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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 of Basic Solutions |
The list of 72 basic solutions, complement number, complement pair number, point totals, and special feature remarks. |
Order 7B List of Basic Solutions |
The 72 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. Use Back button to return |
Examples - 2 |
One example of each of 14 patterns, Orders 12,13,14. Use Back button to return |
Introduction to Order-7The 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 PairsPattern 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 |
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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). 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.
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.
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 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
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 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
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March 01, 2005
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