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22.0 Magic Squares, Higher Order, Composed

22.5 Introduction, 6 x 6 Sub Squares

Comparable with previous section for 4 x 4 Magic Squares, higher order Magic Squares can be composed out of 6 x 6 Magic Squares.

The relation between the integers of the 6 x 6 Magic Sub Squares is however not so elegant as found for 4 x 4 Pan Magic Squares (ref. Section 22.1).

22.6 Magic Squares (12 x 12)
Row Symmetric Sub Squares

In Section 6.11.3 a procedures was developed to generate 6th order Row Symmetric Magic Squares with magic sum s6 = 111.

With some minor modifications subject procedure can be used to find, within the integer range 1 ... 144, a set of 4 Magic Squares - each containing 36 different integers and with magic sum s6 = 435 - as shown below:

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

Other ranges might be possible.

The 4 squares can be arranged in 4! ways, resulting in 4! * n64 Magic Squares of the 12th order with magic sum s12 = 870 (n6 = number of possible Row Symmetric Sub Squares).

22.7 Magic Squares (12 x 12)
Concentric Sub Squares

The four ranges of non consecutive integers found in previous section can be used to generate Concentric Magic Squares - with Pan Magic Center Squares - as shown below:

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

The square shown above corresponds with at least (4!) * (384 * ((8 * (4!)2)4 = 2,35 1026 order 12 Composed Magic Squares (s12 = 870).

22.8 Magic Squares (12 x 12)
Sub Squares with Symmetrical Diagonals

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

Magic Square M of order 12 with the numbers 1 ... 144 can be written as M = A + 12 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4 ... 11.

The balanced series {0, 1, 2, 3, 4 ... 11} can be split into two balanced sub series e.g.

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

which can be used for the construction of four Magic Sub Squares with Symmetric Diagonals, as illustrated below:

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

Attachment 22.8.1 shows the unique sets (8 ea) of order 6 balanced lines for the integers 0 ... 11.

Attachment 22.8.2 shows the resulting order 12 Magic Squares composed of Magic Sub Squares with Symmetrical Diagonals.

Each square shown corresponds with at least (4!) * 7684 = 8,35 1012 order 12 Composed Magic Squares (s12 = 870).

22.9 Magic Squares (18 x 18)
Sub Squares with Symmetrical Diagonals

Magic Square M of order 18 with the numbers 1 ... 324 can be written as M = A + 18 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4 ... 17.

The balanced series {0, 1, 2, 3, 4 ... 17} can be split into three balanced sub series e.g.

{0, 1, 2, 15, 16, 17}, {3, 4, 5, 12, 13, 14} and {6, 7, 8, 9, 10, 11}

which can be used for the construction of nine Magic Sub Squares with Symmetric Diagonals, as illustrated below:

 18 308 3 322 323 1 19 35 291 304 20 306 271 38 52 39 287 288 54 272 286 273 53 37 289 305 34 21 290 36 324 17 309 16 2 307
 15 311 6 319 320 4 22 32 294 301 23 303 274 41 49 42 284 285 51 275 283 276 50 40 292 302 31 24 293 33 321 14 312 13 5 310
 12 314 9 316 317 7 25 29 297 298 26 300 277 44 46 45 281 282 48 278 280 279 47 43 295 299 28 27 296 30 318 11 315 10 8 313
 72 254 57 268 269 55 73 89 237 250 74 252 217 92 106 93 233 234 108 218 232 219 107 91 235 251 88 75 236 90 270 71 255 70 56 253
 69 257 60 265 266 58 76 86 240 247 77 249 220 95 103 96 230 231 105 221 229 222 104 94 238 248 85 78 239 87 267 68 258 67 59 256
 66 260 63 262 263 61 79 83 243 244 80 246 223 98 100 99 227 228 102 224 226 225 101 97 241 245 82 81 242 84 264 65 261 64 62 259
 126 200 111 214 215 109 127 143 183 196 128 198 163 146 160 147 179 180 162 164 178 165 161 145 181 197 142 129 182 144 216 125 201 124 110 199
 123 203 114 211 212 112 130 140 186 193 131 195 166 149 157 150 176 177 159 167 175 168 158 148 184 194 139 132 185 141 213 122 204 121 113 202
 120 206 117 208 209 115 133 137 189 190 134 192 169 152 154 153 173 174 156 170 172 171 155 151 187 191 136 135 188 138 210 119 207 118 116 205

Attachment 22.9.1 shows the unique sets (560 ea) of order 6 balanced lines for the integers 0 ... 17.

Attachment 22.9.2 shows a few of the resulting order 18 Magic Squares composed of nine Magic Sub Squares with Symmetrical Diagonals.

Each square shown corresponds with at least (9!) * 7689 = 3,37 1031 order 18 Composed Magic Squares (s18 = 2925).