Office Applications and Entertaiment, Magic Squares

Vorige Pagina Volgende Pagina 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 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 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 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 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 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 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 4010.


Vorige Pagina Volgende Pagina Index About the Author