Office Applications and Entertainment, Magic Squares

Vorige Pagina Volgende Pagina Index About the Author

25.0 Magic Squares, Higher Order, Overlapping Sub Squares

25.7 Magic Squares, Associated (17 x 17)

Magic Squares of order 17, with two 9th order Overlapping Sub Squares with identical Magic Sums can be constructed based on suitable selected Latin Sub Squares.

Simple Magic Squares M of order 9 with the numbers 1 ... 81 can be written as M = A + 9 * B + [1] where the squares A and B contain only the integers 0, 1, 2 ... 8 as illustrated below for a Simple Magic Square with the center element in the bottom/right corner.

A
6 1 5 4 8 0 2 3 7
4 8 0 2 3 7 6 1 5
2 3 7 6 1 5 4 8 0
5 6 1 0 4 8 7 2 3
0 4 8 7 2 3 5 6 1
7 2 3 5 6 1 0 4 8
1 5 6 8 0 4 3 7 2
8 0 4 3 7 2 1 5 6
3 7 2 1 5 6 8 0 4
B = T(A)
6 4 2 5 0 7 1 8 3
1 8 3 6 4 2 5 0 7
5 0 7 1 8 3 6 4 2
4 2 6 0 7 5 8 3 1
8 3 1 4 2 6 0 7 5
0 7 5 8 3 1 4 2 6
2 6 4 7 5 0 3 1 8
3 1 8 2 6 4 7 5 0
7 5 0 3 1 8 2 6 4
M
61 38 24 50 9 64 12 76 35
14 81 28 57 40 26 52 2 69
48 4 71 16 74 33 59 45 19
42 25 56 1 68 54 80 30 13
73 32 18 44 21 58 6 70 47
8 66 49 78 34 11 37 23 63
20 60 43 72 46 5 31 17 75
36 10 77 22 62 39 65 51 7
67 53 3 29 15 79 27 55 41

For Simple Magic Squares of order 9 the series {0, 1, 2, ... 8} can be replaced by any series {ai, i = 1 ... 9} and {bj, j = 1 ... 9}.

(Pan) Magic Squares M of order 8 with the numbers 1 ... 64 can be written as M = A + 8 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6 and 7 as illustrated below for a Composed Magic Square:

A
6 4 3 1 6 4 3 1
1 3 4 6 1 3 4 6
4 6 1 3 4 6 1 3
3 1 6 4 3 1 6 4
2 0 7 5 2 0 7 5
5 7 0 2 5 7 0 2
0 2 5 7 0 2 5 7
7 5 2 0 7 5 2 0
B = T(A)
6 1 4 3 2 5 0 7
4 3 6 1 0 7 2 5
3 4 1 6 7 0 5 2
1 6 3 4 5 2 7 0
6 1 4 3 2 5 0 7
4 3 6 1 0 7 2 5
3 4 1 6 7 0 5 2
1 6 3 4 5 2 7 0
M
55 13 36 26 23 45 4 58
34 28 53 15 2 60 21 47
29 39 10 52 61 7 42 20
12 50 31 37 44 18 63 5
51 9 40 30 19 41 8 62
38 32 49 11 6 64 17 43
25 35 14 56 57 3 46 24
16 54 27 33 48 22 59 1

For Composed Magic Squares the series {0, 1, 2, 3, 4, 5, 6, 7} has to be split into two sub series with identical sum (14), in the example shown above {0, 2, 5, 7} and {1, 3, 4, 6}.

The series {0, 1, 2, 3, 4, 5, 6, 7} can be replaced by any series {ai, i = 1 ... 8} and {bj, j = 1 ... 8} which can be split into two sub series with identical sum.

Magic Square M of order 17 with the numbers 1 ... 289 can be written as M = A + 17 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4 ... 16.

