Office Applications and Entertainment, Magic Squares

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18.0   Special Magic Squares, Lozenge Squares

18.7   Lozenge Squares (13 x 13)

18.7.1 Associated Magic Squares (La Hire)

Associated Lozenge Squares can be constructed based on Latin Squares (B1/B2) as illustrated below (La Hire):

B1
7 8 9 10 11 12 0 1 2 3 4 5 6
8 9 10 11 12 0 1 2 3 4 5 6 7
9 10 11 12 0 1 2 3 4 5 6 7 8
10 11 12 0 1 2 3 4 5 6 7 8 9
11 12 0 1 2 3 4 5 6 7 8 9 10
12 0 1 2 3 4 5 6 7 8 9 10 11
0 1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 0
2 3 4 5 6 7 8 9 10 11 12 0 1
3 4 5 6 7 8 9 10 11 12 0 1 2
4 5 6 7 8 9 10 11 12 0 1 2 3
5 6 7 8 9 10 11 12 0 1 2 3 4
6 7 8 9 10 11 12 0 1 2 3 4 5
B2 (B1 Mirrored)
6 5 4 3 2 1 0 12 11 10 9 8 7
7 6 5 4 3 2 1 0 12 11 10 9 8
8 7 6 5 4 3 2 1 0 12 11 10 9
9 8 7 6 5 4 3 2 1 0 12 11 10
10 9 8 7 6 5 4 3 2 1 0 12 11
11 10 9 8 7 6 5 4 3 2 1 0 12
12 11 10 9 8 7 6 5 4 3 2 1 0
0 12 11 10 9 8 7 6 5 4 3 2 1
1 0 12 11 10 9 8 7 6 5 4 3 2
2 1 0 12 11 10 9 8 7 6 5 4 3
3 2 1 0 12 11 10 9 8 7 6 5 4
4 3 2 1 0 12 11 10 9 8 7 6 5
5 4 3 2 1 0 12 11 10 9 8 7 6
M = B1 + 13* B2 + 1
86 74 62 50 38 26 1 158 146 134 122 110 98
100 88 76 64 52 27 15 3 160 148 136 124 112
114 102 90 78 53 41 29 17 5 162 150 138 126
128 116 104 79 67 55 43 31 19 7 164 152 140
142 130 105 93 81 69 57 45 33 21 9 166 154
156 131 119 107 95 83 71 59 47 35 23 11 168
157 145 133 121 109 97 85 73 61 49 37 25 13
2 159 147 135 123 111 99 87 75 63 51 39 14
16 4 161 149 137 125 113 101 89 77 65 40 28
30 18 6 163 151 139 127 115 103 91 66 54 42
44 32 20 8 165 153 141 129 117 92 80 68 56
58 46 34 22 10 167 155 143 118 106 94 82 70
72 60 48 36 24 12 169 144 132 120 108 96 84

The resulting Lozenge Square M corresponds with 64 Lozenge Squares, which can be obtained by exchange of row and column n with (14 - n), and include for each square the 180o rotated aspect.

18.7.2 Concentric Magic Squares

The Concentric Center Squares of an order 13 Concentric Lozenge Square have following properties:

  • The 3 x 3, 5 x 5 and 7 x 7 Center Square have only odd numbers
  • The 9 x 9 Concentric Square has 4 x 3 even corner numbers
  • The 11 x 11 Concentric Square has 4 x 7 even corner numbers

as illustrated in following example:

86 74 62 50 38 26 1 158 146 134 122 110 98
100 54 52 46 34 95 99 111 150 148 106 40 70
114 66 166 10 47 53 65 125 127 164 8 104 56
128 68 18 147 31 33 67 143 145 29 152 102 42
142 90 49 37 161 17 73 159 15 133 121 80 28
156 89 51 39 19 165 7 83 151 131 119 81 14
157 91 57 41 21 3 85 167 149 129 113 79 13
2 93 61 55 69 87 163 5 101 115 109 77 168
16 94 63 135 155 153 97 11 9 35 107 76 154
30 78 138 141 139 137 103 27 25 23 32 92 140
44 82 162 160 123 117 105 45 43 6 4 88 126
58 130 118 124 136 75 71 59 20 22 64 116 112
72 96 108 120 132 144 169 12 24 36 48 60 84

It should be noticed that the 13 x 13 Concentric Border is a conversion of the 13 x 13 Associated Border as applied in Section 18.7.1 above.

Concentric Lozenge Squares of order 13 can be constructed based on the remainder of the integers:

  • Generate order 3 Center Squares (odd numbers);
  • Generate order 5 Concentric Squares (odd numbers), based on a selection of the order 3 Center Squares;
  • Generate order 7 Concentric Squares (odd numbers), based on a selection of the order 5 Concentric Squares;
  • Generate order 9 Concentric Squares (4 x 3 even corner numbers), based on a selection of the order 7 Concentric Squares;
  • Generate order 11 Concentric Squares (4 x 7 even corner numbers), based on a selection of the order 9 Concentric Squares.

