Office Applications and Entertainment, Magic Squares

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25.0 Magic Squares, Higher Order, Overlapping Sub Squares

25.1 Introduction

In Sections 22 thru 24 Composed Simple Inlaid Magic Squares have been discussed for miscellaneous orders.

Following sections will summarise the previously found results for Composed Magic Squares with Overlapping Sub Squares, based on their defining linear equations.

In addition to this a suitable construction method will be introduced for higher order Composed Magic Squares with Overlapping Sub Squares.

25.2 Magic Squares (7 x 7)

Magic Squares of order 7, with two 4th order Overlapping Sub Squares with identical Magic Sums have been described in Section 7.7.3.

Examples of such Magic Squares of order 7 are shown in Attachment 7.6.8.

25.3 Magic Squares (9 x 9)

Magic Squares of order 9, with two 5th order Overlapping Sub Squares with identical Magic Sums have been described in Section 9.7.1.

Examples of such Magic Squares of order 7 are shown in Attachment 9.7.3.

25.4 Magic Squares (11 x 11)

A classical Magic Square of order 11, with miscellaneous each other Overlapping Sub Squares has been described in Section 11.2.2.

Examples of such Magic Squares of order 11 are shown in Attachment 14.9.8.1.

Alternatively Magic Squares of order 11, with Magic Sum s11 = 671, composed of following Sub Squares:

  • Two each other overlapping 6th order Compact Pan Magic Sub Squares (s6 = 366)
  • Two 5th order Pan Magic Sub Squares (s5 = 305)

can be constructed, as illustrated below:

s11 = 671
25 9 47 119 102 64 113 70 5 96 21
89 121 67 11 12 66 93 19 120 69 4
27 7 49 117 104 62 76 3 92 16 118
99 111 77 1 22 56 15 115 74 10 91
32 2 54 112 109 57 8 98 14 114 71
94 116 72 6 17 61 28 39 50 83 105
23 46 106 42 88 58 97 80 75 36 20
108 44 78 24 51 59 30 37 52 81 107
79 29 53 110 34 60 95 82 73 38 18
55 100 35 84 31 63 26 41 48 85 103
40 86 33 45 101 65 90 87 68 43 13

Subject Composed Magic Squares can be based on:

  • Suitable selected Latin Sub Squares, resulting in the two 5th order Pan Magic Sub Squares,
  • A guessing routine, based on the defining linear equations as deducted in Section 6.09.3, resulting in the two each other overlapping 6th order Compact Pan Magic Sub Squares.

The possible (unique) order 5 Magic Series for the integers 0 ... 10, as applicable for the Latin Sub Squares mentioned above, are shown in Attachment 25.4.1.

Attachment 25.4.2 shows the corresponding 11th order Magic Squares with 6th order Overlapping Compact Pan Magic Sub Squares (ref. Priem6b).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the aspects of the Sub Squares.

25.5 Magic Squares (13 x 13)

A classical Magic Square of order 13, with miscellaneous each other – asymmetrically - Overlapping Sub Squares has been described in Section 12.6.

Examples of such Magic Squares of order 13 are shown in Attachment 12.6.7.

Alternatively Magic Squares of order 13, with two 7th order Overlapping Sub Squares with identical Magic Sums can be constructed based on suitable selected Latin Sub Squares.

Magic Squares M of order 6 with the numbers 1 ... 36 can be written as M = A + 6 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4 and 5 as illustrated below for a Magic Square with Symmetrical Diagonals:

A
5 1 2 3 4 0
0 4 2 3 1 5
0 1 3 2 4 5
5 1 3 2 4 0
0 4 3 2 1 5
5 4 2 3 1 0
B = T(A)
5 0 0 5 0 5
1 4 1 1 4 4
2 2 3 3 3 2
3 3 2 2 2 3
4 1 4 4 1 1
0 5 5 0 5 0
M
36 2 3 34 5 31
7 29 9 10 26 30
13 14 22 21 23 18
24 20 16 15 17 19
25 11 28 27 8 12
6 35 33 4 32 1

For order 6 Magic Squares with Symmetrical Diagonals, the balanced series {0, 1, 2, 3, 4, 5} can be replaced by any balanced series {ai, i = 1 ... 6} and {bj, j = 1 ... 6}.

Pan Magic Squares M of order 7 with the numbers 1 ... 49 can be written as M = A + 7 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5 and 6 as illustrated below for a Pan Magic Square with the center element in the bottom/right corner.

A
4 2 6 5 3 1 0
6 5 3 1 0 4 2
3 1 0 4 2 6 5
0 4 2 6 5 3 1
2 6 5 3 1 0 4
5 3 1 0 4 2 6
1 0 4 2 6 5 3
B = T(A)
4 6 3 0 2 5 1
2 5 1 4 6 3 0
6 3 0 2 5 1 4
5 1 4 6 3 0 2
3 0 2 5 1 4 6
1 4 6 3 0 2 5
0 2 5 1 4 6 3
M
33 45 28 6 18 37 8
21 41 11 30 43 26 3
46 23 1 19 38 14 34
36 12 31 49 27 4 16
24 7 20 39 9 29 47
13 32 44 22 5 17 42
2 15 40 10 35 48 25

For Pan Magic Squares of order 7 the balanced series {0, 1, 2, 3, 4, 5, 6} can be replaced by any balanced series {ai, i = 1 ... 7} and {bj, j = 1 ... 7}.

