Office Applications and Entertainment, Magic Squares

Vorige Pagina Volgende Pagina Index About the Author

14.0     Special Magic Squares, Prime Numbers

14.13    Consecutive Primes (7)

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.

Suitable Generators (15 Magic Series) can be constructed semi-automatically (ref. CnstrGen15).

A possible Simple Magic Square - constructed with the generator method - is shown below, for the smallest consecutive prime numbers {131 ... 1619} for which an Order 15 Simple Magic Square exists (Mc15 = 12669):

Simple Magic Square
139 409 1609 167 157 1597 1601 149 1607 151 1613 137 1619 131 1583
191 193 1549 211 457 1553 1579 181 1559 179 1571 173 1543 163 1567
229 227 1489 241 587 1493 1499 233 1511 223 1523 199 1531 197 1487
269 1453 1451 281 593 271 257 263 1471 1483 1481 251 1459 239 1447
1399 349 313 1427 577 1409 311 1423 283 1429 293 1439 307 1433 277
1319 1321 359 1327 809 367 353 1361 347 1367 331 1373 337 1381 317
1283 857 401 1289 1291 419 1301 1297 389 397 379 1303 383 373 1307
1229 1231 449 829 1237 443 463 1249 439 1259 433 1277 431 1279 421
1181 1187 467 1201 1193 547 491 823 1217 1223 479 499 487 1213 461
541 1129 521 1117 1123 571 937 1151 509 1153 557 1163 523 1171 503
1069 827 613 1087 1091 607 599 641 569 601 1093 1103 1097 1109 563
1031 673 653 859 1033 1039 1051 1049 619 647 643 1061 631 1063 617
983 991 1013 997 691 743 701 1009 661 683 677 1019 821 1021 659
929 941 971 839 947 757 739 953 719 967 733 761 727 977 709
877 881 811 797 883 853 787 887 769 907 863 911 773 919 751

The Simple Magic Square shown above is essential different from a comparable Magic Square as previously published by Natalia Makarova (ref. A073520).

The Generator Method has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.61.

Each Simple Magic Square shown corresponds with 322560 transformations, as described below:

  • Any line n can be interchanged with line (16 - n). The possible number of transformations is 27 = 128.
    It should be noted that for each square the 180o rotated aspect is included in this collection.

  • Any permutation can be applied to the lines 1, 2, ... 7, provided that the same permutation is applied to the lines 15, 14, ... 9. The possible number of transformations is 7! = 5040.

The resulting number of transformations, excluding the 180o rotated aspects, is 128/2 * 5040 = 322560.

14.13.62 Simple Magic Squares (15 x 15)
         Single Square Inlays (Lower Order)

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.

Examples of order 3 through 7 square inlays, for the consecutive prime numbers {131 ... 1619} with the related Magic Sum s15 = 12669 are shown in Attachment 14.13.62.

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.

The order 6 Concentric Pan Magic Square (Inlay) with Associated Center Square has been discussed in detail in Section 14.4.2.

The square(s) shown correspond with numerous Prime Number Magic Squares with the same Magic Sum and variable values, which can be obtained by selecting other aspects of the inlays and variation of the borders.

14.13.63 Simple Magic Squares (15 x 15)
         Order 3 Magic Square Inlays (4 ea)

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.

Examples of subject Inlaid Magic Squares for the consecutive prime numbers {131 ... 1619} with the related Magic Sum s15 = 12669 are shown below:

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.

Type 1 (top), each resulting square corresponds with 84 = 4096 solutions, which can be obtained by selecting other aspects of the four inlays.

Type 2 (bottom), each resulting square corresponds with 84 * (3!)4 = 5.308.416 solutions, which can be obtained by selecting other aspects of the four inlays and variation of the border (window).

The order 3 Square Inlays of Type 2 might be moved along the Main Diagonals by means of row and column permutations (1, 2, 3 or 4 positions).

