Office Applications and Entertainment, Magic Squares Index About the Author

22.0 Magic Squares, Higher Order, Composed

22.1 Introduction, 4 x 4 Sub Squares

In Section 8.9 a set of 4 Pan Magic Squares of the 4th order was found, each containing 16 different integers,
with magic sum s4 = 130:

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

Based on this set of Pan Magic Squares of the 4th order, Magic Squares of the 8th order could be constructed.

The relation between the numbers of these Pan Magic Squares is as follows:

A
 4 5 n2-5 n2-2 n2-7 n2 2 7 6 3 n2-3 n2-4 n2-1 n2-6 8 1
B
 a1+8 a2+8 a3-8 a4-8 a5-8 a6-8 a7+8 a8+8 a9+8 a10+8 a11-8 a12-8 a13-8 a14-8 a15+8 a16+8
C
 b1+8 b2+8 b3-8 b4-8 b5-8 b6-8 b7+8 b8+8 b9+8 b10+8 b11-8 b12-8 b13-8 b14-8 b15+8 b16+8
D
 c1+8 c2+8 c3-8 c4-8 c5-8 c6-8 c7+8 c8+8 c9+8 c10+8 c11-8 c12-8 c13-8 c14-8 c15+8 c16+8

Based on abovementioned relations, it can be proven that each square contains 16 different integers:

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

The last square contains 16 consecutive distinct integers.

For all Magic Squares of even order, composed out of Pan Magic Squares of the 4th order, comparable relations
can be found and summarized as follows:

 Main Square Sub Square Order 4 Total Order n Sum Sn Quantity Sum S4 Permutations Quantity 4 34 1 34 1 384 8 260 4 130 4! 0,5 1012 12 870 9 290 9! 6,6 1028 16 2056 16 514 16! 4,7 1054 20 4010 25 802 25! 6,3 1089 ... ... ... ... ... ... n n(n2+1)/2 (n/4)2 4*Sn/n (n/4)2! (n/4)2! 384(n/4)2

Next sections show sets of Pan Magic Squares of the 4th order, enabling the construction of 12th, 16th and 20th
order Magic Squares.

22.2 Magic Squares (12 x 12)

For 12th order Magic squares, following set of 9 Pan Magic Squares - each containing 16 different integers -
with magic sum s4 = 290 can be found:

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

These 9 squares can be arranged in 9! ways, resulting in 9! * 3849 = 6,6 1028 Magic Squares of the 12th order
with magic sum s12 = 870.

22.3 Magic Squares (16 x 16)

For 16th order Magic squares, following set of 16 Pan Magic Squares - each containing 16 different integers -
with magic sum s4 = 514 can be found:

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

These 16 squares can be arranged in 16! ways, resulting in 16! * 38416 = 4,7 1054 Magic Squares of the 16th order
with magic sum s16 = 2056.

22.4 Magic Squares (20 x 20)

For 20th order Magic squares, following set of 25 Pan Magic Squares - each containing 16 different integers -
with magic sum s4 = 802 can be found:

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

These 25 squares can be arranged in 25! ways, resulting in 25! * 38425 = 6,3 1089 Magic Squares of the 20th order
with magic sum s20 = 4010.