Office Applications and Entertainment, Magic Squares

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10.2    Construction Methods, Inlaid Magic Squares

10.2.11 Simple Magic Squares (10 x 10)
        Generator Method

Simple Magic Squares of order 10 can be constructed very efficiently with the Generator Principle, as applied for the construction of Bimagic Squares (ref. Section 15).

The Generator Method, as applied for Simple Magic Squares of order 10 can be summarised as follows:

  • Generate Magic Series for the consecutive integers {1 ... 100} and the related Magic Sum (505);
  • Construct Generators with 10 Magic Rows, based on the Magic Series obtained above;
  • Construct Semi Magic Squares, by permutating the numbers within the rows of the Generators;
  • Permutate the rows and columns within the Semi Magic Squares, in order to obtain Magic Squares
    (ref. CnstrSqrs10)).

Suitable Generators for order 10 Magic Squares can be constructed semi-automatically (ref. CnstrGen10).

A possible order 10 Generator, Semi Magic Square and resulting order 10 Simple Magic Square are shown below:

Generator
1 2 3 4 25 92 93 94 95 96
5 6 7 8 9 76 97 98 99 100
10 11 12 13 34 83 84 85 86 87
14 15 16 17 18 67 88 89 90 91
19 20 21 22 46 73 74 75 77 78
23 24 26 27 28 55 79 80 81 82
29 30 31 32 57 63 64 65 66 68
33 35 36 37 38 44 69 70 71 72
39 40 42 51 52 53 54 56 58 60
41 43 45 47 48 49 50 59 61 62
Semi Magic Square
1 2 3 4 25 92 93 94 95 96
100 99 98 97 76 9 8 7 6 5
10 11 12 13 34 83 84 85 86 87
91 90 89 88 67 18 17 16 15 14
19 20 21 22 46 73 74 75 77 78
82 81 80 79 55 28 27 26 24 23
29 30 31 32 57 63 64 65 66 68
72 71 70 69 44 38 37 36 35 33
39 40 42 51 52 53 54 56 58 60
62 61 59 50 49 48 47 45 43 41
Simple Magic Square
1 3 92 94 25 95 96 93 4 2
10 12 83 85 34 86 87 84 13 11
19 21 73 75 46 77 78 74 22 20
29 31 63 65 57 66 68 64 32 30
62 59 48 45 49 43 41 47 50 61
39 42 53 56 52 58 60 54 51 40
72 70 38 36 44 35 33 37 69 71
82 80 28 26 55 24 23 27 79 81
91 89 18 16 67 15 14 17 88 90
100 98 9 7 76 6 5 8 97 99

The Semi Magic Square shown above results in numerous Essential Different Magic Squares. The first twelve sets of potential diagonals are shown in Attachment 10.2.11.

Each Essential Different Magic Square corresponds with 1920 transformations, as described below.

  • Any line n can be interchanged with line (11 - n). The possible number of transformations is 25 = 32.
    It should be noted that for each square the 180o rotated aspect is included in this collection.

  • Any permutation can be applied to the lines 1, 2, 3, 4, 5, provided that the same permutation is applied to the lines 10, 9, 8, 7, 6. The possible number of transformations is 5! = 120.

The resulting number of transformations, excluding the 180o rotated aspects, is 32/2 * 120 = 1920.

10.2.12 Simple Magic Squares (10 x 10)
        Order 5 Magic Square Inlay

Order 10 Simple Magic Squares with order 5 Magic Square Inlay(s) can be constructed with the Generator Method, as illustrated by following example (s5 = 250):

Generator
90 8 98 6 48 7 9 53 86 100
55 3 95 1 96 11 12 56 83 93
49 46 50 54 51 13 14 57 80 91
4 99 5 97 45 15 16 47 88 89
52 94 2 92 10 17 18 58 75 87
19 20 21 22 23 68 81 82 84 85
24 25 26 27 28 65 76 77 78 79
29 30 31 32 33 60 71 72 73 74
34 35 36 37 38 59 61 66 69 70
39 40 41 42 43 44 62 63 64 67
Semi Magic Square
90 8 98 6 48 7 9 53 86 100
55 3 95 1 96 93 83 56 12 11
49 46 50 54 51 57 91 14 80 13
4 99 5 97 45 89 88 47 16 15
52 94 2 92 10 18 17 75 58 87
19 20 21 22 23 68 82 84 81 85
79 77 78 65 76 28 27 26 25 24
29 30 33 60 74 32 31 72 71 73
66 61 59 69 38 70 37 36 35 34
62 67 64 39 44 43 40 42 41 63
Simple Magic Square
90 8 98 6 48 12 93 56 11 83
55 3 95 1 96 16 89 47 15 88
49 46 50 54 51 58 18 75 87 17
4 99 5 97 45 86 7 53 100 9
52 94 2 92 10 80 57 14 13 91
23 22 21 19 20 81 68 84 85 82
76 65 78 79 77 25 28 26 24 27
74 60 33 29 30 71 32 72 73 31
38 69 59 66 61 35 70 36 34 37
44 39 64 62 67 41 43 42 63 40

The construction method is as described in Section 10.2.11 above, with exception of the last step (main diagonals):

  • The top/left to bottom/right diagonal should be completed by permutating
    the columns of the right 5 x 10 rectangle
  • The top/right to bottom/left diagonal should be completed by permutating
    the rows of the top/right 5 x 5 corner and
    the columns of the bottom/left 5 x 5 corner

The applied Square Inlay is an order 5 Associated Magic Square with order 3 Diamond Inlay.

