Order-8 Magic Stars

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

Contents

Introduction to Order-8

Cell designations & some of the features of Order-8 magic stars.

Adjacent Complement Pairs

Pairs that are complements of each other and appear nest to each other in the solutions list.

Values at the points

Characteristics of point values and point total distribution.

Order 8A List of Basic Solutions

The list of 112 basic solutions, complement number, complement pair number, point totals, and special feature remarks.

Order 8B List of Basic Solutions

The 112 basic solutions for this pattern. Same information as for pattern A.

Definitions

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

Examples

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

Examples -2

One example of each of 14 patterns, Orders 12, 13 and 14. Back button to return.

Introduction to Order-8

order8-1.gif (5463 bytes)

The order-8 magic star has 112 basic solutions for each of the two patterns. This despite the fact that the two patterns have points with different names. Order- 7 has the same number of solutions for the two patterns, but the point names are the same. .Both order-9 and order-10 have the same number of solutions for patterns with similar point names and different solution totals for dissimilar patterns. However, order-11 has two patterns with similar cell names and similar total solutions. It also has two patterns with different cell names and similar total solutions

Pattern A consists of two superimposed squares (2 and 4 are the factors of 8). Pattern B is continuous.

Each solution contains the consecutive series of integers from 1 to 16. The sum of this series is 136, there are 8 lines with each cell appearing in 2 lines, so the magic constant (S) is 2 x 136/8 = 34.

rule-w.gif (2726 bytes)

Order-8 Adjacent Complement Pairs

Order-8A has four times as many Adjacent Complement Pairs (8) as order-8B (2).

                Order-8a                     Order-8b 
            Solution #   Pair            Solution #   Pair
            1    2        1              4    5        4   
            6    7        5	        26   27       23
            9   10        6
           16   17       12
           29   30       24
           50   51       43
           78   79       55
           97   98       56

order8-2.gif (7298 bytes)

       Fig. 1A. Index # 1                                  Fig. 1B. complement of fig.A                 Fig. 1C. fig.B. normalized = #2

Above is a graphic example of the first adjacent complement pair.
The middle star is obtained by subtracting each number in star A from 17.Figure 1C is then obtained by rotating B one position to the left and then reflecting it. C is now in the normalized position and by checking the solutions list below (pattern A) we see that it is index number 2.

Values at the points

No solutions in either pattern have the points all even or all odd. (If a solution existed with all even points, another one would exist with all odd points because of the complement feature of all magic stars.)

No solutions in either pattern have the points with numbers 1 to 8 or numbers 9 to 16. Again, if a solution exists with numbers 1 to 8 at the points, another must exist with numbers 9 to 16.

In both patterns the sum of the points range from 38 to 98 with no solution having a point sum of 40 or its complement, 96.

Order-8A The corners of the two squares always have the same sum. Because the valleys are common to both squares, the sum of the 12 cells in each square is always the same.
Order-8B The opposite small triangles, such as a, b, n and e, k, l always sum to the same value.

Distribution of Point Totals

Point     Pattern          Point     Pattern        Point     Pattern  
Total     A    B           Total     A    B         Total     A    B 
 38       2    2            48       1    1          58       6    6
 40       0    0            50       4    4          60       7    7
 42       2    2            52       5    5          62       5    5
 44       1    1            54       3    3          64       1    1
 46       5    5            56       2    2          66       3    3
                                                     68      18   18
 90       5    5            80       2    2          70       3    3
 92       1    1            82       3    3          72       1    1
 94       2    2            84       5    5          74       5    5
 96       0    0            86       4    4          76       7    7
 98       2    2            88       1    1          78       6    6
Order-8 is different then most orders in that the solutions that have
a point total of 68, have a complement that also has a point total of 68. 
The point totals that are lower then 68 have complements that are 
higher then 68, thus the high range is a mirror image of the low range.
All solutions of pattern A, with point total of 68 have the corners of
each of the two squares summing to the constant (S).
Top.gif (1256 bytes)rule-w.gif (2726 bytes)

