Order-7 Magic Stars

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Contents

Introduction to Order-7 Cell designations & some of the features of Order-7 magic stars.
Distribution of Point Totals Characteristics of the point totals for order-7.
Order-7A List The list of 72 basic solutions, complement number, complement pair number, point totals, and special feature remarks.
Order-7B List The 72 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-7

The order-7 magic star has 72 basic solutions for each of the two patterns. Note that both patterns have the same point names. Because seven is a prime number, both patterns are continuous.

Order-7 is an anomaly because it has less basic solutions then order-6. In all other cases (at least to order-14), as the order increases, so does the number of solutions.

Each solution contains the consecutive series of integers from 1 to 14. The sum of this series is 105, there are 7 lines with each cell appearing in 2 lines, so the magic constant (S) is (2 x 105)/7 = 30.

Adjacent Complement Pairs

Pattern 7A                  Pattern 7B
Solution      pair #        Solution      pair #
numbers                     numbers 
  2   3         2             1   2         1
  5   6         3             4   5         3
  7   8         4            12  13        10
 11  12         7            20  21        17
 39  40        30            41  42        30
I call a pair an Adjacent Complement Pair if a solution and its complement (after normalization) are next to each other in the solution list.

Pattern 7A has an unusual concentration of complement pairs at the start of the list. The first 8 solutions consist of four pairs, and three of them are adjacent pairs (out of a total of 5 in the entire list).

Distribution of Point Totals

Both patterns 7a and 7b have only one solution for a = 3 (in both cases # 55) and no solutions for a = 7.

Both patterns have two solutions with all odd numbers at the points, and two solutions with all even numbers.
Both patterns have two solutions with numbers 1 to 7 at the points and therefore two solutions with numbers 8 to 14 at the points.

Point     Pattern          Point     Pattern        Point     Pattern  
Total     A    B           Total     A    B         Total     A    B  
 28       2    2            37       0    0          46       0    0
 29       0    0            38       0    0          47       0    0
 30       0    0            39       1    1          48       2    2
 31       1    1            40       2    2          49       4    4
 32       2    2            41       2    2          50       1    1
 33       1    1            42       3    3          51       3    3
 34       1    1            43       2    2          52       3    3
 35       2    2            44       2    2
 36       2    2            45       0    0
Total solutions for the low half of the count is             36   36 
Note that the totals range over 25 numbers but seven of these have no solutions.

The high totals, which are from the other member of the complement pair, are the same as these, but in reverse order. i.e. 53 matches with 52, 54 with 51,. . ., 77 with 28.

Here's a coincidence; the point total of solution 7-A, # 51 is 51!

The only solution in each pattern with a = 3.

                 

