Office Applications and Entertaiment, Magic Squares

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22.0 Magic Squares, Higher Order, Composed

22.5 Introduction, Misc. Sub Squares

In Section 9.8 a set of 9 Magic Squares of the 3th order was found, each containing 9 consecutive integers, with corresponding Magic Sum.

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

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

22.6 Magic Squares (12 x 12)

For 12th 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 Ci (i = 1 ... 16), the resulting number of Magic Squares of the 12th order with Magic Sum 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 12th order with Magic Sum 870 will be:

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

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

22.7 Magic Squares (15 x 15)

For 15th 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 Ci (i = 1 ... 25), and 28800 possible squares for Pan Magic Square B, the resulting number of Magic Squares of the 15th order with Magic Sum 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 Ci (i = 1 ... 9) the resulting number of Magic Squares of the 15th order with Magic Sum 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.

22.8 Magic Squares (16 x 16)

For 16th 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 16th order with Magic Sum 2056 can be determined for following 4 Cases:

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

In the first Case - B and Ci Pan Magic as in the example above - the resulting square C will be Pan Magic.

If, on the other hand, B and Ci are Associated, the resulting square C will be Associated as well.

22.9 Magic Squares (18 x 18)

For 18th 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 Ci (i = 1 ... 36), and 1.740.800 possible squares (Medjig Solutions) for Magic Square B the resulting number of Magic Squares of the 18th order with Magic Sum 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 Ci (i = 1 ... 9) the resulting number of Magic Squares of the 18th order with Magic Sum 2925 will be 8 * 1.740.8009 = 1,17 1057.

22.10 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:

Main Square
Order

Square B

Square Ci

Attachment

Order

Type

Order

i

Type

15 x 15

5 x 5
3 x 3

Misc. Inalys
Simple

3 x 3
5 x 5

1 ... 25
1 ... 9

Simple
Misc. Inalys

Att 18.4.01 Sht. 1
Att 18.4.01 Sht. 2

20 x 20

5 x 5
4 x 4

Misc. Inalys
Pan Magic

4 x 4
5 x 5

1 ... 25
1 ... 16

Pan Magic
Misc. Inalys

Att 18.4.02 Sht. 1
Att 18.4.02 Sht. 2

21 x 21

7 x 7
3 x 3

Misc. Inalys
Simple

3 x 3
7 x 7

1 ... 49
1 ... 9

Simple
Misc. Inalys

Att 18.4.03 Sht. 1
Att 18.4.03 Sht. 2

24 x 24

6 x 6
4 x 4

Medjig, Conc.
Pan Magic

4 x 4
6 x 6

1 ... 36
1 ... 16

Pan Magic
Medjig, Conc.

Att 18.4.04 Sht. 1
Att 18.4.04 Sht. 2

25 x 25

5 x 5

Misc. Inalys

5 x 5

1 ... 25

Misc. Inalys

Att 18.4.05

Each composed square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the inlays.


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