Order 8A List of Basic Solutions

Sol.# |------- Cell value ---- Upper case indicates points -------| Comple  Pair  Point   Remarks   
 #    A   b   c   D   e   f   G   h   i   J   k   l   M   N   O   P  ment    #    total
  1   1   3  14  16   2   7   9   4   8  13   5  15   6  12  11  10     2     1    78 Adjacent complement pair
  2   1   3  14  16   2  12   4   9  13   8  10  15  11   7   6   5     1     1    58
  3   1   3  16  14   2  11   7   4   8  15   5  13   6  10   9  12    72     2    74
  4   1   3  16  14   6  12   2  13  11   8  10  15   7   5   4   9    77     3    50
  5   1   4  13  16   2  10   6   9   5  14  11   8   7  12   3  15     8     4    74
  6   1   4  13  16   6   5   7  11   2  14   9  10  12   3  15   8     7     5    76 Adjacent complement pair
  7   1   4  13  16   7   8   3  15   6  10  12  11   5   9   2  14     6     5    60
  8   1   4  13  16   9   6   3  12   8  11   7  15  10   2  14   5     5     4    62
  9   1   5  12  16   3   8   7   2  10  15   4  14   6  13  11   9    10     6    78 Adjacent complement pair
 10   1   5  12  16   3  13   2   7  15  10   9  14  11   8   6   4     9     6    58
 11   1   5  15  13   8  11   2   6  12  14   3  16   4   7  10   9    82     7    60
 12   1   5  15  13  12   7   2  16   6  10   9  14   4   3   8  11    95     8    52
 13   1   5  16  12   6  13   3  15   7   9  14  10   8   4   2  11   102     9    50
 14   1   6  15  12   2  16   4   5  14  11   9  13   7  10   3   8   105    10    56
 15   1   6  15  12   2  16   4   7  10  13  11   9  14   3   8   5    81    11    60
 16   1   7  10  16   4  11   3   2  14  15   5  13   8  12   9   6    17    12    70 Adjacent complement pair
 17   1   7  10  16   4  12   2   3  15  14   6  13   9  11   8   5    16    12    66
 18   1   7  11  15   8   5   6   4  10  14   3  16   2  13  12   9    60    13    72
 19   1   7  16  10   2  13   9   6   5  14  11   8   4  12   3  15    62    14    68 Corners of each square = S
 20   1   8  11  14   6  12   2   3  16  13   5  15   7  10   9   4    84    15    60
 21   1   8  16   9   5  13   7  12  11   4  15  14  10   3   6   2   110    16    42
 22   1   8  16   9   6   7  12   3   4  15   5  13   2  10  14   11   75    17    74
 23   1   8  16   9  10   4  11  15   2   6  13  14   5   3  12   7   106    18    54
 24   1   8  16   9  10  13   2   6  11  15   4  14   5   3  12   7    61    19    54
 25   1   9  11  13   2   7  12   8  10   4  15  14   5  16   3   6    31    20    60
 26   1   9  11  13   8   7   6   2  12  14   3  16   5  10  15   4    57    21    68
 27   1   9  11  13   8   7   6  14  12   2  15  16   5  10   3   4    99    22    44
 28   1   9  16   8   6  15   5   4  11  14   7  12  10   2  13   3    87    23    56
 29   1  10   7  16   4   2  12   3   6  13   9  11   8  15  14   5    30    24    84 Adjacent complement pair
 30   1  10   7  16   6   8   4  11  14   5  15  13   9  12   3   2    29    24    52
 31   1  10   9  14   7   2  11   3   8  12   6  15   5  13  16   4    25    20    76
 32   1  10   9  14   7   5   8  13  11   2  16  15   6  12   4   3    93    25    50
 33   1  10   9  14   8   5   7  11  12   4  13  16   2  15   3   6    69    26    52
 34   1  10  11  12  14   2   6  13   7   8   9  16   5   4  15   3    56    27    54
 35   1  10  14   9   5   7  13  11   8   2  16  15   3  12   4   6   100    28    50