Routine Priem3a1 generated 2912 (364 unique) order 3 Center Squares with odd numbers (s1 = 255).
Attachment 18.7.1 shows 48 of the 364 unique order 3 Center Squares.

Attachment 18.7.3 shows the first occurring order 5 Concentric Squares with odd numbers for each of the order 3 Center Squares with odd numbers shown in Attachment 18.7.1 (ref. Priem5c1).

Attachment 18.7.4 shows the first occurring order 7 Concentric Squares with odd numbers for each of the order 5 Concentric Squares with odd numbers shown in Attachment 18.7.3 (ref. Priem7d1).

Attachment 18.7.5 shows the first occurring order 9 Concentric Squares with 4 x 3 even corner numbers for each of the order 7 Concentric Squares with odd numbers shown in Attachment 18.7.4 (ref. Priem9c1).

Attachment 18.7.6 shows the first occurring order 11 Concentric Squares with 4 x 7 even corner numbers for each of the order 9 Concentric Squares with 4 x 3 even corner numbers shown in Attachment 18.7.5 (ref. MgcSqr11a1).

Attachment Lozenge 13.1 shows the corresponding order 13 Concentric Lozenge Squares, obtained by combining the 13 x 13 border with the order 11 Concentric Squares shown in Attachment 18.7.6.

Notes:

  1. By means of permutation of the border pairs of the main and sub squares, each of the order 13 Concentric Lozenge Squares shown correspond with (3!)2 * (5!)2 * (5! * 2!)2 * (6! * 3!)2 * (10!)2 = 7,33 1030 squares.
  2. While varying the border corner points of the main and sub squares, this number has at least to be multiplied with 76 * 60 * 84 * 72 * 44 = 1,2 * 109 resulting in 8,9 * 1039 squares, for each unique order 3 Center Square.

Both values are only a fraction of the total possible number of order 13 Concentric Lozenge Squares.

18.8   Lozenge Squares (15 x 15)

18.8.1 Associated Magic Squares (La Hire)

Associated Lozenge Squares can be constructed based on Latin Squares (B1/B2) as illustrated below (La Hire):

B1
8 9 10 11 12 13 14 0 1 2 3 4 5 6 7
9 10 11 12 13 14 0 1 2 3 4 5 6 7 8
10 11 12 13 14 0 1 2 3 4 5 6 7 8 9
11 12 13 14 0 1 2 3 4 5 6 7 8 9 10
12 13 14 0 1 2 3 4 5 6 7 8 9 10 11
13 14 0 1 2 3 4 5 6 7 8 9 10 11 12
14 0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 0
2 3 4 5 6 7 8 9 10 11 12 13 14 0 1
3 4 5 6 7 8 9 10 11 12 13 14 0 1 2
4 5 6 7 8 9 10 11 12 13 14 0 1 2 3
5 6 7 8 9 10 11 12 13 14 0 1 2 3 4
6 7 8 9 10 11 12 13 14 0 1 2 3 4 5
7 8 9 10 11 12 13 14 0 1 2 3 4 5 6
B2 (B1 Mirrored)
7 6 5 4 3 2 1 0 14 13 12 11 10 9 8
8 7 6 5 4 3 2 1 0 14 13 12 11 10 9
9 8 7 6 5 4 3 2 1 0 14 13 12 11 10
10 9 8 7 6 5 4 3 2 1 0 14 13 12 11
11 10 9 8 7 6 5 4 3 2 1 0 14 13 12
12 11 10 9 8 7 6 5 4 3 2 1 0 14 13
13 12 11 10 9 8 7 6 5 4 3 2 1 0 14
14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
0 14 13 12 11 10 9 8 7 6 5 4 3 2 1
1 0 14 13 12 11 10 9 8 7 6 5 4 3 2
2 1 0 14 13 12 11 10 9 8 7 6 5 4 3
3 2 1 0 14 13 12 11 10 9 8 7 6 5 4
4 3 2 1 0 14 13 12 11 10 9 8 7 6 5
5 4 3 2 1 0 14 13 12 11 10 9 8 7 6
6 5 4 3 2 1 0 14 13 12 11 10 9 8 7
M = B1 + 15* B2 + 1
114 100 86 72 58 44 30 1 212 198 184 170 156 142 128
130 116 102 88 74 60 31 17 3 214 200 186 172 158 144
146 132 118 104 90 61 47 33 19 5 216 202 188 174 160
162 148 134 120 91 77 63 49 35 21 7 218 204 190 176
178 164 150 121 107 93 79 65 51 37 23 9 220 206 192
194 180 151 137 123 109 95 81 67 53 39 25 11 222 208
210 181 167 153 139 125 111 97 83 69 55 41 27 13 224
211 197 183 169 155 141 127 113 99 85 71 57 43 29 15
2 213 199 185 171 157 143 129 115 101 87 73 59 45 16
18 4 215 201 187 173 159 145 131 117 103 89 75 46 32
34 20 6 217 203 189 175 161 147 133 119 105 76 62 48
50 36 22 8 219 205 191 177 163 149 135 106 92 78 64
66 52 38 24 10 221 207 193 179 165 136 122 108 94 80
82 68 54 40 26 12 223 209 195 166 152 138 124 110 96
98 84 70 56 42 28 14 225 196 182 168 154 140 126 112

The resulting Lozenge Square M corresponds with 128 Lozenge Squares, which can be obtained by exchange of row and column n with (16 - n), and include for each square the 180o rotated aspect included.