Potential order 3 Simple Magic Square Inlays (unique) for the consecutive prime numbers (131 ... 1619), resulting in numerous valid sets of four, are shown in Attachment 14.13.63.

14.13.64 Simple Magic Squares (15 x 15)
         Order 4 Pan Magic Square Inlays (4 ea)

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.

Examples of subject Inlaid Magic Squares for the consecutive prime numbers {131 ... 1619} with the related Magic Sum s15 = 12669 are shown below:

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.

Type 1 (top), each resulting square corresponds with 2 * 3843 = 113.246.208 solutions, which can be obtained by selecting other aspects of the four inlays.

Type 2 (bottom), each resulting square corresponds with 3844 * (4!)4 = 7,214 1015 solutions, which can be obtained by selecting other aspects of the four inlays and variation of the border (window).

The order 4 Square Inlays of Type 2 might be moved along the Main Diagonals by means of row and column permutations (1, 2 or 3 positions).

Potential order 4 Pan Magic Square Inlays (unique, s4 = 1500 ... 2916) for the consecutive prime numbers (89 ... 1367), resulting in numerous valid sets of four, are shown in Attachment 14.13.64.

14.13.65 Inlaid Magic Squares (15 x 15)
         Order 3 (Semi) Magic Square Inlays (9 ea)

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.

An example of subject Inlaid Magic Squares for the consecutive prime numbers {131 ... 1619} with the related Magic Sum s15 = 12669 is shown below:

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:

  • One Magic Center Square selected from the Potential Square Inlays shown in Attachment 14.13.63
  • Eight Semi Magic Border Squares (7 Magic Lines) selected from a (comparable) collection of Semi Magic Squares as discussed in Section 14.13.35

and result in a 9th order Magic Square Inlay with Magic Sum s9 = 7677.

The resulting square shown above corresponds with 8 * 127 * 2 * (6!) * (3!)2 = 1,5 1013 solutions, which can be obtained by selecting other aspects of the nine inlays and variation of the (eccentric) border.

14.13.66 Inlaid Magic Squares (15 x 15)
         Miscellaneous Order 9 Square Inlays

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.

An example of such an Inlaid Magic Square for the consecutive prime numbers {131 ... 1619} with the related Magic Sum s15 = 12669 is shown below:

Simple Magic, Order 9 Inlay with Diamond Inlays, s9 = 7821
701 1061 971 769 631 1049 821 827 991 1409 1499 151 1399 193 197
839 829 911 1069 727 967 787 809 883 1367 317 199 1373 1381 211
941 659 719 1021 643 1039 1151 887 761 191 1423 1427 229 149 1429
881 797 619 929 653 919 1103 983 937 181 1433 1439 173 139 1483
1129 1109 1123 1051 863 709 617 607 613 163 1451 1471 179 137 1447
691 683 733 673 1063 977 1093 1097 811 167 131 1453 157 1459 1481
1013 953 739 601 1091 743 857 877 947 1327 227 1321 389 223 1361
773 823 997 677 1033 661 751 1087 1019 1307 239 263 1319 233 1487
853 907 1009 1031 1117 757 641 647 859 1301 241 1297 449 1303 257
1289 251 271 1619 269 1613 277 1609 1283 1607 293 433 283 1291 281
421 1171 1543 401 409 467 431 1531 1523 439 577 587 1549 1201 419
331 307 1597 311 1601 337 1579 347 353 1217 1213 349 1231 1583 313
397 359 367 1571 1567 379 1559 383 599 373 1153 593 1259 1553 557
1229 1249 499 503 541 563 523 491 521 1163 1193 1223 1187 547 1237
1181 1511 571 443 461 1489 479 487 569 457 1279 463 1493 1277 509

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.