 36   1  11  12  10   6   3  15   4   7   8   9  16   2  14  13   5    76    29    68
 37   1  11  12  10   6   3  15   8   7   4  13  16   2  14   9   5    71    30    60
 38   1  11  13   9   6   5  14   2   8  10   7  16   3  12  15   4    70    31    68
 39   1  11  13   9   6   5  14  10   8   2  15  16   3  12   7   4    91    32    52
 40   1  11  13   9   8  15   2   4  16  12   7  14   3  10   5   6   103    33    48
 41   1  11  13   9  12   7   6  15  10   3  14  16   5   4   8   2   112    34    38
 42   1  11  15   7  13   8   6  14   9   5  12  16   4   2  10   3   111    35    38
 43   1  12  13   8   5   6  15   3   9   7  10  16   2  14  11   4    74    36    62
 44   1  12  13   8   9  14   3   4  16  11   7  15   2  10   6   5   109    37    46
 45   1  12  14   7   6   8  13  10   9   2  16  15   3  11   5   4   101    38    46
 46   1  12  16   5  13   7   9  11   6   8  10  15   3   2  14   4    86    39    46
 47   1  12  16   5  13  10   6   8   9  11   7  15   3   2  14   4    94    40    46
 48   1  13   9  11  16   5   2  15  10   7  12  14   3   6   8   4   108    41    42
 49   1  13  11   9   7  10   8   6  15   5  12  16   2  14   4   3    92    42    46
 50   2   1  16  15   3   9   7   8   5  14   6  12  11   4  13  10    51    43    76 Adjacent complement pair
 51   2   1  16  15   5  11   3  12   9  10   8  14   6   7   4  13    50    43    60
 52   2   3  15  14   6   4  10  16   1   7  13  12   8   5   9  11    80    44    66
 53   2   3  16  13   1  15   5   4  11  14   6  12   9   8   7  10    90    45    68
 54   2   3  16  13   1  15   5   4  11  14   6  12  10   7   8   9    89    46    68
 55   2   3  16  13   6   5  10  14   1   9  11  12   4   8   7  15    58    47    68
 56   2   4  15  13   3   6  12   7   1  14   8  10  11   5  16   9    34    27    82
 57   2   5  14  13   1   8  12   6   9   7  10  15   3  16   4  11    26    21    68
 58   2   5  14  13   1  11   9  12   3  10  16   6  15   4   7   8    55    47    68
 59   2   5  16  11   1  13   9   6   4  15  10   7  14   3  12   8    66    48    74
 60   2   6  10  16   1  14   3   7  13  11  12   9  15   8   5   4    18    13    64
 61   2   6  11  15   4   7   8   1   9  16   3  13   5  14  12  10    24    19    82
 62   2   6  12  14  11   4   5  15   1  13  10   9   3   8   7  16    19    14    68
 63   2   6  14  12   1  16   5   4  15  10   9  13   8  11   3   7   107    49    58
 64   2   6  16  10   3  12   9   1  11  13   4  15   8   7  14   5    88    50    68
 65   2   6  16  10   4  13   7   1  12  14   3  15   5   9  11   8    83    51    66
 66   2   7  10  15  12   1   6  16   4   8  11  13   9   3  14   5    59    48    62
 67   2   7  11  14  10   1   9  15   6   4  12  16   8   5  13   3    85    52    58
 68   2   7  15  10   9   3  12  16   1   5  14  13   6   4  11   8   104    53    58
 69   2   9   8  15   7   1  11   4   5  14   6  12   3  16  13  10    33    26    84
 70   2   9  10  13   1   6  14   4  11   5  12  15   7  16   8   3    38    31    68
 71   2   9  10  13   4   1  16   6   5   7  11  14   8  12  15   3    37    30    76
 72   2   9  13  10   6  15   3   1  14  16   4  12   8   7  11   5     3     2    62
 73   2  11   8  13  15   1   5  14   9   6  10  16   4   7  12   3    96    54    52
 74   2  11  12   9   4   5  16   1   7  10   8  14   3  15  13   6    43    36    74
