Office Applications and Entertainment, Magic Squares

23.0 Magic Squares, Higher Order, Composed

23.1 Introduction, Misc. Sub Squares (1)

In Section 9.9.1 a set of 9 Magic Squares of the 3^th order was found, each containing 9 consecutive integers, with corresponding Magic Sum.

Based on this set of 3^th order Magic Squares, Magic Squares of the 9^th order could be constructed.

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.

23.2 Magic Squares (12 x 12)

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

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

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

MC's 312 339 69 150 177 42 420 231 366 285 123 96 15 204 258 393

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 consecutive integers - with corresponding Magic Sum, can be found:

B 4 9 2 3 5 7 8 1 6

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

MC's 226 546 98 162 290 418 482 34 354

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.

23.3 Magic Squares (15 x 15)

For 15^th order Magic Squares, following set of 25 Magic Squares - each containing 9 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	MC's 312 150 123 636 474 96 609 447 285 258 420 393 231 69 582 204 42 555 528 366 663 501 339 177 15
C

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

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 18.4.01 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 consecutive integers - with corresponding Magic Sum, can be found:

B 4 9 2 3 5 7 8 1 6	MC's 440 1065 190 315 565 815 940 65 690
C

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

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 18.4.01 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.

23.4 Magic Squares (16 x 16)

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

B 12 13 3 6 7 2 16 9 14 11 5 4 1 8 10 15	MC's 738 802 162 354 418 98 994 546 866 674 290 226 34 482 610 930
C

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

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.

23.5 Magic Squares (18 x 18)

For 18^th order Magic Squares, following set of 36 Magic Squares - each containing 9 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	MC's 690 933 15 501 150 636 447 204 744 258 879 393 798 312 366 609 663 177 69 555 123 852 906 420 825 582 717 231 42 528 96 339 960 474 285 771
C

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

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 consecutive integers - with corresponding Magic Sum, can be found:

B 4 9 2 3 5 7 8 1 6	MC's 759 1839 327 543 975 1407 1623 111 1191
C

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

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.

23.6 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.