Order-7A List

Sol. |------- Basic Solutions (Upper case = peaks)-------|  Comp  Pair  Point
 #   A   b   c   D   e   f   G   h   i   J   k   L   M   N  lement  #   Total    Remarks  Order-7A
 1   1   2  13  14   3   5   8   9   6   7  11  10   4  12     4    1   56
 2   1   2  13  14   3   7   6   4   8  12  11   5   9  10     3    2   57  adjacent complement pairs
 3   1   2  13  14   4   7   5  11   8   6  12  10   3   9     2    2   48
 4   1   2  13  14   4   9   3   6  10  11  12   5   8   7     1    1   49
 5   1   3  12  14   5   9   2   4  11  13   8   6   7  10     6    3   53  adjacent complement pairs
 6   1   3  12  14   7   4   5  11   6   8  10   9   2  13     5    3   52
 7   1   4  11  14   5   9   2   3  12  13   7   6   8  10     8    4   54  adjacent complement pairs
 8   1   4  11  14   8   3   5  12   6   7  10   9   2  13     7    4   51
 9   1   4  13  12   2   5  11   3   6  10   9   7   8  14    30    5   63
10   1   4  13  12   6   7   5   3   8  14  10   2   9  11    14    6   54
11   1   5  10  14   2   3  11   8   7   4   9  12   6  13    12    7   61  adjacent complement pairs
12   1   5  10  14   6   8   2   7  12   9  13   3  11   4    11    7   44
13   1   5  11  13   2   3  12   8   6   4  14   7  10   9    45    8   56
14   1   5  11  13   2   9   6   8  12   4   7  14   3  10    10    6   51
15   1   5  11  13   2   9   6   8  12   4  14   7  10   3    44    9   44
16   1   5  11  13   7   6   4  14   9   3  12  10   2   8    46   10   41
17   1   5  13  11   3   6  10   4   7   9  14   2  12   8    55   11   53
18   1   5  13  11   7   2  10   8   3   9  12   4   6  14    24   12   55
19   1   6  10  13   5   3   9   2   7  12   8   4  11  14    25   13   64
20   1   6  12  11   2   8   9   4  10   7  14   3  13   5    50   14   49  points are all odd numbers
21   1   7  12  10   4  11   5   9  13   3  14   6   8   2    63   15   35
22   1   7  12  10   6  11   3   9  13   5  14   4   8   2    62   16   33
23   1   7  13   9   2  11   8   6  12   4  14   5  10   3    67   17   40
24   1   7  13   9   8   2  11  10   3   6  12   5   4  14    18   12   50
25   1   8   7  14   9   5   2  10  12   6  13   3  11   4    19   13   41
26   1   8  11  10   4   9   7   5  12   6  14   2  13   3    49   18   42
27   1   8  11  10  12   6   2  14   9   5  13   4   3   7    64   19   32
28   1   8  12   9   6   2  13  10   4   3  14   5   7  11    47   20   49  points are all odd numbers
29   1   8  12   9   6  11   4  10  13   3  14   5   7   2    69   21   31
30   1   9   6  14  11   2   3  13  10   4  12   5   8   7     9    5   42
31   1   9   7  13  12   3   2  14  10   4  11   6   5   8    41   22   39
32   1   9  12   8   5  11   6   7  13   4  14   3  10   2    66   23   34
33   1   9  12   8  11   4   7  14   6   3  13   5   2  10    60   24   36
34   1  10  11   8  12   3   7  13   6   4  14   2   5   9    68   25   36
35   1  10  13   6   8  11   5   9  12   4  14   2   7   3    72   26   28  Points are numbers 1 to 7