 75   2  12   4  16   9   1   8  11  10   5  14  13   6  15   7   3    22    17    62
 76   2  13  10   9   8   1  16   6   5   7  11  14   4  12  15   3    36    29    68
 77   3   1  14  16   2   7   9   6   4  15   5  11  10   8  13  12     4     3    86
 78   3   1  16  14   2   5  13   8   4   9   7  15   6  10  11  12    79    55    78 Adjacent complement pair
 79   3   1  16  14   2  10   8  13   9   4  12  15  11   5   6   7    78    55    58
 80   3   2  14  15   5   4  10  16   1   7  13  11   9   6   8  12    52    44    70
 81   3   2  15  14   1  10   9   7   6  12   8  11   5  13   4  16    15    11    76
 82   3   5  11  15   6   9   4   2  12  16   1  14   7  10  13   8    11     7    76
 83   3   5  16  10   4  13   7   1  11  15   2  14   6   8  12   9    65    51    70
 84   3   6   9  16   2  12   4   1  14  15   5  11  10  13   8   7    20    15    76
 85   3   6  10  15   1   5  13  11   2   8  16   7   9  14   4  12    67    52    78
 86   3   6  10  15   4   1  14   5   2  13   7  11   8  12  16   9    46    39    90
 87   3   6  13  12   2  11   9   1   8  16   5  10   4  15   7  14    28    23    80
 88   3   6  13  12   2  11   9   1  14  10   5  16   4  15   7   8    64    50    68
 89   3   6  13  12   2  16   4   1  14  15   5  11   9  10   7   8    54    46    68
 90   3   6  13  12   2  16   4   1  14  15   5  11  10   9   8   7    53    45    68
 91   3   7   9  15   2   1  16   6   4   8  11  12  10  13  14   5    39    32    84
 92   3   7  11  13   2   5  14   1   4  15   6  10   9  12  16   8    49    42    90
 93   3   8   7  16   2   1  15   6   4   9  12  10  11  14  13   5    32    25    86
 94   3   8  10  13   2   5  14   1   4  15   7   9   6  16  12  11    47    40    90
 95   4   2  12  16   3   8   7  11   1  15  10   5  13   6   9  14    12     8    84
 96   4   2  16  12   3   8  11   7   1  15   6   9  10   5  14  13    73    54    84
 97   4   2  16  12   3   8  11  15   1   7  14   9  10   5   6  13    98    56    68 Adjacent complement pair
 98   4   3  16  11   2   9  12  14   1   7  15   8  10   6   5  13    97    56    68
 99   4   6   8  16   1   2  15   5   3  11  10   9  12  13  14   7    27    22    92
100   4   6   9  15   1   2  16   7   3   8  12  10  13  11  14   5    35    28    86
101   4   7   8  15   1   2  16   5   3  10  11   9  12  13  14   6    45    38    90
102   5   1  12  16   7   3   8  10   2  14   4  11   9   6  15  13    13     9    86
103   5   1  13  15   2   9   8   4   6  16   3  10  12   7  14  11    40    33    88
104   5   1  16  12   3   4  15  10   2   7   8  14   6   9  11  13    68    53    78
105   5   2  11  16   4   8   6   3  12  13   1  15  10   9  14   7    14    10    80
106   5   2  13  14   7   1  12   9   3  10   4  15   6   8  16  11    23    18    82
107   5   3  11  15   4   8   7   2  13  12   1  16   9  10  14   6    63    49    78
108   6   1  12  15   2   7  10   5   3  16   4   8  11   9  13  14    48    41    94
109   6   1  13  14   3   8   9   4   5  16   2  10  11   7  15  12    44    37    90
110   7   1  12  14   4   5  11   6   2  15   3   9   8  10  13  16    21    16    94
111   7   3   9  15   4   2  13   6   1  14   5   8  11  10  16  12    42    35    98
112   8   4   6  16   1   3  14   7   2  11  10   5  12  15   9  13    41    34    98
Top.gif (1256 bytes)rule-w.gif (2718 bytes)

