Office Applications and Entertainment, Magic Squares

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14.0     Special Magic Squares, Prime Numbers

14.13    Consecutive Primes (5)

14.13.51 Simple Magic Squares (14 x 14)

Prime Number Simple Magic Squares of order 14 can be constructed with the Generator Principle, as applied in previous sections.

Suitable Generators (14 Magic Series) can be constructed semi-automatically (ref. CnstrGen14).

A possible Simple Magic Square - constructed with the generator method - is shown below, for the smallest consecutive prime numbers {89 ... 1367} for which an Order 14 Simple Magic Square exists (Mc14 = 9660):

Simple Magic Square
1367 1361 107 101 89 97 1327 109 859 193 1307 1319 103 1321
1229 199 1217 1223 197 223 211 241 857 1193 229 1201 1213 227
311 331 1069 1091 1093 1087 337 353 1061 1063 359 347 349 809
1279 167 1249 1231 151 157 1259 1237 179 191 929 173 1277 181
1303 127 1291 1301 113 131 1297 149 163 139 1289 937 1283 137
277 1123 283 271 1129 1117 281 1097 307 1109 317 953 293 1103
239 1181 257 233 1187 1171 251 829 1151 313 1153 263 269 1163
401 1013 1009 419 1019 431 421 983 439 661 443 433 991 997
449 461 463 457 977 967 919 941 503 971 659 479 467 947
487 907 509 491 911 887 499 743 883 523 541 881 877 521
547 863 569 557 853 839 563 643 821 827 577 607 823 571
773 797 601 593 811 787 599 613 733 761 769 619 587 617
631 751 653 641 757 727 647 701 673 683 691 709 719 677
367 379 383 1051 373 1039 1049 1021 1031 1033 397 739 409 389

The Generator Method has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.51.

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

  • Any line n can be interchanged with line (15 - 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 14, 13, ... 8. The possible number of transformations is 7! = 5040.

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

14.13.52 Simple Magic Squares (14 x 14)
         Single Square Inlays (Lower Order)

Order 14 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 {89 ... 1367} with the related Magic Sum s14 = 9660 are shown in Attachment 14.13.52.

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.

14.13.53 Simple Magic Squares (14 x 14)
         Order 3 Magic Square Inlays (4 ea)

Comparable with the method discussed in Section 14.13.27 it is possible to construct Order 14 Simple Magic Squares with four order 3 Simple Magic Square Inlays.

Examples of subject Inlaid Magic Squares for the consecutive prime numbers {89 ... 1367} with the related Magic Sum s14 = 9660 are shown below:

Simple Magic Type 1, Order 3 Inlays (4 ea)
743 347 419 761 317 449 1361 929 109 1327 103 1367 107 1321
179 503 827 197 509 821 1307 1013 1303 1301 137 1319 131 113
587 659 263 569 701 257 1291 139 149 1289 151 1297 1031 1277
641 443 479 733 499 337 1279 971 1283 1259 157 1249 167 163
359 521 683 127 523 919 173 1237 1223 181 1231 1063 191 1229
563 599 401 709 547 313 193 1217 211 199 1201 1171 1213 1123
607 863 673 617 859 857 631 853 811 691 619 97 839 643
1021 991 1033 1009 389 1019 433 397 997 421 727 409 431 383
1129 1103 1109 269 1117 311 277 251 1097 283 1093 281 1069 271
491 541 877 809 907 883 881 887 613 911 577 89 601 593
1163 227 229 1193 1187 1181 233 223 241 239 1153 293 1151 947
439 983 967 953 977 457 571 463 487 467 937 557 941 461
1091 1051 1039 1061 331 1087 353 307 379 373 823 367 1049 349
647 829 661 653 797 769 677 773 757 719 751 101 739 787
s3
1509 1527
1563 1569
Simple Magic Type 2, Order 3 Inlays (4 ea)
743 347 419 1361 929 1327 103 107 1321 109 1367 761 317 449
179 503 827 1307 1303 1013 131 1301 137 113 1319 197 509 821
587 659 263 1277 139 1289 1291 151 149 1297 1031 569 701 257
607 863 617 631 97 811 853 643 619 839 691 859 673 857
1021 991 1033 431 397 433 727 421 383 997 409 389 1009 1019
491 593 809 541 887 911 577 613 881 89 601 907 877 883
1163 227 1193 239 223 233 1153 1151 947 241 293 1187 229 1181
1129 1093 269 251 277 283 271 1097 1069 1103 281 1117 1109 311
647 829 653 677 719 773 751 757 787 739 101 797 661 769
1091 1051 1061 307 823 373 353 379 349 1049 367 331 1039 1087
439 941 953 983 463 571 937 487 461 467 557 977 967 457
641 443 479 1283 971 1259 167 1279 157 163 1249 733 499 337
359 521 683 173 1231 191 1223 1063 1229 1237 181 127 523 919
563 599 401 199 1201 193 1123 211 1171 1217 1213 709 547 313

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 (89 ... 1367), resulting in numerous valid sets of four, are shown in Attachment 14.13.53.

14.13.54 Simple Magic Squares (14 x 14)
         Order 4 Pan Magic Square Inlays (4 ea)

Comparable with the method discussed in Section 14.13.27 it is possible to construct Order 14 Simple Magic Squares with four order 4 Pan Magic Square Inlays.

