Office Applications and Entertainment, Magic Squares

24.0 Magic Squares, Higher Order, Composed

24.1 Introduction, Misc. Sub Squares (2)

In Section 9.9.2 Magic Squares of the 9^th order could be constructed based on a set of 9 Magic Squares of the 3^th order, each containing 9 non-consecutive integers, with corresponding Magic Sum.

Next sections show comparable sets of (Pan) Magic Squares, enabling the construction of 12^th, 15^th, 16^th, 18^th and a few higher order Magic Squares.

24.2 Magic Squares (12 x 12)

For 12^th order Magic Squares, following set of 16 Magic Squares - each containing 9 non-consecutive integers - with corresponding Magic Sum, can be found:

A1 4 9 2 3 5 7 8 1 6

A2 48 128 16 32 64 96 112 0 80

B 12 13 3 6 7 2 16 9 14 11 5 4 1 8 10 15

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

MC's 228 231 201 210 213 198 240 219 234 225 207 204 195 216 222 237

With 8 possible squares for each square C_i (i = 1 ... 16), the resulting number of Magic Squares of the 12^th order with Magic Sum s12 = 870 will be:

     either 384 * 8¹⁶ = 1,08 10¹⁷ for Pan    Magic Square B;
     or    7040 * 8¹⁶ = 1,98 10¹⁸ for Simple Magic Square B.

It can be noticed that if B is Associated, the resulting square C will be Associated as well.

Alternatively, following set of 9 Magic Squares - each containing 16 non-consecutive integers - with corresponding Magic Sum, can be found:

A1 12 13 3 6 7 2 16 9 14 11 5 4 1 8 10 15

A2 99 108 18 45 54 9 135 72 117 90 36 27 0 63 81 126

B 4 9 2 3 5 7 8 1 6

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

MC's 286 306 278 282 290 298 302 274 294

With 8 possible squares for square B, the resulting number of Magic Squares of the 12^th order with Magic Sum s12 = 870 will be:

     either 8 *  384⁹ = 1,45 10²⁴ for Pan    Magic Squares C_i (i = 1 ... 9);
     or     8 * 7040⁹ = 3,40 10³⁵ for Simple Magic Squares C_i (i = 1 ... 9).

It can be noticed that if C_i is Associated, the resulting square C will be Associated as well.

24.3 Magic Squares (15 x 15)

For 15^th order Magic Squares, following set of 25 Magic Squares - each containing 9 non-consecutive integers - with corresponding Magic Sum, can be found:

B 12 6 5 24 18 4 23 17 11 10 16 15 9 3 22 8 2 21 20 14 25 19 13 7 1

A1 4 9 2 3 5 7 8 1 6

A2 75 200 25 50 100 150 175 0 125

MC's 336 318 315 372 354 312 369 351 333 330 348 345 327 309 366 324 306 363 360 342 375 357 339 321 303

C

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

With 8 possible squares for each square C_i (i = 1 ... 25), and 28800 possible squares for Pan Magic Square B, the resulting number of Magic Squares of the 15^th order with Magic Sum s15 = 1695 will be 28800 * 8²⁵ = 1,09 10²⁷.

Att 24.6.01 Sht. 1, provides some additional examples of order 15 Magic Squares, composed of 25 order 3 Sub Squares for miscellaneous types Square B. For enumeration base reference is made to Section 5.8.

Alternatively, following set of 9 Magic Squares - each containing 25 non-consecutive integers - with corresponding Magic Sum, can be found:

B 4 9 2 3 5 7 8 1 6

A1 12 6 5 24 18 4 23 17 11 10 16 15 9 3 22 8 2 21 20 14 25 19 13 7 1

A2 99 45 36 207 153 27 198 144 90 81 135 126 72 18 189 63 9 180 171 117 216 162 108 54 0

MC's 560 585 550 555 565 575 580 545 570

C

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

With 8 possible squares for square B and 28800 possible squares for each Pan Magic Squares C_i (i = 1 ... 9) the resulting number of Magic Squares of the 15^th order with Magic Sum s15 = 1695 will be 8 * 28800⁹ = 1,09 10⁴¹.