Order 8B List of Basic Solutions

Sol.# |------- Cell value ---- Upper case indicates points -------| Comple  Pair  Point   Remarks   
 #    A   b   c   D   e   f   G   h   I   j   K   l   M   n   O   P  ment    #    total
  1   1   3  16  14   2  13   5  11  15   7  10   4   9  12   6   8    75     1    68
  2   1   3  16  14   5   8   7  11  13   4  12   2   9  15   6  10    81     2    72
  3   1   3  16  14   7   8   5  15  11  10   6   4   9  13   2  12    82     3    60
  4   1   5  12  16   3  11   4  10  15   7   9   2  13   8   6  14     5     4    78  Adjacent complement pair
  5   1   5  12  16   9   6   3  15  11  10   4   7   8  14   2  13     4     4    58
  6   1   5  13  15   3  14   2  16  11  12   8   4   6   9   7  10    66     5    60
  7   1   5  13  15   7   8   4  14  11  10   6   2  12   9   3  16    19     6    68
  8   1   5  13  15   8   7   4  14  11   3  12   2   6  16   9  10    64     7    68
  9   1   5  13  15   8   7   4  14  11   9   6   2  12  10   3  16    28     8    68
 10   1   5  13  15  10   7   2  16  11   9   4   6   8  14   3  12    65     9    56
 11   1   7  11  15   4   3  12   6   9   8  13   5  10  14   2  16    37    10    78
 12   1   7  11  15  10   3   6   8  13   2   9   5  12  16   4  14    69    11    74
 13   1   7  14  12  10   9   3   8  16   2   6   5  15  13   4  11    29    12    68
 14   1   7  15  11   5   8  10   3  14   2  13   6  12  16   4   9    88    13    74
 15   1   7  15  11   6   9   8  16   3  13  12   4   2  14   5  10   103    14    52
 16   1   7  16  10   2  13   9   3  15   5  12   8  11  14   4   6    74    15    68
 17   1   8  11  14   2   5  13   6   7  10  15   4   9  12   3  16    36    16    78
 18   1   8  12  13  10   7   4   6  16   3   5   9  14  15   2  11    20    17    66
 19   1   9   8  16  12   4   2  10  13   3   6   7  11  15   5  14     7     6    68
 20   1   9  11  13  10   7   4   5  16   2   6   8  15  14   3  12    18    17    70
 21   1   9  11  13  12   3   6  15   4   8  10   2   7  14   5  16    55    18    62
 22   1   9  13  11   7   4  12  10   3   8  16   2   6  15   5  14    30    19    68
 23   1   9  14  10   3  15   6  12   7   8  16   4   2  13  11   5    47    20    58
 24   1   9  16   8   2  14  10  11   4  13  15   5   3  12   6   7    76    21    54
 25   1   9  16   8   4  12  10   2  13   3  14   7  11  15   5   6    89    22    68  
 26   1  10   7  16   5   9   4  14   6   8  15   2   3  11  12  13    27    23    70 Adjacent complement pair
 27   1  10   7  16   6   8   4  15   5   9  14   3   2  12  11  13    26    23    66
 28   1  10   7  16  12   4   2   9  13   3   6   8  11  15   5  14     9     8    68
 29   1  10   9  14   8   7   5   3  16   4   6  12  13  15   2  11    13    12    68
 30   1  10   9  14   8   7   5  13   6   4  16   2   3  15  12  11    22    19    68
 31   1  10  11  12  14   2   6  13   5   7   8   4   9  15   3  16    48    24    60
 32   1  10  16   7   6  13   8   2  14   3  11   9  12  15   4   5    84    25    62
 33   1  11   7  15   5   6   8  12   3  10  16   4   2  13   9  14    39    26    68
 34   1  11  12  10  16   5   3  14   6   8   4   7   9  15   2  13    94    27    48
 35   1  11  15   7   5  14   8  12   3  10  16   4   2  13   9   6    38    28    52
 36   1  12   5  16   9   6   3  15   4  11  10   7   2  13   8  14    17    16    58