36   1  11   8  10  12   3   5  14   9   2  13   4   6   7    65   27   35
37   1  11  13   5  10   6   9  12   7   2  14   3   4   8    70   28   32
38   1  11  13   5  10   8   7  12   9   2  14   3   4   6    71   29   28  Points are numbers 1 to 7
39   2   1  14  13   3  10   4   5   9  12  11   6   7   8    40   30   52  adjacent complement pairs
40   2   1  14  13   4   6   7  10   5   8  12   9   3  11    39   30   53
41   2   3  12  13   1   5  11   4   6   9   8  10   7  14    31   22   66
42   2   3  14  11   1   6  12   4   5   9  10   8   7  13    53   31   62
43   2   3  14  11   1  13   5   4  12   9  10   8   7   6    56   32   48
44   2   4  10  14   1   3  12   7   6   5  13   8  11   9    15    9   61
45   2   4  10  14   1   9   6   7  12   5  13   8  11   3    13    8   49
46   2   4  10  14   3   6   7   1   9  13   8   5  12   11   16   10   64
47   2   5  11  12   1   7  10   3   9   8  13   4  14   6    28   20   56  points are all even numbers
48   2   5  14   9   1  13   7   3  12   8   6  11   4  10    57   33   51
49   2   6  10  12   3   1  14   7   4   5  11   8   9  13    26   18   63
50   2   7  11  10   5   1  14   9   3   4  13   6   8  12    20   14   56  points are all even numbers
51   2   9  14   5   1  13  11   3  12   4  10   7   8   6    59   34   43
52   2   9  14   5   1  13  11   4  12   3  10   8   7   6    58   35   42
53   2  10   5  13  12   1   4  14   9   3  11   6   7   8    42   31   43
54   2  11   8   9  13   1   7  14   6   3  12   4   5  10    61   36   40
55   3   9  11   7   8   1  14  10   2   4  12   5   6  13    17   11   52
56   4   1  12  13   5   3   9  11   2   8  14   7   6  10    43   32   57
57   4   1  14  11   2  12   5   3   9  13  10   6   8   7    48   33   54
58   4   2  14  10   1   6  13   5   3   9  11   8   7  12    52   35   63
59   4   2  14  10   1   6  13   5   3   9  12   7   8  11    51   34   62
60   5   1  11  13   4   3  10   6   2  12   9   8   7  14    33   24   69
61   5   1  14  10   2   7  11   4   3  12   9   8   6  13    54   36   65
62   5   3   8  14   1   2  13   6   4   7   9  11  10  12    22   16   72
63   5   3   8  14   1   2  13   6   4   7  11   9  12  10    21   15   70
64   5   3   9  13   1   6  10   2   7  11   4  12   8  14    27   19   73
65   5   3  12  10   1   6  13   2   4  11   7   9   8  14    36   27   70
66   5   4   8  13   2   1  14   6   3   7  10   9  11  12    32   23   71
67   5   4   9  12   3   1  14   8   2   6  13   7  11  10    23   17   65
68   6   2  12  10   3   4  13   5   1  11   9   8   7  14    34   25   69
69   6   3   7  14   1   2  13   5   4   8   9  10  12  11    29   21   74
70   6   3   8  13   1   4  12   2   5  11   9   7  14  10    37   28   73
71   8   3   6  13   1   4  12   2   5  11   7   9  14  10    38   29   77  points are numbers 8 to 14
72   8   4   6  12   3   1  14   5   2   9   7  10  11  13    35   26   77  points are numbers 8 to 14

