Office Applications and Entertainment, Magic Squares

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22.0  Magic Squares, Higher Order, Inlaid

22.11 Introduction

In previous sections Composed Simple Inlaid Magic Squares have been discussed for miscellaneous orders.

Following sections will show some examples of Composed Inlaid Magic Squares with Associated Borders.

22.12 Magic Squares (8 x 8)

Inlaid Magic Squares of order 8, with four 3th order Embedded Magic Squares with different Magic Sums and an Associated Border, have been described in Section 8.8.5.

Examples of such Inlaid Magic Squares of order 8 are shown in Attachment 8.6.14.

22.13 Magic Squares (9 x 9)

Inlaid Magic Squares of order 9, with four Embedded Magic Squares with different Magic Sums and an Associated Border, have been described in Section 9.7.6.

Examples of such Inlaid Magic Squares of order 9 are shown in Attachment 9.7.7.

Alternatively, the Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on resp. order 3 and 4 Magic Lines for the integers 0 ... 8 as shown in Attachment 18.6.1.

Attachment 18.6.2 shows a few 9th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4), however with two each 3th order Semi Magic Center Squares (6 magic lines).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border

22.14 Magic Squares (11 x 11)

Inlaid Magic Squares of order 11, with Embedded Magic Squares of order 4 and 5 (overlapping) with different Magic Sums and an Associated Border, have been described in Section 11.3.2.

Examples of such Inlaid Magic Squares of order 11 are shown in Attachment 14.9.9b.

22.15 Magic Squares (12 x 12)

The 12th order Inlaid Magic Square shown below, is composed out of an Associated Border and four each 5th order Pan Magic Center Squares with Magic Sums s(1) = 291, s(2) = 423, s(3) = 302, and s(4) = 434.

1 139 140 141 143 135 3 11 9 8 7 133
84 14 40 53 66 118 122 100 89 78 34 72
96 65 114 22 38 52 77 30 130 98 88 60
108 46 50 64 113 18 106 86 76 29 126 48
132 112 17 42 58 62 28 125 102 94 74 24
36 54 70 110 16 41 90 82 26 124 101 120
25 23 45 56 67 111 131 105 92 79 27 109
121 68 115 15 47 57 80 31 123 107 93 13
97 39 59 69 116 19 99 95 81 32 127 37
85 117 20 43 51 71 33 128 103 87 83 49
73 55 63 119 21 44 91 75 35 129 104 61
12 138 137 136 134 142 10 2 4 5 6 144
291 423
302 434

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

s(1) = 5 * s12 / 6 - s(4)
s(2) = 5 * s12 / 6 - s(3)

With s12 = 870 the Magic Sum of the 12th order Inlaid Magic Square.

The 5th order Pan Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on 5th order Magic Lines for the integers 0 ... 11 as shown in Attachment 18.3.1.

Attachment 18.3.2 shows a few 12th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border

22.16 Magic Squares (16 x 16)

The 16th order Inlaid Magic Square shown below, is composed out of an Associated Border and four each 7th order Ultra Magic Center Squares with Magic Sums s(1) = 952, s(2) = 840, s(3) = 959 and s(4) = 847.

69 80 68 71 73 75 78 66
252 17 175 99 246 216 138 61
60 250 221 129 63 19 166 104
108 54 24 170 109 241 223 131
140 111 243 214 136 58 29 161
172 141 49 31 163 102 248 218
220 168 106 253 209 143 51 22
28 211 134 56 26 173 97 255
178 190 187 185 183 180 192 181
225 95 147 6 40 122 205 12
10 45 113 207 227 86 152 204
198 232 90 157 1 47 115 156
159 3 38 120 202 237 81 124
125 193 239 83 150 8 42 92
88 154 13 33 127 195 230 44
35 118 200 234 93 145 15 236
21 32 162 110 251 217 135 52
213 247 212 144 50 30 171 105
165 59 25 167 100 256 210 142
133 98 254 219 137 55 20 176
101 132 64 18 174 107 249 215
53 169 103 244 224 130 62 27
245 222 139 57 23 164 112 242
76 65 77 74 72 70 67 79
240 82 158 11 41 119 196 229
7 36 128 194 238 91 153 37
203 233 87 148 16 34 126 85
146 14 43 121 199 228 96 117
116 208 226 94 155 9 39 149
89 151 4 48 114 206 235 197
46 123 201 231 84 160 2 5
191 179 182 184 186 189 177 188
952 840
959 847