A few more examples of Simple Magic Squares with Order 9 Composed Square Inlays are shown in Attachment 14.13.65.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.67 Inlaid Magic Squares (15 x 15)
         Order 5 and 6 Diamond Inlays

An example of an Inlaid Magic Square - with Order 5 and 6 Diamond Inlays - for the consecutive prime numbers {131 ... 1619} with the related Magic Sum s15 = 12669 is shown below:

Simple Magic, Order 5 and 6 Diamond Inlays
131 1619 137 883 139 1103 1609 149 1607 151 1613 1601 157 1597 173
163 1583 1579 167 1259 1163 659 1571 1229 179 1307 191 181 1181 257
1567 193 197 1237 1291 773 587 523 223 1549 227 991 1553 199 1559
211 1523 619 571 673 823 953 1231 499 1511 1543 479 239 263 1531
229 631 503 733 1223 599 541 941 997 1297 269 1493 1481 233 1499
509 937 1217 1171 977 677 809 1193 569 487 1277 1129 313 251 1153
241 1303 661 1213 907 1187 1117 739 617 863 271 1487 281 1489 293
1483 317 1093 1109 643 601 1249 877 1301 1061 311 1031 277 1033 283
1471 307 337 1279 557 1013 1283 593 349 1459 353 331 1451 1453 433
1447 421 1439 367 491 1201 641 1433 383 1151 379 347 1289 1321 359
1429 373 1427 409 389 683 401 419 1381 397 1423 691 1399 439 1409
1327 431 467 1319 1361 449 463 443 1097 457 1373 563 1367 461 1091
1051 1123 1063 647 1069 577 547 1087 613 521 607 709 1049 967 1039
653 1021 1019 743 719 1009 983 701 947 727 919 787 751 929 761
757 887 911 821 971 811 827 769 857 859 797 839 881 853 829

The Diamond Inlays - with Magic Sums s5 = 4465 and s6 = 5358 - have been borrowed from the the Order 11 Inlaid Magic Square as discussed in Section 14.13.29.

The resulting square shown above corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.68 Composed Magic Squares (15 x 15)
         Order 7 and 8 Magic Sub Squares

When the Magic Sum s15 is a multiple of 15 (e.g. 86055), order 15 (Semi) Magic Squares might be composed of:

  • One 7th order Simple Magic Corner Square with Magic Sum s7 = 7 * s15 / 15 = 40159 (top/left)
  • One 8th order Simple Magic Corner Square with Magic Sum s8 = 8 * s15 / 15 = 45896 (bottom/right)
  • Two Magic Rectangles order 7 x 8 with s7 = 40159 and s8 = 45896

An example is shown below for the consecutive prime numbers {4783 ... 6709} with the related Magic Sums mentioned above.

Composed Simple Magic Square
4999 6469 4909 6571 5507 5483 6221 5399 6091 5791 5387 5167 6421 5003 6637
6301 5839 4957 5851 5861 5147 6203 6011 5449 5503 5233 5743 6359 5021 6577
6007 5779 6373 4789 5843 6581 4787 5521 5501 5281 5179 6343 5867 5683 6521
5641 4861 6709 5737 6689 4931 5591 5113 5153 5701 6337 6379 5081 6481 5651
5419 5431 6361 4793 5981 5857 6317 6389 6367 6323 5273 5107 5009 6491 4937
6043 5101 6067 5717 4799 6311 6121 5023 5051 5087 6397 6427 6449 6529 4933
5749 6679 4783 6701 5479 5849 4919 6703 6547 6473 6353 4993 4973 4951 4903
5827 5471 5953 5693 5581 6287 5347 6247 5059 4969 6673 6101 5077 5171 6599
5869 5639 5527 5563 5923 6257 5381 5653 5623 5011 6661 5351 5927 5119 6551
6113 5569 5657 5647 5557 6173 5443 5821 6037 6277 4813 6079 5813 6089 4967
5897 5519 5689 5441 5659 6073 5881 5227 6229 6691 4801 5417 6131 6569 4831
5413 5903 5783 5807 5801 5309 6143 6269 4987 5039 6653 5879 5261 5189 6619
5323 6133 5407 6053 5741 5303 6199 5531 5711 5099 6607 4943 6163 5279 6563
6217 5333 6211 5393 5437 5297 6271 5939 5987 6151 4871 6553 5477 6029 4889
5237 6329 5669 6299 6197 5197 5231 5209 6263 6659 4817 5573 6047 6451 4877