           

Order-7B List

Sol. |------- Basic Solutions (Upper case = peaks)-------|  Comp  Pair  Point
 #   A   b   c   D   e   f   G   h   i   J   k   L   M   N  lement  #   Total    Remarks  Order-7B
 1   1   3  12  14   2   8   6  13   7   4   5   9  10  11     2    1   55  adjacent complement pairs
 2   1   3  12  14   2  10   4  13   8   5   7   6  11   9     1    1   50
 3   1   4  14  11   2  10   7  12   6   5   8   3  13   9    49    2   49  points are all odd numbers
 4   1   6   9  14   2  11   3  10  13   4  12   5   8   7     5    3   42  adjacent complement pairs
 5   1   6   9  14   5   3   8  13   2   7   4  10  11  12     4    3   63
 6   1   6  10  13   3  12   2  11   8   9   4   7   5  14    28    4   51
 7   1   6  10  13   4  11   2  12   7   9   3   8   5  14    31    5   52
 8   1   6  11  12   5   9   4  13   3  10   2   7   8  14    26    6   56
 9   1   6  13  10   4  14   2  12  11   5   9   3   7   8    60    7   36
10   1   6  14   9   3   8  10  11   2   7   5   4  12  13    50    8   56
11   1   6  14   9   3  13   5  11  12   2  10   4   7   8    68    9   36
12   1   7   8  14   3   2  11  10   5   4   6  12   9  13    13   10   64  adjacent complement pairs
13   1   7   8  14   5   9   2  12  10   6  13   3  11   4    12   10   41
14   1   7   9  13   3   8   6  10   2  12   5   4  11  14    25   11   61
15   1   7   9  13   3  12   2  10  14   4  11   6   5   8    44   12   39
16   1   7  14   8   6   4  12  13   3   2   5   9  10  11    55   13   53
17   1   7  14   8   6  12   4  13  10   3  11   2   9   5    70   14   32
18   1   8  10  11   7   3   9  13   6   2   4  14   5  12    29   15   54
19   1   8  11  10   3  12   5   9  14   2  13   4   6   7    64   16   35
20   1   8  11  10   7   4   9  13   3   5   2  12   6  14    21   17   57  adjacent complement pairs
21   1   8  12   9   7  11   3  13   4  10   2   6   5  14    20   17   48
22   1   9  12   8   6  14   2  11   7  10   5   3   4  13    47   18   41
23   1   9  13   7   6  14   3  11  12   4   8   5   2  10    67   19   32
24   1   9  14   6   8  12   4  13  10   3  11   2   7   5    72   20   28  Points are numbers 1 to 7
25   1  10   5  14   8   6   2  12   7   9  13   3  11   4    14   11   44
26   1  10  12   7   9   6   8  13   4   5   2  11   3  14     8    6   49
27   1  10  14   5   8   4  13  12   2   3   6   7   9  11    39   21   49  points are all odd numbers
28   1  11   4  14   9   5   2  12   3  13   7   6   8  10     6    4   54
29   1  11   5  13   9   2   6  12   8   4   7  14   3  10    18   15   51
30   1  11   5  13   9   2   6  12   8   4  14   7  10   3    46   22   44
31   1  11   8  10   9   4   7  12   5   6   3  13   2  14     7    5   53
32   1  11   8  10   9   4   7  12   5   6  14   2  13   3    40   23   42
33   1  12   7  10  11   4   5  13   9   3  14   6   8   2    66   24   35
34   1  12   7  10  11   6   3  13   9   5  14   4   8   2    65   25   33
35   1  12   8   9  11   6   4  13  10   3  14   5   7   2    69   26   31
36   1  12   9   8  11   5   6  13   7   4  14   3  10   2    63   27   34
37   1  12  14   3  11   6  10  13   2   5   7   4   8   9    61   28   40
38   1  12  14   3  11   9   7  13   8   2  10   4   5   6    71   29   28  points are numbers 1 to 7
39   2  3   13  12   1   9   8  11   5   6   7   4  14  10    27   21   56
40   2  4   11  13   1   7   9  10   3   8   6   5  14  12    32   23   63
41   2  5   10  13   1  12   4   9  14   3  11   6   7   8    42   30   43  adjacent complement pairs
42   2  5   10  13   6   4   7  14   1   8   3   9  12  11    41   30   62
43   2  5   12  11   6  10   3  14   4   9   1   8   7  13    48   31   53
44   2  6    8  14   5   4   7  12   1  10   3   9  11  13    15   12   66
45   2  6   12  10   1  14   5   8  13   4  11   3   7   9    62   32   40
46   2  6   13   9   3   7  11  10   1   8   4   5  12  14    30   22   61
47   2  9    8  11   6   1  12  10   3   5   4  13   7  14    22   18   64
48   2  9   11   8  10   5   7  14   3   6   1  12   4  13    43   31   52
49   2  9   13   6   7   3  14  11   1   4   5   8  10  12     3    2   56
50   2  10   4  14   9   1   6  12   7   5  13   8  11   3    10    8   49
51   2  12   6  10  13   3   4  14   1  11   5   8   7   9    59   33   51
52   2  12  10   6  13   3   8  14   1   7   9   4  11   5    58   34   43
53   2  12  10   6  13   4   7  14   1   8   3   9   5  11    56   35   48
54   2  12  10   6  13   4   7  14   1   8   9   3  11   5    57   36   42
55   3   2  12  13   1  10   6  11   8   5   9   4  14   7    16   13   52
56   4   2  14  10   3  11   6  12   5   7   1   8   9  13    53   35   57
57   4   3  11  12   6   5   7  14   1   8   2   9  13  10    54   36   63
58   4   3  12  11   6   5   8  14   1   7   2   9  13  10    52   34   62
59   4   9  10   7  12   3   8  14   2   6   1  13   5  11    51   33   54
60   5   2   9  14   3   6   7  11   4   8   1  12  10  13     9    7   69
61   5   2  13  10   1   8  11   9   3   7   4   6  14  12    37   28   65
62   5   3   9  13   7   4   6  14   2   8   1  12  11  10    45   32   65
63   5   3  10  12   1   6  11   8   2   9   4   7  14  13    36   27   71
64   5   4   7  14   6   2   8  12   1   9   3  11  13  10    19   16   70
65   5   4   9  12   2   6  10   8   1  11   3   7  13  14    34   25   72
66   5   4  11  10   2   6  12   8   1   9   3   7  13  14    33   24   70
67   5   7   4  14   6   2   8   9   1  12   3  11  10  13    23   19   73
68   6   1   9  14   4   5   7  12   3   8   2  11  13  10    11    9   69
69   6   4   9  11   2   5  12   7   1  10   3   8  13  14    35   26   74
70   6   5   9  10   4   2  14   8   1   7   3  11  12  13    17   14   73
71   8   2   7  13   1   5  11   6   3  10   4   9  14  12    38   29   77  points are numbers 8 to 14
72   8   5   7  10   4   2  14   6   1   9   3  11  12  13    24   20   77  points are numbers 8 to 14

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