 37   1  12   6  15   9   3   7  11   4  13   8  10   5  14   2  16    11    10    58
 38   1  12   7  14   6   5   9   3  10   2  16   4  11  13   8  15    35    28    84
 39   1  12   7  14   6   5   9  11   2  10  16   4   3  13   8  15    33    26    68
 40   1  12  10  11  15   3   5  13   4   9   6   8   7  16   2  14    85    29    50
 41   1  12  11  10  16   6   2  15   5   9   4   7   8  14   3  13    95    30    46
 42   1  12  13   8   4  15   7   6   9  10  11  14   3  16   5   2   105    31    46
 43   1  12  13   8   4  16   6   7   9  11  10  14   3  15   5   2   110    32    44
 44   1  12  15   6   8  16   4  13   5  11  10   9   2  14   7   3   111    33    38
 45   1  12  16   5   7  14   8  11   3  15   9  10   4  13   2   6   112    34    38
 46   1  12  16   5  13  14   2   9  11   3   7   8  10  15   6   4   108    35    46
 47   1  13   5  15   4   9   6   3  12   2  16   8   7  14  11  10    23    20    78
 48   1  13   6  14  10   2   8   4   9   3  12   7  11  15   5  16    31    24    76
 49   1  13   8  12  16   4   2  10   9   3   6   7  11  15   5  14    87    36    60
 50   1  13   9  11  14   5   4  15   2   8  10   6   3  16   7  12   104    37    50
 51   1  13  11   9   4  15   6   5  10   8  12  14   3  16   7   2    97    38    50
 52   1  13  14   6   9  15   4   5  12   3  10  11   8  16   7   2    99    39    50
 53   1  13  16   4   8  12  10   2   9   3  14   7  11  15   5   6    86    40    60
 54   1  13  16   4  14  11   5  10   6   2  12   3   9  15   8   7    98    41    52
 55   1  14   3  16   8   6   4   5  11   2  13   9   7  15  10  12    21    18    74
 56   1  14   7  12  10   8   4  11   5  13   6  15   2  16   3   9   100    42    42
 57   1  14  11   8  16   6   4  13   3  10   5   9   7  15   2  12   102    43    42
 58   1  14  12   7   6   8  13   5   2  11  15  10   4  16   3   9    72    44    54
 59   1  14  12   7  15   8   4   5  11   2   6  10  13  16   3   9    91    45    54
 60   1  14  13   6   8  16   4   5  11   3  12  10   7  15   9   2   101    46    52
 61   1  15   7  11  12   9   2   4  13   3   6  14  10  16   5   8    92    47    56
 62   1  15  14   4   8  12  10   3   6   9  11  13   7  16   2   5   106    48    46
 63   1  15  14   4  11  12   7   3   9   6   8  13  10  16   2   5   109    49    46
 64   2   4  12  16   1  10   7  15   8  14  11   3   5   9   6  13     8     7    68
 65   2   4  12  16   3  10   5  11  14   8   9   1  13   7   6  15    10     9    80
 66   2   4  12  16   8   3   7  13  10   5  11   1   9  14   6  15     6     5    76
 67   2   4  16  12   6   7   9   8  13   5  10   1  15  11   3  14    70    50    78
 68   2   5  16  11   1  13   9   6  14   4  15   3  10  12   8   7    79    51    76
 69   2   6  10  16   1  14   3  12  13  15   5   9   8   7   4  11    12    11    62
 70   2   6  12  14   1  16   3  10  15  13   5  11   8   9   4   7    67    50    58
 71   2   6  16  10   3  12   9  14   5  15  11   1   8   7   4  13    93    52    62
 72   2   7  12  13   1   6  14   5   8   9  16   3  10  11   4  15    58    44    82
 73   2   7  14  11   3  15   5  16   6  12  13   1   4   8  10   9    96    53    60
 74   2  10  14   8   4  15   7   1  16   3  11   9  13  12   6   5    16    15    68
 75   2  10  15   7  13   6   8   5  11   1   9   4  16  14   3  12     1     1    68
