Office Applications and Entertainment, Magic Squares

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27.0 Magic Squares, Higher Order, Composed Border (1)

27.1 Introduction

In previous sections Inlaid Magic Squares with Associated Borders have been discussed for miscellaneous orders.

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

27.2 Magic Squares (8 x 8)

Composed Magic Squares of order 8, as constructed in Section 22.1, can be transformed into Inlaid Magic Squares with a Composed Border as illustrated below:

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

As discussed in Section Section 22.1, both Magic Squares shown above correspond with 0,5 1012 Magic Squares.

27.3 Magic Squares (9 x 9)

Inlaid Magic Squares of order  9, with a Composed Border, have been described in Section 9.6.4.

27.4 Magic Squares (10 x 10)

Inlaid Magic Squares of order 10, with a Composed Border, based on the Medjig method of construction, have been described in Section 10.1.3.

Alternatively Composed Magic Squares, as deducted and discussed in Section 10.3.1, can be transformed into Inlaid Magic Squares with a Composed Border as illustrated below:

Composed
71 47 40 44 83 82 63 23 28 24
39 45 52 66 22 26 27 67 80 81
35 49 56 62 20 21 34 74 75 79
57 61 54 30 77 73 78 38 19 18
53 41 36 72 92 12 13 84 91 11
68 51 46 37 14 4 5 95 98 87
58 32 42 70 15 93 100 2 7 86
31 59 69 43 16 6 3 97 96 85
64 55 50 33 76 99 94 8 1 25
29 65 60 48 90 89 88 17 10 9
Inlaid
71 47 83 82 63 23 28 24 40 44
39 45 22 26 27 67 80 81 52 66
53 41 92 12 13 84 91 11 36 72
68 51 14 4 5 95 98 87 46 37
58 32 15 93 100 2 7 86 42 70
31 59 16 6 3 97 96 85 69 43
64 55 76 99 94 8 1 25 50 33
29 65 90 89 88 17 10 9 60 48
35 49 20 21 34 74 75 79 56 62
57 61 77 73 78 38 19 18 54 30

As the 10th order Composed Magic Square shown above (left), with Magic Sum s10 = 505, is composed out of:

  • One 4th order Associated Corner Square, Magic Sum s4 = 202 (top/left)
  • One 6th order Bordered Magic Corner Square, Magic Sum s6 = 303 (bottom/right)
  • Two order 4 x 6 Associated Magic Rectangles (s4 = 202 and s6 = 303)

the Composed Border of the resulting Inlaid Magic Square (right) is Associated.

27.5 Magic Squares (11 x 11)

Inlaid Magic Squares of order 11, with a Composed Border, have been described in Section 11.1.3.

27.6 Magic Squares (12 x 12)

Composed Magic Squares of order 12, as constructed in Section 22.2, can be transformed into Inlaid Magic Squares with a Composed Border as illustrated below:

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

As discussed in Section Section 22.2, both Magic Squares shown above correspond with 6,6 1028 Magic Squares.

Alternatively Composed Magic Squares of order 12 can be transformed into Inlaid Magic Squares with a Composed Border as illustrated below:

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

The Composed Center Square C of the (right) Inlaid Magic Square with Composed Border contains the consecutive integers ci = {41 ... 104} = 40 + {1 ... 64}.

Consequently C can be replaced by a Magic Square C = A + [40], with A any regular Magic Square and ai = {1 ... 64}, as illustrated below for one of Walter Trumps Bimagic Squares.

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

The Magic / Bimagic Sums of the order 8 Bimagic Center Square are 580 / 44780.

The number of order 12 Inlaid Magic Squares with order 8 Bimagic Center Square can be calculated as follows:

  • The number of possible borders = 5! * 3845 = 105
  • The Number of regular order 8 Bimagic Squares = 192 * 136244 = 26158848

The resulting number of order 12 Inlaid Magic Squares with order 8 Bimagic Center Square is consequently 2,62 1022.

27.7 Magic Squares (16 x 16)

Comparable transformations can be applied on Composed Magic Squares of order 16 (ref. Section 22.3), as illustrated below:

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

As discussed in Section Section 22.3, the Magic Square shown above correspond with 4,7 1054 Magic Squares.

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

The Composed Center Square C of the Inlaid Magic Square shown above contains the consecutive integers ci = {57 ... 200} = 56 + {1 ... 144}.

Consequently C can be replaced by a Magic Square C = A + [56], with A any regular Magic Square and ai = {1 ... 144}, as illustrated below for one of Walter Trumps Bimagic Squares.

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

The Magic / Bimagic Sums of the order 12 Bimagic Center Square are 1542 / 218882.

The number of possible borders for the Inlaid Magic Squares shown above is 7! * 3847 = 6,2 1021.

Comparable 16 x 16 Inlaid Magic Squares can be constructed based on:

  • the Composed Border as applied above and
  • the order 6 Sub Squares of the Composed Magic Squares as discussed in Section 22.5

of which a few examples are shown in Attachment 26.5.

27.8 Magic Squares (20 x 20)

Comparable transformations can be applied on Composed Magic Squares of order 20 (ref. Section 22.4), as illustrated below:

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

As discussed in Section Section 22.4, the Magic Square shown above correspond with 6,3 1089 Magic Squares.

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