Office Applications and Entertainment, Magic Squares | ||
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14.0 Special Magic Squares, Prime Numbers
14.13.61 Simple Magic Squares (15 x 15)
Prime Number Simple Magic Squares of order 15 can be constructed with the Generator Principle, as applied in previous sections.
Simple Magic Square
The Simple Magic Square shown above is essential different from a comparable Magic Square as previously published by Natalia Makarova (ref. A073520).
The resulting number of transformations, excluding the 180o rotated aspects, is 128/2 * 5040 = 322560.
14.13.62 Simple Magic Squares (15 x 15)
Order 15 Simple Magic Squares with single square inlay(s) of lower order
can be constructed with the generator method as discussed in detail in
Section 14.13.22 through Section 14.13.26.
The order 3, 4 and 5 Square Inlays might be moved along the Main Diagonal (top/left to bottom/right) by means of row and column permutations.
14.13.63 Simple Magic Squares (15 x 15)
Comparable with the method discussed in Section 14.13.27
it is possible to construct Order 15 Simple Magic Squares with four order 3 Simple Magic Square Inlays.
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Simple Magic Type 1, Order 3 Inlays (4 ea)
743 347 419 761 317 449 1493 167 1601 1579 1597 1583 163 173 1277 179 503 827 197 509 821 1499 1571 1559 193 1567 191 1553 181 1319 587 659 263 569 701 257 1511 157 1613 1609 1607 149 151 1619 1217 641 443 479 823 307 439 1471 1549 1523 211 1531 199 1543 227 1283 359 521 683 139 523 907 239 1459 241 1439 409 1427 1453 1423 1447 563 599 401 607 739 223 233 1489 229 1481 251 1451 1483 1487 1433 751 983 733 787 971 757 977 991 859 853 811 769 773 857 797 1171 1213 1223 1201 463 1181 1193 1187 499 457 491 1009 461 433 487 829 967 809 937 953 839 863 941 877 911 919 881 883 929 131 1361 1381 1429 1399 1409 1367 277 269 283 1373 271 281 887 389 293 1153 1163 1129 1151 1103 1117 547 571 577 557 727 541 593 1123 617 1291 1327 1303 1097 1321 1297 331 311 337 313 353 1307 349 1301 431 719 661 653 1033 1031 673 1021 1039 1013 677 1049 997 947 1019 137 1093 613 1069 709 1091 1063 631 601 1061 643 619 647 1051 1087 691 1229 1289 1249 1259 1231 1279 383 367 397 373 467 1237 379 421 1109 s3
1509 1527 1563 1569 Simple Magic Type 2, Order 3 Inlays (4 ea)
743 347 419 1493 163 1277 167 1583 1579 1597 173 1601 761 317 449 179 503 827 1319 1499 1559 193 1553 181 1567 191 1571 197 509 821 587 659 263 1511 157 1217 1607 1609 1613 151 1619 149 569 701 257 751 983 733 977 797 811 853 787 857 859 773 769 991 971 757 1171 1213 1223 433 1187 487 457 461 491 1009 1193 499 1201 463 1181 829 967 809 941 863 877 919 937 883 881 911 131 929 953 839 1361 1381 1429 277 271 293 1373 281 1399 283 269 389 887 1409 1367 1153 1163 1129 547 557 577 571 727 593 617 541 1123 1151 1103 1117 1291 1327 1303 331 313 337 1301 349 1307 353 311 431 1097 1321 1297 719 661 653 1033 1039 1021 1049 137 677 947 1019 1013 997 1031 673 1093 613 1069 631 1051 647 619 643 601 691 1087 709 1061 1091 1063 1229 1289 1249 1237 367 383 421 379 397 467 373 1109 1259 1231 1279 641 443 479 1471 1549 1523 211 1531 199 1543 1283 227 823 307 439 359 521 683 239 1423 1427 1447 241 409 1453 1439 1459 139 523 907 563 599 401 229 1433 233 1481 1451 1483 251 1487 1489 607 739 223
Miscellaneous (main) diagonal sets are possible for both square types.
14.13.64 Simple Magic Squares (15 x 15)
Comparable with the method discussed in Section 14.13.27
it is possible to construct Order 15 Simple Magic Squares with four order 4 Pan Magic Square Inlays.