Potential Order 5 Magic Squares with Diamond Inlays might be constructed for the integers {1 ... 100} with routine MgcSqr0212.

Alternatively Order 5 Simple Magic, Associated, Pan Magic or Ultra Magic Squares can be used as Square Inlay(s).

10.2.13 Simple Magic Squares (10 x 10)
        Order 6 Magic Square Inlay

Order 10 Simple Magic Squares with order 6 Magic Square Inlay(s) can be constructed with the Generator Method, as illustrated by following example (s6 = 300):

Generator
5 55 84 17 43 96 9 10 89 97
86 54 7 92 48 13 11 12 82 100
20 40 99 2 58 81 15 18 78 94
83 57 4 95 45 16 21 22 71 91
8 52 87 14 46 93 23 24 68 90
98 42 19 80 60 1 25 26 66 88
27 28 29 30 31 44 75 77 79 85
32 33 34 35 36 47 65 73 74 76
37 38 39 41 49 50 51 59 69 72
3 6 53 56 61 62 63 64 67 70
Semi Magic Square
5 55 84 17 43 96 97 89 9 10
86 54 7 92 48 13 100 82 11 12
20 40 99 2 58 81 15 18 94 78
83 57 4 95 45 16 22 21 71 91
8 52 87 14 46 93 23 24 68 90
98 42 19 80 60 1 26 25 88 66
75 28 29 30 31 44 79 85 77 27
76 65 34 74 73 47 35 36 32 33
51 59 72 38 39 50 41 69 49 37
3 53 70 63 62 64 67 56 6 61
Simple Magic Square
5 55 84 17 43 96 94 15 78 18
86 54 7 92 48 13 71 22 91 21
20 40 99 2 58 81 9 97 10 89
83 57 4 95 45 16 11 100 12 82
8 52 87 14 46 93 68 23 90 24
98 42 19 80 60 1 88 26 66 25
75 30 28 29 31 44 77 79 27 85
76 74 65 34 73 47 32 35 33 36
51 38 59 72 39 50 49 41 37 69
3 63 53 70 62 64 6 67 61 56

The construction method is as described in Section 10.2.11 above, with exception of the last step (main diagonals):

  • The top/left to bottom/right diagonal should be completed by permutating
    the columns of the right 4 x 10 rectangle
  • The top/right to bottom/left diagonal should be completed by permutating
    the rows of the top/right 4 x 6 corner and
    the columns of the bottom/left 6 x 4 corner

The applied Square Inlay is an order 6 Most Perfect Magic Square.

Potential Order 6 Most Perfect Magic Squares might be constructed for the integers {1 ... 100} with routine MgcSqr0213.

Alternatively other types of Order 6 Magic Squares (ref. Sections 6) can be used as Square Inlay(s).

10.2.14 Simple Magic Squares (10 x 10)
        Order 7 Magic Square Inlay

Order 10 Simple Magic Squares with order 7 Magic Square Inlay(s) can be constructed with a comparable method, as illustrated by following example (s7 = 350):

Semi Magic Square
81 24 30 34 70 40 71 98 55 2
23 25 31 89 67 41 74 5 53 97
79 50 68 15 36 73 29 52 4 99
80 76 64 19 38 45 28 56 96 3
21 26 62 90 39 66 46 92 57 6
22 77 63 83 35 43 27 51 9 95
44 72 32 20 65 42 75 18 49 88
58 59 60 48 17 78 37 54 87 7
86 84 82 93 47 61 33 10 1 8
11 12 13 14 91 16 85 69 94 100
Simple Magic Square
81 24 30 34 70 40 71 98 55 2
23 25 31 89 67 41 74 5 53 97
79 50 68 15 36 73 29 52 4 99
80 76 64 19 38 45 28 56 96 3
21 26 62 90 39 66 46 92 57 6
22 77 63 83 35 43 27 51 9 95
44 72 32 20 65 42 75 18 49 88
37 58 78 59 60 48 17 54 87 7
33 86 61 84 82 93 47 10 1 8
85 11 16 12 13 14 91 69 94 100

The construction method can be summarised as follows:

  • The 'Border' can be constructed based on the Magic Sum s7 and the defining equations
    as incorporated in procedure MgcSqr0214
  • The Order 7 Simple Magic Square Inlay should be constructed based on the remaining integers
  • The top/right to bottom/left diagonal should be completed by permutating
    the rows of the top/right 3 x 7 corner (if required)
    the columns of the bottom/left 7 x 3 corner

The Order 7 Simple Magic Square Inlay might be constructed with the Generator Method as discussed above for order 10 Simple Magic Squares (ref. Section 10.2.11).

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