15   2   5  20   3   9  13  17  19  12  16   4379  3508  154
10878  10   7  11  14   8   4  16   5   1  20   2   3  17   6   9  18  15  19  12  13   5287  4256  154
10879  10   8   9  15   3   7  17   4   5  16   1   6  19   2  11  14  20  12  18  13   3815  3103  154
10880  10   9   6  17   5   2  18   1   7  16   3   8  15   4  13  11  19  14  12  20   4717  3761  152
10881  10   9   7  16   5   1  20   4   6  12   3   8  19   2  11  14  17  13  15  18   4718  3762  154
10882  10   9   8  15   7   3  17   1   4  20   2   6  14   5  13  12  19  16  11  18   4823  3850  152

This page was originally posted June 1998
It was last updated March 27, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2009 by Harvey D. Heinz