Magic Square M of order 13 with the numbers 1 ... 169 can be written as M = A + 13 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4 ... 12.

The series {0, 1, 2, 3, 4 ... 12} can be split into two balanced sub series - with identical sum (36) - around the center element e.g.:

     {0, 1, 2, 10, 11, 12}, {6}, {3, 4, 5, 7, 8, 9}

which can be used for the construction of the two order 6 Sub Squares and - if combined with the common center element 6 - the two each other overlapping order 7 Sub Squares, as illustrated below:

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

The possible (unique) order 6 and 7 Balanced Series for the integers 0 ... 12, as described above, are shown in Attachment 25.5.1.

Attachment 25.5.2 shows a few 13th order Magic Squares with 7th order Overlapping Pan Magic Sub Squares.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the Latin Sub Squares.

25.6 Magic Squares (15 x 15)

A classical Magic Square of order 15, with miscellaneous each other Overlapping Sub Squares has been described in Section 11.2.2.

Examples of such Magic Squares of order 15 are shown in Attachment 14.9.8.2.

Alternatively Magic Squares of order 15, with two 8th order Overlapping Sub Squares with identical Magic Sums can be constructed based on suitable selected Latin Sub Squares as illustrated below.

Pan Magic Squares M of order 7 with the numbers 1 ... 49 can be written as M = A + 7 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5 and 6 as illustrated below.

A
0 5 3 1 6 4 2
1 6 4 2 0 5 3
2 0 5 3 1 6 4
3 1 6 4 2 0 5
4 2 0 5 3 1 6
5 3 1 6 4 2 0
6 4 2 0 5 3 1
B = T(A)
0 1 2 3 4 5 6
5 6 0 1 2 3 4
3 4 5 6 0 1 2
1 2 3 4 5 6 0
6 0 1 2 3 4 5
4 5 6 0 1 2 3
2 3 4 5 6 0 1
M
1 13 18 23 35 40 45
37 49 5 10 15 27 32
24 29 41 46 2 14 19
11 16 28 33 38 43 6
47 3 8 20 25 30 42
34 39 44 7 12 17 22
21 26 31 36 48 4 9

For Pan Magic Squares of order 7 the series {0, 1, 2, 3, 4, 5, 6} can be replaced by any series {ai, i = 1 ... 7} and {bj, j = 1 ... 7}.

(Pan) Magic Squares M of order 8 with the numbers 1 ... 64 can be written as M = A + 8 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6 and 7 as illustrated below for a Composed Magic Square:

A
6 4 3 1 6 4 3 1
1 3 4 6 1 3 4 6
4 6 1 3 4 6 1 3
3 1 6 4 3 1 6 4
2 0 7 5 2 0 7 5
5 7 0 2 5 7 0 2
0 2 5 7 0 2 5 7
7 5 2 0 7 5 2 0
B = T(A)
6 1 4 3 2 5 0 7
4 3 6 1 0 7 2 5
3 4 1 6 7 0 5 2
1 6 3 4 5 2 7 0
6 1 4 3 2 5 0 7
4 3 6 1 0 7 2 5
3 4 1 6 7 0 5 2
1 6 3 4 5 2 7 0
M
55 13 36 26 23 45 4 58
34 28 53 15 2 60 21 47
29 39 10 52 61 7 42 20
12 50 31 37 44 18 63 5
51 9 40 30 19 41 8 62
38 32 49 11 6 64 17 43
25 35 14 56 57 3 46 24
16 54 27 33 48 22 59 1

For Composed Magic Squares the series {0, 1, 2, 3, 4, 5, 6, 7} has to be split into two sub series with identical sum (14), in the example shown above {0, 2, 5, 7} and {1, 3, 4, 6}.

The series {0, 1, 2, 3, 4, 5, 6, 7} can be replaced by any series {ai, i = 1 ... 8} and {bj, j = 1 ... 8} which can be split into two sub series with identical sum.

Magic Square M of order 15 with the numbers 1 ... 225 can be written as M = A + 15 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4 ... 14.

The balanced series {0, 1, 2, 3, 4 ... 14} can be split into two sub series - with identical sum (49) - around the center element e.g.:

     {0, 1, 6, 9, 10, 11, 12}, {7}, {2, 3, 4, 5, 8, 13, 14}

which can be used for the construction of the two order 7 Sub Squares and - if combined with the common center element 7 - the two each other overlapping order 8 Sub Squares, as illustrated below:

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

The possible (unique) order 15 Balanced Series for the integers 0 ... 14, as described above, are shown in Attachment 25.6.1.

Attachment 25.6.2 shows a few 15th order Associated Magic Squares with 8th order Overlapping Sub Squares (Composed).

Attachment 25.6.3 shows a few 15th order Associated Magic Squares with 8th order Overlapping Sub Squares (Composed, Magic Middle and Center Squares).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the Latin Sub Squares.

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