The order 7 Simple Magic Sub Square applied above is composed of:

  • Two each order 4 Simple Magic Squares with Magic Sum s4 = 22948 and
  • Two each order 3 Semi   Magic Squares with Magic Sum s3 = 17211

The order 8 Simple Magic Sub Square applied above is composed of 4 each order 4 Simple Magic Squares with Magic Sum s4 = 22948.

The resulting square shown above corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.69 Composed Magic Squares (15 x 15)
         Order 6 and 9 Magic Sub Squares

When the Magic Sum s15 is a multiple of 15 (e.g. 86055), order 15 (Semi) Magic Squares might be composed of:

  • One 6th order Simple Magic Corner Square with Magic Sum s6 = 6 * s15 / 15 = 34422 (bottom/right)
  • One 9th order Simple Magic Corner Square with Magic Sum s9 = 9 * s15 / 15 = 51633 (top/left)
  • Two Magic Rectangles order 6 x 8 with s6 = 34422 and s9 = 51633

An example is shown below for the consecutive prime numbers {4783 ... 6709} with the related Magic Sums mentioned above.

Composed Simple Magic Square
6323 6257 6287 5857 5171 5189 5153 6217 5179 6271 5501 6301 5009 6373 4967
5381 5351 5297 5801 6101 6131 6091 5281 6199 5113 5443 5233 6173 6011 6449
5407 6089 5399 5701 5393 6073 6037 5413 6121 5717 5531 6277 5011 6367 5519
5683 5741 5689 5711 5743 5851 5657 5737 5821 5639 5669 5261 6263 5021 6569
6379 5051 6353 5059 6389 5081 5953 6269 5099 5641 5693 5197 6311 5003 6577
5437 5471 6047 5923 5477 5417 6029 5779 6053 5441 5839 5119 6329 4993 6701
5521 5939 5503 5563 5557 5849 5903 5791 6007 6299 6197 5569 4973 6427 4957
5623 5591 5827 5807 5651 5813 5573 5867 5881 5039 5309 6317 6337 6469 4951
5879 6143 5231 6211 6151 6229 5237 5279 5273 6473 6451 6359 5227 4969 4943
5987 6163 5087 5653 5167 5479 6547 4877 6673 5749 6679 4783 5209 5659 6343
5023 5323 6247 5581 6113 5347 6421 4919 6659 6043 5101 6067 6133 4999 6079
5449 6551 4937 5107 4987 6397 5077 6529 6599 5419 5431 6361 5869 6553 4789
5783 4931 6571 6481 6563 4933 4903 6607 4861 5507 5483 6221 5897 6521 4793
5527 6637 4871 4909 4889 6619 6661 6689 4831 5861 5147 6203 5333 5387 6491
6653 4817 6709 6691 6703 5647 4813 4801 4799 5843 6581 4787 5981 5303 5927

The order 6 Simple Magic Sub Square applied above is composed of 4 each order 3 Semi Magic Squares (7 magic lines) with Magic Sum s3 = 17211.

The Order 9 Simple Magic Sub Square (s9 = 51633) has been constructed with the Generator Principle as applied in previous sections,

The resulting square shown above corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.70 Summary

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

CnstrGen15

Attachment 14.13.61

3

Square Inlays, Cons. Primes {131 ... 1619}

Prime1363

Attachment 14.13.63

4

Square Inlays, Cons. Primes {131 ... 1619}

Prime1364

Attachment 14.13.64

-

-

-

-

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.

Vorige Pagina Volgende Pagina Index About the Author