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

s(1) = 7 * s16 / 8 - s(4)
s(2) = 7 * s16 / 8 - s(3)

With s16 = 2056 the Magic Sum of the 16th order Inlaid Magic Square.

Attachment 18.4.2 shows a few 16th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4), however with Pan Magic Center Squares (Non Symmetric).

The 7th order Pan Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on 7th order Magic Lines for the integers 0 ... 15 as shown in Attachment 18.4.1.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.

22.17 Magic Squares (20 x 20)

The 20th order Inlaid Magic Square shown below, originally published by John Hendricks, is composed out of an Associated Border and four each 9th order Pan Magic Center Squares with Magic Sums s(1), s(2), s(3) and s(4).

295 283 292 288 291 294 282 296 284 300
46 270 37 313 65 389 59 238 147 201
226 393 45 229 159 218 267 21 310 77
146 209 279 38 307 61 390 57 233 145
206 78 387 41 230 157 213 265 29 319
266 141 210 277 33 305 69 399 58 227
26 317 73 385 49 239 158 207 261 30
306 225 149 219 278 27 301 70 397 53
66 39 318 67 381 50 237 153 205 269
386 47 221 150 217 273 25 309 79 398
120 104 116 102 114 111 108 112 103 115
130 377 93 325 9 359 178 247 181 346
13 345 169 259 198 127 361 90 337 166
189 139 378 87 321 10 357 173 245 246
338 7 341 170 257 193 125 369 99 186
241 190 137 373 85 329 19 358 167 126
97 333 5 349 179 258 187 121 370 366
165 249 199 138 367 81 330 17 353 86
379 98 327 1 350 177 253 185 129 326
347 161 250 197 133 365 89 339 18 6
395 271 24 308 76 392 42 223 154 220
75 388 56 232 142 203 274 40 311 64
315 212 262 23 314 80 391 44 228 156
35 63 394 60 231 144 208 276 32 302
275 160 211 264 28 316 72 382 43 234
215 304 68 396 52 222 143 214 280 31
155 236 152 202 263 34 320 71 384 48
235 22 303 74 400 51 224 148 216 272
55 54 240 151 204 268 36 312 62 383
286 298 289 293 290 287 299 285 297 281
131 364 88 336 12 342 163 254 200 15
8 356 172 242 183 134 380 91 324 335
192 122 363 94 340 11 344 168 256 95
323 14 360 171 244 188 136 372 82 375
260 191 124 368 96 332 2 343 174 135
84 328 16 352 162 243 194 140 371 195
176 252 182 123 374 100 331 4 348 255
362 83 334 20 351 164 248 196 132 175
354 180 251 184 128 376 92 322 3 355
101 117 105 119 107 110 113 109 118 106
1719 1899
1710 1890

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

s(1) = 9 * s20 / 10 - s(4)
s(2) = 9 * s20 / 10 - s(3)

With s20 = 4010 the Magic Sum of the 20th order Inlaid Magic Square.

Attachment 18.5.2 shows a few 20th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4).

Attachment 18.5.3 shows a few more 20th order Inlaid Magic Squares for miscellaneous possible Magic Sums s(3) and s(4), however with Simple Magic Center Squares for which one third of the diagonals sum to the Magic Sum.

The 9th order (Pan) Magic Center Squares can be constructed by means of suitable selected Latin Squares as illustrated in Attachment 18.3.3, based on 9th order Magic Lines for the integers 0 ... 19 as shown in Attachment 18.5.1.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.


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