Att 24.6.01 Sht. 2, provides some additional examples of order 15 Magic Squares, composed of 9 order 5 Sub Squares for miscellaneous types Square C. For enumeration base reference is made to Section 5.8.

24.4 Magic Squares (16 x 16)

For 16^th order Magic Squares, following set of 16 (Pan) Magic Squares - each containing 16 non-consecutive integers - with corresponding Magic Sum, can be found:

A1 5 4 14 11 10 15 1 8 3 6 12 13 16 9 7 2

A2 64 48 208 160 144 224 0 112 32 80 176 192 240 128 96 16

B 12 13 3 6 7 2 16 9 14 11 5 4 1 8 10 15

MC's 528 532 492 504 508 488 544 516 536 524 500 496 484 512 520 540

C

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

The resulting number of Magic Squares of the 16^th order with Magic Sum s16 = 2056 can be determined for following 4 Cases:

     Square B Pan    Magic, Squares C_i (i = 1 ... 16) Pan    Magic:  384 *  384¹⁶ = 8,58 10⁴³
     Square B Simple Magic, Squares C_i (i = 1 ... 16) Pan    Magic: 7040 *  384¹⁶ = 1,57 10⁴⁵
     Square B Pan    Magic, Squares C_i (i = 1 ... 16) Simple Magic:  384 * 7040¹⁶ = 1,40 10⁶⁴
     Square B Simple Magic, Squares C_i (i = 1 ... 16) Simple Magic: 7040 * 7040¹⁶ = 2,56 10⁶⁵

If B and C_i are Pan Magic, the resulting square C will be Pan Magic as well.

If B and C_i are Associated, the resulting square C will be Associated as well.

24.5 Magic Squares (18 x 18)

For 18^th order Magic Squares, following set of 36 Magic Squares - each containing 9 non-consecutive integers - with corresponding Magic Sum, can be found:

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

A1 4 9 2 3 5 7 8 1 6

A2 108 288 36 72 144 216 252 0 180

MC's 510 537 435 489 450 504 483 456 516 462 531 477 522 468 474 501 507 453 441 495 447 528 534 480 525 498 513 459 438 492 444 471 540 486 465 519

C

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

With 8 possible squares for each square C_i (i = 1 ... 36), and 1.740.800 possible squares (Medjig Solutions) for Magic Square B the resulting number of Magic Squares of the 18^th order with Magic Sum s18 = 2925 will be 1.740.800 * 8³⁶ = 6,58 10²⁸.

Alternatively, following set of 9 Magic Squares - each containing 36 non-consecutive integers - with corresponding Magic Sum, can be found:

B 4 9 2 3 5 7 8 1 6

A1 26 35 1 19 6 24 17 8 28 10 33 15 30 12 14 23 25 7 3 21 5 32 34 16 31 22 27 9 2 20 4 13 36 18 11 29

A2 225 306 0 162 45 207 144 63 243 81 288 126 261 99 117 198 216 54 18 180 36 279 297 135 270 189 234 72 9 171 27 108 315 153 90 252

MC's 969 999 957 963 975 987 993 951 981

C

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

With 8 possible squares for square B and 1.740.800 possible squares (Medjig Solutions) for each Magic Squares C_i (i = 1 ... 9) the resulting number of Magic Squares of the 18^th order with Magic Sum s18 = 2925 will be 8 * 1.740.800⁹ = 1,17 10⁵⁷.

24.6 Magic Squares, Misc. Orders

Magic Squares composed out of Sub Squares with different Magic Sums are also referred to as Inlaid Magic Squares.

A few more examples of miscellaneous types of Composed Magic Squares are summarized in following table:

Each composed square shown corresponds with numerous solutions, which can be obtained by selecting other possible solutions for square B or C_i.