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Simple Magic Type 1, Order 4 Inlays (4 ea)
421 463 883 1093 347 281 1039 1129 331 163 1571 167 1597 1601 1583 739 1237 277 607 919 1249 227 401 1553 1493 1543 1549 191 181 503 547 337 1009 967 359 269 1051 1117 1579 1489 1567 173 179 467 1559 1153 823 691 193 1171 997 479 149 257 151 1619 157 1607 1609 1613 457 439 797 947 433 569 827 1063 1481 1459 251 1427 1471 239 809 563 1181 223 673 677 1213 283 719 829 1399 229 1483 1487 211 1499 523 373 863 881 619 383 1013 877 1523 1511 197 1433 743 1531 199 1097 647 757 139 1163 727 769 233 241 1447 263 853 1429 1453 1451 379 1301 1321 773 1307 1297 1319 1303 1327 349 409 431 389 397 367 1423 1373 271 1439 1409 1367 1361 1381 353 317 311 751 307 313 293 733 887 911 859 701 941 953 929 907 661 971 857 709 811 839 1229 1151 1223 1217 499 1187 491 1201 541 557 1193 1019 521 509 131 1031 977 1049 1021 683 631 1033 991 659 643 653 761 937 983 617 1283 419 1291 1289 1259 449 1231 577 487 443 1279 787 461 1277 137 1091 1061 1103 571 1123 1109 593 599 601 587 613 821 641 1087 1069 s4
2860 2796 2640 2892 Simple Magic Type 2, Order 4 Inlays (4 ea)
421 463 883 1093 331 167 1571 163 1601 1597 1583 347 281 1039 1129 739 1237 277 607 1549 1553 1543 1493 191 181 503 919 1249 227 401 547 337 1009 967 1579 1489 1559 1567 179 173 467 359 269 1051 1117 1153 823 691 193 257 157 1607 151 1619 1609 1613 1171 997 479 149 379 1307 1321 773 1327 349 397 431 409 367 1297 1303 1301 1319 389 1423 751 307 1381 353 317 311 293 271 1373 313 1439 1361 1367 1409 733 859 941 887 811 907 971 661 709 701 839 857 929 953 911 1229 1151 1223 1217 557 541 499 1019 521 131 491 509 1187 1193 1201 1031 761 977 1021 683 653 1033 1049 937 659 617 983 631 643 991 1283 1279 1291 1289 577 1231 443 137 461 1277 487 787 449 419 1259 1091 1061 1109 601 571 613 587 593 641 599 1069 1103 1123 1087 821 457 439 797 947 1481 251 1459 1427 1471 809 239 433 569 827 1063 563 1181 223 673 829 1483 229 1399 1487 211 1499 677 1213 283 719 523 373 863 881 1523 1511 197 1433 743 1531 199 619 383 1013 877 1097 647 757 139 241 1447 263 853 1429 1451 1453 1163 727 769 233
Miscellaneous (main) diagonal sets are possible for both square types.
14.13.65 Inlaid Magic Squares (15 x 15)
Comparable with the method mentioned in Section 14.13.63 above it is possible to construct Order 13 Simple Magic Squares with nine order 3 (Semi) Magic Square Inlays.
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Simple Magic, Order 3 Inlays (9 ea)
1163 1493 251 727 991 571 557 1151 773 229 241 1427 1423 239 1433 1613 191 1103 283 439 1567 563 347 1571 277 271 1409 1373 263 1399 131 1223 1553 1279 859 151 1361 983 137 1367 281 379 1327 1321 317 599 1091 443 967 769 823 673 1063 1249 233 1453 1447 197 211 1451 227 383 1523 709 853 997 1009 433 1543 179 1471 1499 163 1481 199 1307 659 167 883 937 739 1303 1489 193 181 1483 1511 173 157 1487 307 1213 1117 929 947 953 761 1061 389 1319 293 349 1291 1429 311 1033 223 1381 617 593 1619 1301 269 641 1277 331 313 1259 1439 373 1297 1201 139 1283 1289 257 149 881 1181 1231 337 509 449 1229 1237 409 419 1607 359 1609 353 401 367 1049 1601 1597 431 1031 1039 397 661 677 1123 643 1097 907 1109 1093 683 757 911 751 647 919 691 503 463 479 1579 457 461 1559 467 863 1549 1583 521 877 421 887 1019 1171 577 1187 587 653 613 1051 971 619 631 701 607 1153 1129 1459 1193 487 491 499 1531 523 1217 569 1021 977 601 1013 541 547 941 1069 719 733 743 1087 787 797 857 829 809 821 839 827 811 s3
2907 2289 2481 2133 2559 2985 2637 2829 2211
The nine Square Inlays are selected as follows:
and result in a 9th order Magic Square Inlay with Magic Sum s9 = 7677.
14.13.66 Simple Magic Squares (15 x 15)
Order 15 Simple Magic Squares with Order 9 square Inlays can be constructed based on
Order 9 Composed and Inlaid Magic Squares, as discussed in detail in Section 14.13.7.
Simple Magic, Order 9 Inlay with Diamond Inlays, s9 = 7821
The Order 9 Inlay - with Diamond Inlays s4 = 3476 and s5 = 4345 - is based on
the consecutive prime numbers {601 ... 1151} with the related Magic Sum s9 = 7821.
The obtained results regarding the miscellaneous types of Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table: |
Order
Main Characteristics
Subroutine
Results
-
-
-
-
15
Consecutive Primes, Simple Magic
3
Square Inlays, Cons. Primes {131 ... 1619}
4
Square Inlays, Cons. Primes {131 ... 1619}
-
-
-
-
Following sections will explain the concept of Prime Number Magic Squares composed of Twin Primes
and illustrate how subject squares can be generated with comparable routines as described in previous sections.
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