The balanced series {0, 1, 2, 3, 4 ... 16} can be split into two sub series - with identical sum (64) - around the center element e.g.:

     {0, 1, 5, 9, 10, 12, 13, 14}, {8}, {2, 3, 4, 6, 7, 11, 15, 16}

which can be used for the construction of the two order 8 Sub Squares and - if combined with the common center element 8 - the two each other overlapping order 9 Sub Squares, as illustrated below:

A
10 0 9 8 13 14 1 5 12 4 2 15 11 4 2 15 11
8 13 14 1 5 12 10 0 9 11 15 2 4 11 15 2 4
1 5 12 10 0 9 8 13 14 2 4 11 15 2 4 11 15
9 10 0 14 8 13 12 1 5 15 11 4 2 15 11 4 2
14 8 13 12 1 5 9 10 0 6 3 16 7 6 3 16 7
12 1 5 9 10 0 14 8 13 7 16 3 6 7 16 3 6
0 9 10 13 14 8 5 12 1 3 6 7 16 3 6 7 16
13 14 8 5 12 1 0 9 10 16 7 6 3 16 7 6 3
5 12 1 0 9 10 13 14 8 2 3 6 7 16 15 4 11
13 10 9 0 13 10 9 0 6 7 16 15 4 11 8 2 3
0 9 10 13 0 9 10 13 15 4 11 8 2 3 6 7 16
10 13 0 9 10 13 0 9 3 8 2 16 6 7 11 15 4
9 0 13 10 9 0 13 10 16 6 7 11 15 4 3 8 2
14 12 5 1 14 12 5 1 11 15 4 3 8 2 16 6 7
1 5 12 14 1 5 12 14 2 3 8 7 16 6 4 11 15
12 14 1 5 12 14 1 5 7 16 6 4 11 15 2 3 8
5 1 14 12 5 1 14 12 4 11 15 2 3 8 7 16 6
B = T(A)
10 8 1 9 14 12 0 13 5 13 0 10 9 14 1 12 5
0 13 5 10 8 1 9 14 12 10 9 13 0 12 5 14 1
9 14 12 0 13 5 10 8 1 9 10 0 13 5 12 1 14
8 1 10 14 12 9 13 5 0 0 13 9 10 1 14 5 12
13 5 0 8 1 10 14 12 9 13 0 10 9 14 1 12 5
14 12 9 13 5 0 8 1 10 10 9 13 0 12 5 14 1
1 10 8 12 9 14 5 0 13 9 10 0 13 5 12 1 14
5 0 13 1 10 8 12 9 14 0 13 9 10 1 14 5 12
12 9 14 5 0 13 1 10 8 6 15 3 16 11 2 7 4
4 11 2 15 6 7 3 16 2 7 4 8 6 15 3 16 11
2 15 4 11 3 16 6 7 3 16 11 2 7 4 8 6 15
15 2 11 4 16 3 7 6 6 15 8 16 11 3 7 4 2
11 4 15 2 7 6 16 3 7 4 2 6 15 8 16 11 3
4 11 2 15 6 7 3 16 16 11 3 7 4 2 6 15 8
2 15 4 11 3 16 6 7 15 8 6 11 3 16 4 2 7
15 2 11 4 16 3 7 6 4 2 7 15 8 6 11 3 16
11 4 15 2 7 6 16 3 11 3 16 4 2 7 15 8 6
M = A + 17 * B + [1]
181 137 27 162 252 219 2 227 98 226 3 186 165 243 20 220 97
9 235 100 172 142 30 164 239 214 182 169 224 5 216 101 241 22
155 244 217 11 222 95 179 150 32 156 175 12 237 88 209 29 254
146 28 171 253 213 167 234 87 6 16 233 158 173 33 250 90 207
236 94 14 149 19 176 248 215 154 228 4 187 161 245 21 221 93
251 206 159 231 96 1 151 26 184 178 170 225 7 212 102 242 24
18 180 147 218 168 247 91 13 223 157 177 8 238 89 211 25 255
99 15 230 23 183 138 205 163 249 17 229 160 174 34 246 92 208
210 166 240 86 10 232 31 185 145 105 259 58 280 204 50 124 80
82 198 44 256 116 130 61 273 41 127 85 152 107 267 60 275 191
35 265 79 201 52 282 113 133 67 277 199 43 122 72 143 110 272
266 48 188 78 283 65 120 112 106 264 139 289 194 59 131 84 39
197 69 269 45 129 103 286 62 136 75 42 114 271 141 276 196 54
83 200 40 257 117 132 57 274 284 203 56 123 77 37 119 262 144
36 261 81 202 53 278 115 134 258 140 111 195 68 279 73 46 135
268 49 189 74 285 66 121 108 76 51 126 260 148 118 190 55 281
193 70 270 47 125 104 287 64 192 63 288 71 38 128 263 153 109

The possible (unique) order 17 Balanced Series for the integers 0 ... 16, as described above, are shown in Attachment 25.7.2.

