Office Applications and Entertainment, Magic Squares of Subtraction

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11.0 Squares of Subtraction (11 x 11)

11.1 Generator Method (11 x 11)

The number of complement free Euler Series for order 11 (Additive) Magic Squares (s11 = 671) with Residuum 61 would be inconvenient high.

A more controllable collection can be obtained by limiting the collection to complement free Bimagic Euler Series (s2 = 54351).

This results in a collection of 30396 magic series (s11 = 671, Res11 = 61) and enables, for order 11 Magic Squares which are also Squares of Subtraction, a comparable construction as described in Section 9.1:

  • Based on the magic series mentioned above, Generators can be constructed with CnstrGen11:
    complement free sets of five complement free rows,
    which can be amended with five rows with their complements (blue) and
    one symmetrical row with the remaining nine integers (green)
  • Based on the Generators obtained above, Semi Magic Squares with 11 magic rows and columns can be constructed by permutating the numbers within the rows of the Generators;
  • By permutation of the rows and columns within the Semi Magic Squares, (Additive) Magic Squares which are also Squares of Subtraction can be obtained (ref. CnstrSqrs11b).

Subject procedure is illustrated below for the first occurring Semi Magic Square based on the first occurring (completed) Generator.

Generator
1 14 29 43 53 63 77 81 94 104 112
2 12 30 44 54 64 71 83 95 102 114
3 15 32 40 52 57 75 88 89 105 115
5 13 31 36 50 66 73 87 96 103 111
6 16 24 42 48 62 76 85 99 100 113
121 108 93 79 69 59 45 41 28 18 10
120 110 92 78 68 58 51 39 27 20 8
119 107 90 82 70 65 47 34 33 17 7
117 109 91 86 72 56 49 35 26 19 11
116 106 98 80 74 60 46 37 23 22 9
4 21 25 38 55 61 67 84 97 101 118
Sem Magic Square
1 14 29 43 53 63 77 81 94 112 104
2 12 30 44 54 64 71 83 114 102 95
3 15 32 40 52 57 75 105 89 115 88
31 36 5 13 50 66 87 111 103 73 96
99 62 76 85 100 48 42 113 16 24 6
121 79 108 93 45 69 59 10 18 41 28
120 110 51 78 68 92 58 39 20 27 8
119 107 90 82 34 65 33 17 7 47 70
91 117 109 49 86 26 56 19 72 11 35
80 98 116 106 74 60 46 9 37 22 23
4 21 25 38 55 61 67 84 101 97 118
Simple Magic Square
1 14 29 53 63 43 94 112 81 104 77
2 12 30 54 64 44 114 102 83 95 71
3 15 32 52 57 40 89 115 105 88 75
91 117 109 86 26 49 72 11 19 35 56
119 107 90 34 65 82 7 47 17 70 33
120 110 51 68 92 78 20 27 39 8 58
4 21 25 55 61 38 101 97 84 118 67
121 79 108 45 69 93 18 41 10 28 59
99 62 76 100 48 85 16 24 113 6 42
31 36 5 50 66 13 103 73 111 96 87
80 98 116 74 60 106 37 22 9 23 46

Notes

  1. Based on the (sub) collection of magic series described above, numerous Generators can be generated, within a reasonable time.
  2. The construction of Semi Magic Squares (s11 = 671, Res11 = 61) is illustrated and described in Exhibit 11.

12.0 Squares of Subtraction (12 x 12)

12.1 Generator Method (12 x 12)

Most Perfect Magic Squares, as deducted and discussed in Section 12.2.3, might have twelve rows with Res12 = 72.

Consequently subject squares can be used as Generators for the construction of order 12 Magic Squares which are also Squares of Subtraction:

  • Suitable Generators (s12 = 870, Res12 = 72) can be constructed with procedure (ref. MostPerf12);
  • Based on the Generators obtained above, Semi Magic Squares with 12 magic rows and columns can be constructed by permutating the numbers within the rows of the Generators (ref. SemiSqrs12);
  • By permutation of the rows and columns within the Semi Magic Squares, (Additive) Magic Squares which are also Squares of Subtraction can be obtained (ref. CnstrSqrs12b).

Subject procedure is illustrated below for the first occurring (suitable) Generator.

Generator
3 118 87 70 99 22 135 34 51 82 39 130
137 32 53 80 41 128 5 116 89 68 101 20
1 120 85 72 97 24 133 36 49 84 37 132
138 31 54 79 42 127 6 115 90 67 102 19
9 112 93 64 105 16 141 28 57 76 45 124
134 35 50 83 38 131 2 119 86 71 98 23
10 111 94 63 106 15 142 27 58 75 46 123
140 29 56 77 44 125 8 113 92 65 104 17
12 109 96 61 108 13 144 25 60 73 48 121
139 30 55 78 43 126 7 114 91 66 103 18
4 117 88 69 100 21 136 33 52 81 40 129
143 26 59 74 47 122 11 110 95 62 107 14
Semi Magic Square
3 118 87 70 99 22 135 34 51 82 39 130
137 32 53 80 41 128 5 116 89 68 20 101
1 120 85 72 97 24 133 36 49 84 132 37
138 31 54 79 42 127 6 115 90 19 67 102
9 112 93 64 105 16 141 28 57 76 124 45
134 35 50 83 38 131 2 86 98 119 71 23
10 111 94 63 106 15 142 46 58 123 27 75
140 29 77 56 44 113 8 92 125 104 65 17
12 109 61 13 144 60 48 96 121 73 108 25
43 78 18 114 103 55 7 126 66 30 91 139
100 69 88 129 40 117 136 21 52 33 4 81
143 26 110 47 11 62 107 74 14 59 122 95
Simple Magic Square
3 118 87 70 135 99 22 82 39 130 34 51
137 32 53 80 5 41 128 68 20 101 116 89
1 120 85 72 133 97 24 84 132 37 36 49
138 31 54 79 6 42 127 19 67 102 115 90
9 112 93 64 141 105 16 76 124 45 28 57
43 78 18 114 7 103 55 30 91 139 126 66
140 29 77 56 8 44 113 104 65 17 92 125
12 109 61 13 48 144 60 73 108 25 96 121
10 111 94 63 142 106 15 123 27 75 46 58
143 26 110 47 107 11 62 59 122 95 74 14
100 69 88 129 136 40 117 33 4 81 21 52
134 35 50 83 2 38 131 119 71 23 86 98

Notes

  1. Although suitable Generators and resulting Semi Magic Squares (s12 = 870, Res12 = 72) can be generated quite fast, correct sets of diagonals occur very rarely.
  2. Order 12 (non additive) Composed Magic Squares of Subtraction will be discussed in following Section 12.5.

12.2 Other Residuum Values

Attachment 12.2 shows a few Magic Squares of Subtraction with s12 = 870 and different residuum values, which could be obtained with the method described above.

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