 76   2  12   6  14   5   4  11   1  10   3  16   8   9  15   7  13    24    21    82
 77   2  12   9  11  16   3   4   8  10   1   7   6  13  15   5  14    90    54    66
 78   2  13  11   8  16   7   3   6  12   1   5   9  14  15   4  10    83    55    58
 79   2  14  11   7   5  13   9   1  10   4  15  12   6  16   8   3    68    51    60
 80   2  14  15   3  13  12   6   5   9   1  11   8  10  16   7   4   107    56    52
 81   3   1  14  16   2   9   7  15  11  13   8   6   5  12   4  10     2     2    64
 82   3   1  14  16   4   9   5  13  15   7   8   2  11  10   6  12     3     3    76
 83   3   2  16  13   6   8   7  10  15   4   9   1  14  11   5  12    78    55    78
 84   3   7  15   9   4  11  10   1  16   2  12   8  13  14   5   6    32    25    74
 85   3   9   7  15   8   1  10   4  11   2  13   5  12  14   6  16    40    29    86
 86   3   9  14   8   4  15   7   5  13   1  16   2  11  10  12   6    53    40    76
 87   3  10   9  12  14   2   6   7  11   1   8   4  15  13   5  16    49    36    76
 88   3  10  14   7   9  12   6   2  16   1   8  11  13  15   5   4    14    13    62
 89   3  10  15   6   2  14  12   1  11   5  16   8   9  13   7   4    25    22    68
 90   3  11   8  12  16   2   4   9  10   1   7   5  13  14   6  15    77    54    70
 91   4   1  15  14   5   7   8   9  16   3  10   2  13  12   6  11    59    45    82
 92   4   2  13  15   8   5   6  10  16   1   9   3  12  14   7  11    61    47    80
 93   4   5  10  15  11   1   7  14   8   3  12   2   6  16   9  13    71    52    74
 94   4  10   5  15   9   2   8   3  13   1  11   6  14  12   7  16    34    27    88
 95   4  10   6  14   8   3   9   2  13   1  12   5  15  11   7  16    41    30    90
 96   4  14   5  11  10   1  12   2   6   3  15   9   8  16   7  13    73    53    76
 97   5   3  12  14   1   9  10   6  15   2  16   4   8  13  11   7    51    38    86
 98   5   3  15  11   4   7  12   6  13   1  16   2  10  14   9   8    54    41    84
 99   5   4  12  13   2   8  11   3  16   1  15   6  10  14   9   7    52    39    86
100   5   7   9  13   6   3  12   4  11   2  15   1  14  10   8  16    56    42    94
101   5   7  12  10   2  14   8   4  15   1  16   3  11   9  13   6    60    46    84
102   5   8   6  15   7   2  10   4  12   1  14   3  13  11   9  16    57    43    94
103   5  11   4  14  10   1   9   8   6   2  16   3   7  13  12  15    15    14    84
104   5  11   8  10   9   1  14   2   7   3  15   4  13  12   6  16    50    37    86
105   6   3  11  14   1   7  12   4  15   2  16   5   9  13  10   8    42    31    90
106   6   4  14  10   1   8  15   3  12   5  16   2  13   9   7  11    62    48    90
107   6   4  16   8   3  12  11   5  14   2  15   1  13   9  10   7    80    56    84
108   6   5   8  15   3   4  12   1  16   2  13   9  11  14   7  10    46    35    90
109   7   1  11  15   3   4  12   5  16   2  13   6  10  14   8   9    63    49    90
110   7   3  10  14   2   6  12   4  15   1  16   5   9  13  11   8    43    32    92
111   7   8   4  15   3   6  10   2  14   1  16   5  11   9  13  12    44    33    98
112   8   7   6  13   4   2  15   1  11   3  16   5  12  10   9  14    45    34    98
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Harvey Heinz   harveyheinz@shaw.ca
This page last updated August 25, 2009
Copyright 1998 by Harvey D. Heinz