Attachment 25.7.3 shows a few 17th order Associated Magic Squares with 9th order Overlapping Sub Squares and
                  Order 8 Composed Magic Corner Squares.

Attachment 25.7.4 shows a few 17th order Associated Magic Squares with 9th order Overlapping Sub Squares.
                  Order 8 Composed Magic Corner Squares (Magic Middle and Center Squares).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the Latin Sub Squares.

25.8 Magic Squares (17 x 17)
     Most Perfect Magic Sub Squares (8 x 8)

Alternatively the series {0, 1, 2, 3, 4 ... 16} can be split into two balanced sub series - with identical sum (64) - around the center element e.g.:

     {0, 1, 2, 3, 13, 14, 15, 16}, {8}, {4, 5, 6, 7, 9, 10, 11, 12}

which can be used for the construction of the two order 8 Sub Squares and - if combined with the common center element 8 - the two each other overlapping order 9 Sub Squares.

The application of balanced sub series enables the construction of order 8 Pan Magic, Compact and Complete Sub Squares (Most Perfect).

Pan Magic Squares M of order 8 with the numbers 1 ... 64 can be written as M = A + 8 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6 and 7 as illustrated below for a Most Perfect Pan Magic Square

A
0 1 2 3 7 6 5 4
7 6 5 4 0 1 2 3
0 1 2 3 7 6 5 4
7 6 5 4 0 1 2 3
0 1 2 3 7 6 5 4
7 6 5 4 0 1 2 3
0 1 2 3 7 6 5 4
7 6 5 4 0 1 2 3
B = R(A)
4 3 4 3 4 3 4 3
5 2 5 2 5 2 5 2
6 1 6 1 6 1 6 1
7 0 7 0 7 0 7 0
3 4 3 4 3 4 3 4
2 5 2 5 2 5 2 5
1 6 1 6 1 6 1 6
0 7 0 7 0 7 0 7
M
33 26 35 28 40 31 38 29
48 23 46 21 41 18 43 20
49 10 51 12 56 15 54 13
64 7 62 5 57 2 59 4
25 34 27 36 32 39 30 37
24 47 22 45 17 42 19 44
9 50 11 52 16 55 14 53
8 63 6 61 1 58 3 60

The balanced series {0, 1, 2, 3, 4, 5, 6, 7} can be replaced by any balanced series {ai, i = 1 ... 8} and {bj, j = 1 ... 8}.

The possible (unique) order 8 and 9 Balanced Series for the integers 0 ... 16, as described above, are shown in Attachment 25.7.1.

Attachment 25.7.5 shows a few 17th order Magic Squares with 9th order Overlapping Sub Squares and
                  Order 8 Most Perfect Pan Magic Corner Squares (Pan Magic, Compact and Complete).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the Latin Sub Squares.

25.9 Magic Squares, Associated (19 x 19)

Although Euler postulated that sets of suitable Latin Squares did not exist for oddly even orders n ≡ 2 (mod 4), R. C. Bose and S. Shrinkhade proved the contrary for all n >= 10 (1959/1960).

Consequently also Magic Squares of order 19, with two 10th order Overlapping Sub Squares with identical Magic Sums, can be constructed based on suitable selected Latin Sub Squares.

Simple Magic Squares M of order 9 with the numbers 1 ... 81 can be written as M = A + 9 * B + [1] where the squares A and B contain only the integers 0, 1, 2 ... 8 as illustrated below.

A
0 4 8 7 2 3 5 6 1
7 2 3 5 6 1 0 4 8
5 6 1 0 4 8 7 2 3
8 0 4 3 7 2 1 5 6
3 7 2 1 5 6 8 0 4
1 5 6 8 0 4 3 7 2
4 8 0 2 3 7 6 1 5
2 3 7 6 1 5 4 8 0
6 1 5 4 8 0 2 3 7
B = T(A)
0 7 5 8 3 1 4 2 6
4 2 6 0 7 5 8 3 1
8 3 1 4 2 6 0 7 5
7 5 0 3 1 8 2 6 4
2 6 4 7 5 0 3 1 8
3 1 8 2 6 4 7 5 0
5 0 7 1 8 3 6 4 2
6 4 2 5 0 7 1 8 3
1 8 3 6 4 2 5 0 7
M
1 68 54 80 30 13 42 25 56
44 21 58 6 70 47 73 32 18
78 34 11 37 23 63 8 66 49
72 46 5 31 17 75 20 60 43
22 62 39 65 51 7 36 10 77
29 15 79 27 55 41 67 53 3
50 9 64 12 76 35 61 38 24
57 40 26 52 2 69 14 81 28
16 74 33 59 45 19 48 4 71

For Simple Magic Squares of order 9 the series {0, 1, 2, ... 8} can be replaced by any series {ai, i = 1 ... 9} and {bj, j = 1 ... 9}.

