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I had started to build my prime square using the previous analysis as a start point. The only thing to clarify is that I choose the order 15, because we just have a 12 order prime magic square.
So, I build this snake square:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
30 29 28 27 26 25 24 23 22 21 20 19 18 17 16
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
60 59 58 57 56 55 54 53 52 51 50 49 48 47 46
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
90 89 88 87 86 85 84 83 82 81 80 79 78 77 76
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
120 119 118 117 116 115 114 113 112 111 110 109 108 107 106
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
150 149 148 147 146 145 144 143 142 141 140 139 138 137 136
151 152 153 154 155 156 157 158 159 160 161 162 163 164 165
180 179 178 177 176 175 174 173 172 171 170 169 168 167 166
181 182 183 184 185 186 187 188 189 190 191 192 193 194 195
210 209 208 207 206 205 204 203 202 201 200 199 198 197 196
211 212 213 214 215 216 217 218 219 220 221 222 223 224 225
At this point I should execute some changes in line, but I still had not
hypothesized what Muncey could have fact at this point, therefore I passed
directly to the correspondent prime square:
1 3 5 7 11 13 17 19 23 29 31 37 41 43 47 9308
113 109 107 103 101 97 89 83 79 73 71 67 61 59 53 8370
127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 7200
281 277 271 269 263 257 251 241 239 233 229 227 223 211 199 5964
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 4616
463 461 457 449 443 439 433 431 421 419 409 401 397 389 383 3240
467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 1808
659 653 647 643 641 631 619 617 613 607 601 599 593 587 577 348
661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 -1090
863 859 857 853 839 829 827 823 821 811 809 797 787 773 769 -2682
877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 -4252
1069 1063 1061 1051 1049 1039 1033 1031 1021 1019 1013 1009 997 991 983 -5764
1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 -7320
1291 1289 1283 1279 1277 1259 1249 1237 1231 1229 1223 1217 1213 1201 1193 -9306
1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 -10680
24 96 72 60 66 30 58 48 0 -10 -46 -50 -74 -96 -86 -68 -52
Note that I don't report the sum of columns, rows, and diagonals, but the
difference with magic constant.
How you can see, only one columns is at the magic value, while the others are
near to the expected value.
At this point, if you want to make one column to the right magic constant, you have to change some values of that column with other values of other columns (possibly in the same rows for not changing the rows state - this is only necessary if the rows was at the right magic value-).
I built an interactive program for doing this changes (the program is grown as I need subsequently, so it is not good coded!). In less time I make all the columns to the right magic state, because founds the right shifts was easy (only the latest columns make me more times).
Ok, I must apply the some method to the rows, but now the difference with the expected value may reaches 10000. But if you changes many numbers, you can reduce those values to less difference, so it is easy to find the right changes.
Now all rows and columns are to the magic state, but the diagonal are not right. So I built a procedure that swap two rows (or two columns) with some scheme or in random manner until the two diagonals become to the magic state (if we change the position of two rows, the magic state of rows and columns don't change, while the diagonal state will changes).
Ok, but if we changes rows while we look for one diagonal become to the magic
state using the procedure that automatically search the right value as in the
previous step? Yes, but the other diagonals?
I build this procedure and make it to search more solution until the other
diagonal become to the magic state. This find out the solution:
Prime Magic Square of order 15
1087 953 1093 167 499 97 829 1237 79 73 809 967 787 911 47
463 461 457 491 691 503 89 163 23 1019 1223 1399 937 523 1193
1 1301 307 1163 1319 1259 17 19 877 29 547 181 1187 1217 211
661 1063 271 887 263 317 331 1361 239 173 389 1097 1213 199 1171
1069 673 677 311 313 257 251 337 1367 419 1013 373 397 769 1409
659 293 647 643 641 439 709 617 613 607 601 1201 593 599 773
1021 3 883 103 839 631 1117 823 283 1151 149 577 41 1427 587
127 653 1303 7 1277 1321 157 719 727 1229 947 227 61 743 137
139 859 5 421 101 701 433 521 449 941 1381 991 751 1181 761
1297 131 1283 1307 179 151 1373 241 347 1327 71 59 1423 67 379
1291 1091 487 269 1049 1109 1033 431 881 349 229 409 223 401 383
113 109 1061 1279 1103 1039 509 929 569 733 1153 757 191 37 53
281 1289 107 1051 11 13 919 1031 1129 541 353 557 367 1009 977
563 277 857 853 443 971 1249 1123 1231 233 31 43 997 193 571
863 479 197 683 907 827 619 83 821 811 739 797 467 359 983
Ok, try now with order 17: build snake square, make rows, make columns, find
automatically the solution.
Prime Magic Square of order 17
1 43 5 7 1721 1709 379 827 1823 419 1319 1663 463 1361 911 659 1381
139 137 131 127 113 1549 1553 103 101 1571 1579 1583 1597 79 1607 1109 1613
149 151 157 163 167 1789 1801 181 191 1831 197 1321 1867 1097 1103 647 1879
337 331 317 1531 311 109 1459 1559 1567 839 263 1433 631 919 1423 1609 53
347 349 353 359 367 373 1699 1697 389 1669 1667 409 683 421 1627 1621 661
557 547 541 313 521 1453 503 283 829 1307 1847 1861 41 461 1367 1373 887
1013 1009 997 991 983 977 971 967 953 947 31 937 1327 1601 47 881 59
1753 1759 1777 1783 1787 173 227 1811 23 97 89 467 929 677 457 439 443
769 761 757 751 743 739 719 727 733 709 701 691 1657 1637 71 907 1619
1259 1277 1279 1283 1289 1291 1297 1301 1303 193 1181 199 211 223 643 229 233
1747 1733 1741 1723 11 13 17 383 1693 29 941 857 397 3 1151 1129 1123
1493 1489 1487 1483 1481 307 293 1471 281 277 853 619 859 1153 251 241 653
797 787 773 809 811 821 823 19 1451 1447 1439 37 1429 1427 877 61 883
1019 1021 1051 1039 1033 1049 1031 1061 1063 1069 1087 1091 73 641 179 67 1117
1249 1217 1231 1229 1223 1237 1213 1201 1193 1187 401 269 271 257 431 433 449
563 569 571 577 587 593 599 601 607 613 617 1171 1163 1871 1873 1877 239
1499 1511 1523 523 1543 509 107 499 491 487 479 83 1093 863 673 1409 1399
At this point I can say that snake construction was useful only for make the
columns near to the magic state, but for rows the difference are big (however
snake construction are important for order 12 because even the rows are more
near to the magic state, but we see this more after).
But can we build a program that automatically construct a prime magic square?
If we build it we can easier try to find square of other order.
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Magic Square | 
Tognon Stefano Research
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