18.8.2 Concentric Magic Squares

The Concentric Center Squares of an order 15 Concentric Lozenge Square have following properties:

  • The 3 x 3, 5 x 5 and 7 x 7 Center Square have only odd numbers
  • The 9 x 9 Concentric Square has 4 even corner numbers
  • The 11 x 11 Concentric Square has 4 x 5 even corner numbers

as illustrated in following example:

114 100 86 72 58 44 30 1 212 198 184 170 156 142 128
130 116 102 88 122 60 31 17 3 214 200 186 172 158 96
146 90 120 188 202 89 87 85 83 81 10 94 204 136 80
162 148 134 220 161 123 63 61 59 57 55 218 92 78 64
178 164 108 153 193 41 43 95 189 191 39 73 118 62 48
194 180 125 151 49 215 23 99 207 21 177 75 101 46 32
210 181 119 149 51 25 219 9 111 201 175 77 107 45 16
211 197 117 129 53 27 5 113 221 199 173 97 109 29 15
2 213 105 71 79 93 115 217 7 133 147 155 121 13 224
18 4 91 69 179 205 203 127 19 11 47 157 135 222 208
34 20 150 67 187 185 183 131 37 35 33 159 76 206 192
50 36 152 8 65 103 163 165 167 169 171 6 74 190 176
66 52 22 38 24 137 139 141 143 145 216 132 106 174 160
82 68 124 138 104 166 195 209 223 12 26 40 54 110 144
98 126 140 154 168 182 196 225 14 28 42 56 70 84 112

It should be noticed that the two exterior Concentric Borders (order 13 and 15) are deducted from the Associated Borders as applied in Section 18.8.1 above.

Concentric Lozenge Squares of order 15 can be constructed based on the remainder of the integers:

  • Generate order 3 Center Squares (odd numbers);
  • Generate order 5 Concentric Squares (odd numbers), based on a selection of the order 3 Center Squares;
  • Generate order 7 Concentric Squares (odd numbers), based on a selection of the order 5 Concentric Squares;
  • Generate order 9 Concentric Squares (4 even corner numbers), based on a selection of the order 7 Concentric Squares;
  • Generate order 11 Concentric Squares (4 x 5 even corner numbers), based on a selection of the order 9 Concentric Squares.

Routine Priem3a2 generated 3976 (497 unique) order 3 Center Squares with odd numbers (s1 = 339).
Attachment 18.8.1 shows 48 of the 497 unique order 3 Center Squares.

Attachment 18.8.3 shows the first occurring order 5 Concentric Squares with odd numbers for each of the order 3 Center Squares with odd numbers shown in Attachment 18.8.1 (ref. Priem5c2).

Attachment 18.8.4 shows the first occurring order 7 Concentric Squares with odd numbers for each of the order 5 Concentric Squares with odd numbers shown in Attachment 18.8.3 (ref. Priem7d2).

Attachment 18.8.5 shows the first occurring order 9 Concentric Squares with 4 even corner numbers for each of the order 7 Concentric Squares with odd numbers shown in Attachment 18.8.4 (ref. Priem9c2).

Attachment 18.8.6 shows the first occurring order 11 Concentric Squares with 4 x 5 even corner numbers for each of the order 9 Concentric Squares with 4 even corner numbers shown in Attachment 18.8.5 (ref. MgcSqr11a2).

Attachment Lozenge 15.1 shows the corresponding order 15 Concentric Lozenge Squares, obtained by combining the order 13 and 15 borders with the order 11 Concentric Squares shown in Attachment 18.8.6.

Notes:

  1. By means of permutation of the border pairs of the main - and sub squares, each of the order 15 Concentric Lozenge Squares shown correspond with (3!)2 * (5)2 * (7!)2 * (5! * 4!)2 * (8! * 3!)2 * (12!)2 = 1,5 1048 squares.
  2. While varying the border corner points of the main - and sub squares, this number has at least to be multiplied with 104 * 88 * 112 * 108 * 88 * 52 = 9,7 * 109 resulting in 1,43 * 1058 squares, for each unique order 3 Center Square.

Both values are only a fraction of the total possible number of order 15 Concentric Lozenge Squares.

18.9   Summary

The obtained results regarding the miscellaneous types of Lozenge Squares as deducted and discussed in previous sections are summarized in following table:

Order

Characteristics

Subroutine

Examples

Total Number

Notes

13

Concentric

-

Lozenge 13.1

-

Note 1

Associated, Latin Square Based

-

Section 18.7.1

-

-

15

Concentric

-

Lozenge 15.1

-

Note 2

Associated, Latin Square Based

-

Section 18.8.1

-

-

Note 1: Constructed border  order 13
Note 2: Constructed borders order 13 and 15


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