Simple Magic Squares M of order 10 with the numbers 1 ... 100 can be written as M = A + 10 * B + [1] where the squares A and B contain only the integers 0, 1, 2 ... 9 as illustrated below.

A
0 1 2 3 4 5 6 7 8 9
2 4 9 7 0 3 5 1 6 8
9 7 1 5 6 4 8 3 0 2
1 0 8 6 9 2 7 4 5 3
8 6 5 0 7 1 3 2 9 4
6 3 4 9 2 8 0 5 1 7
5 9 7 8 3 0 2 6 4 1
3 8 0 4 5 7 1 9 2 6
7 5 6 2 1 9 4 8 3 0
4 2 3 1 8 6 9 0 7 5
B
0 1 2 3 4 5 6 7 8 9
4 9 1 5 6 0 2 8 7 3
2 0 7 8 9 1 5 4 3 6
5 2 9 4 8 3 1 6 0 7
7 5 4 1 3 2 9 0 6 8
1 8 0 7 5 6 4 3 9 2
9 3 6 0 1 7 8 2 5 4
6 4 8 2 7 9 3 5 1 0
8 6 3 9 0 4 7 1 2 5
3 7 5 6 2 8 0 9 4 1
M
1 12 23 34 45 56 67 78 89 100
43 95 20 58 61 4 26 82 77 39
30 8 72 86 97 15 59 44 31 63
52 21 99 47 90 33 18 65 6 74
79 57 46 11 38 22 94 3 70 85
17 84 5 80 53 69 41 36 92 28
96 40 68 9 14 71 83 27 55 42
64 49 81 25 76 98 32 60 13 7
88 66 37 93 2 50 75 19 24 51
35 73 54 62 29 87 10 91 48 16

For Simple Magic Squares of order 10 the series {0, 1, 2, ... 9} can be replaced by any series {ai, i = 1 ... 10} and {bj, j = 1 ... 10}.

Magic Square M of order 19 with the numbers 1 ... 361 can be written as M = A + 19 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4 ... 18.

The series {0, 1, 2, 3, 4 ... 18} can be split into two sub series - with identical sum (81) - around the center element e.g.:

     {0, 1, 2, 3, 13, 14, 15, 16, 17}, {9}, {4, 5, 6, 7, 8, 10, 11, 12, 18}

which can be used for the construction of the two order 9 Sub Squares and - if combined with the common center element 9 - the two each other overlapping order 10 Sub Squares, as illustrated below:

A
13 2 3 1 17 15 9 0 16 14 0 13 17 16 2 3 14 15 1
16 14 15 2 1 9 13 17 3 0 16 2 3 14 15 1 0 13 17
3 17 0 13 14 16 1 9 2 15 14 15 1 0 13 17 16 2 3
14 9 16 17 3 0 2 15 13 1 17 0 13 3 16 2 1 14 15
15 3 13 9 2 17 0 14 1 16 3 16 2 1 14 15 17 0 13
17 15 14 0 16 1 3 2 9 13 1 14 15 17 0 13 3 16 2
1 0 17 15 9 2 16 13 14 3 13 17 0 2 3 16 15 1 14
9 16 1 14 15 13 17 3 0 2 2 3 16 15 1 14 13 17 0
2 13 9 16 0 3 14 1 15 17 15 1 14 13 17 0 2 3 16
0 1 2 3 13 14 15 16 17 9 18 12 11 10 8 7 6 5 4
4 8 18 12 6 7 10 11 5 18 11 5 10 7 4 12 9 8 6
12 6 7 10 11 5 4 8 18 6 4 7 18 8 11 10 5 12 9
10 11 5 4 8 18 12 6 7 7 10 8 12 6 9 11 18 4 5
18 4 8 7 12 6 5 10 11 8 9 6 7 5 12 4 10 11 18
7 12 6 5 10 11 18 4 8 12 5 10 4 18 6 9 8 7 11
5 10 11 18 4 8 7 12 6 5 8 11 6 4 7 18 12 9 10
8 18 4 6 7 12 11 5 10 11 6 9 5 12 10 8 4 18 7
6 7 12 11 5 10 8 18 4 4 7 18 8 9 5 6 11 10 12
11 5 10 8 18 4 6 7 12 10 12 4 9 11 18 5 7 6 8
B
3 16 14 15 2 17 0 9 13 1 4 12 10 18 7 5 8 6 11
17 15 3 9 0 13 16 1 2 14 8 6 11 4 12 10 18 7 5
15 13 17 2 16 9 3 14 1 0 18 7 5 8 6 11 4 12 10
9 3 15 0 1 16 17 2 14 13 12 10 4 7 5 18 6 11 8
1 17 0 16 14 15 13 3 9 2 6 11 8 12 10 4 7 5 18
16 14 13 1 3 2 9 0 15 17 7 5 18 6 11 8 12 10 4
14 2 9 13 17 3 1 15 0 16 10 4 12 5 18 7 11 8 6
2 0 16 17 9 1 14 13 3 15 11 8 6 10 4 12 5 18 7
13 9 1 14 15 0 2 17 16 3 5 18 7 11 8 6 10 4 12
0 1 2 3 13 14 15 16 17 9 18 12 11 10 8 7 6 5 4
0 16 14 17 3 1 13 2 15 7 12 18 6 4 11 10 5 9 8
13 2 15 0 16 14 17 3 1 11 7 8 10 5 9 18 12 4 6
17 3 1 13 2 15 0 16 14 12 4 11 5 7 18 8 9 6 10
16 14 0 3 1 17 2 15 13 18 11 4 9 6 7 5 8 10 12
2 15 13 16 14 0 3 1 17 6 9 7 8 11 10 12 4 18 5
3 1 17 2 15 13 16 14 0 8 10 6 18 12 5 4 11 7 9
14 0 16 1 17 3 15 13 2 4 5 10 7 9 12 6 18 8 11
15 13 2 14 0 16 1 17 3 10 6 5 12 8 4 9 7 11 18
1 17 3 15 13 2 14 0 16 5 8 9 4 18 6 11 10 12 7
M = A + 19 * B + [1]
71 307 270 287 56 339 10 172 264 34 77 242 208 359 136 99 167 130 211
340 300 73 174 2 257 318 37 42 267 169 117 213 91 244 192 343 147 113
289 265 324 52 319 188 59 276 22 16 357 149 97 153 128 227 93 231 194
186 67 302 18 23 305 326 54 280 249 246 191 90 137 112 345 116 224 168
35 327 14 314 269 303 248 72 173 55 118 226 155 230 205 92 151 96 356
322 282 262 20 74 40 175 3 295 337 135 110 358 132 210 166 232 207 79
268 39 189 263 333 60 36 299 15 308 204 94 229 98 346 150 225 154 129
48 17 306 338 187 33 284 251 58 288 212 156 131 206 78 243 109 360 134
250 185 29 283 286 4 53 325 320 75 111 344 148 223 170 115 193 80 245
1 21 41 61 261 281 301 321 341 181 361 241 221 201 161 141 121 101 81
5 313 285 336 64 27 258 50 291 152 240 348 125 84 214 203 105 180 159
260 45 293 11 316 272 328 66 38 216 138 160 209 104 183 353 234 89 124
334 69 25 252 47 304 13 311 274 236 87 218 108 140 352 164 190 119 196
323 271 9 65 32 330 44 296 259 351 219 83 179 120 146 100 163 202 247
46 298 254 310 277 12 76 24 332 127 177 144 157 228 197 238 85 350 107
63 30 335 57 290 256 312 279 7 158 199 126 349 233 103 95 222 143 182
275 19 309 26 331 70 297 253 49 88 102 200 139 184 239 123 347 171 217
292 255 51 278 6 315 28 342 62 195 122 114 237 162 82 178 145 220 355
31 329 68 294 266 43 273 8 317 106 165 176 86 354 133 215 198 235 142

Attachment 25.9.1 shows a few of the 747 (343 unique) possible order 19 Series for the integers 0 ... 18, as described above.

Attachment 25.9.2 shows the corresponding 19th order Composed Magic Squares with 10th order Overlapping Sub Squares.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the Latin Sub Squares.

Vorige Pagina Volgende Pagina Index About the Author