Examples of subject Inlaid Magic Squares for the consecutive prime numbers {89 ... 1367} with the related Magic Sum s14 = 9660 are shown below:

Simple Magic Type 1, Order 4 Inlays (4 ea)
677 1231 89 823 439 1259 97 977 271 167 1291 883 173 1283
103 809 691 1217 107 967 449 1249 1319 139 941 191 151 1327
1321 587 733 179 1289 409 947 127 1301 163 1303 881 157 263
719 193 1307 601 937 137 1279 419 1361 199 919 109 113 1367
443 1097 101 1019 631 1187 131 907 181 197 1237 1013 239 1277
149 971 491 1049 227 811 727 1091 223 1213 211 997 1223 277
1229 311 887 233 1297 521 797 241 229 1031 269 1171 1193 251
839 281 1181 359 701 337 1201 617 257 1153 283 857 1163 431
991 929 661 479 877 509 503 863 541 911 467 523 859 547
307 1151 1129 293 331 1123 313 317 1117 1103 347 653 1109 367
433 421 401 1039 1021 461 953 1033 457 1009 463 499 983 487
787 557 853 683 773 829 571 827 607 563 577 821 569 643
599 769 757 593 647 761 641 619 709 751 739 673 659 743
1063 353 379 1093 383 349 1051 373 1087 1061 613 389 1069 397
s4
2820 2772
2660 2856
Simple Magic Type 2, Order 4 Inlays (4 ea)
677 1231 89 823 1283 173 271 1291 167 883 439 1259 97 977
103 809 691 1217 151 1319 1327 139 191 941 107 967 449 1249
1321 587 733 179 1301 881 163 157 1303 263 1289 409 947 127
719 193 1307 601 1367 919 109 199 1361 113 937 137 1279 419
991 523 661 479 541 547 911 929 859 503 877 509 467 863
307 1151 1129 293 367 1103 347 653 1109 1117 331 1123 313 317
433 499 401 1039 421 487 463 983 457 1009 1021 461 953 1033
787 643 853 683 557 569 607 821 577 563 773 829 571 827
599 751 757 593 743 673 709 739 659 769 647 761 641 619
1063 613 379 1093 353 1061 1051 389 1069 397 383 349 1087 373
443 1097 101 1019 181 197 1013 1237 239 1277 631 1187 131 907
149 971 491 1049 211 1223 1213 997 277 223 227 811 727 1091
1229 311 887 233 1031 251 1193 269 229 1171 1297 521 797 241
839 281 1181 359 1153 257 283 857 1163 431 701 337 1201 617

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 or 2 positions).

Potential order 4 Pan Magic Square Inlays (unique) for the consecutive prime numbers (89 ... 1367), resulting in numerous valid sets of four, are shown in Attachment 14.13.54.

14.13.56 Composed Magic Squares (14 x 14)
         Order 7 Semi Magic Sub Squares (4 ea)

Order 14 (Semi) Magic Squares might be composed of order 7 (Semi) Magic Sub Squares when the Magic Sum s14 is a multiple of 2.

An example is shown below for the consecutive prime numbers {163 ... 1471} with the related Magic Sums s14 = 10966 and s7 = 5483

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

The Composed Simple Magic Square shown above corresponds with 4 * (7!)4 = 2,581 1015 squares for the applied diagonal elements (highlighted).

Order 12 Simple Magic Squares composed of (Semi) Magic Sub Squares can be transformed into Four Way V type ZigZag Magic Squares of order 12 as illustrated below:

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

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

14.13.57 Composed Magic Squares (14 x 14)
         Order 6 and 8 Magic Sub Squares

When the Magic Sum s14 is a multiple of 14 (e.g. s14 = 9660), order 14 (Semi) Magic Squares might be composed of:

  • One 8th order Simple Magic Corner Square with Magic Sum s8 = 8 * s14 / 14 = 5520 (bottom/right)
  • One 6th order Simple Magic Corner Square with Magic Sum s6 = 6 * s14 / 14 = 4140 (top/left)
  • Two Magic Rectangles order 6 x 8 with s6 = 4140 and s8 = 5520

An example is shown below for the consecutive prime numbers {89 ... 1367} with the related Magic Sums mentioned above.

Composed Simple Magic Square
389 971 857 823 661 439 1367 1361 1301 701 239 223 191 137
919 863 431 859 607 461 509 353 281 887 1019 1049 1193 229
727 313 673 587 1187 653 569 337 613 733 877 883 907 601
593 521 503 997 739 787 641 691 677 709 797 401 853 751
571 1063 1153 317 227 809 563 821 811 479 379 757 547 1163
941 409 523 557 719 991 491 577 457 631 829 827 449 1259
647 487 349 1061 269 1327 271 251 1129 1109 347 293 1087 1033
419 683 467 1039 211 1321 1097 1123 257 283 1013 1069 311 367
977 929 643 241 1237 113 101 103 1277 1279 151 167 1213 1229
421 383 659 617 1009 1051 1291 1283 97 89 1249 1231 149 131
773 769 947 307 1171 173 433 463 983 881 937 761 619 443
233 373 1031 1091 1303 109 911 953 397 499 359 541 839 1021
743 599 331 967 181 1319 127 193 1217 1223 263 277 1103 1117
1307 1297 1093 197 139 107 1289 1151 163 157 1201 1181 199 179

The order 8 Simple Magic Sub Square applied above is composed of 4 each order 4 Simple  Magic Squares with Magic Sum s4 = 2760.

The order 6 Simple Magic Sub Square applied above (s6 = 4140) is a Concentric Magic Square with an order 4 Simple Magic Center Square (s4 = 2760).

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

14.13.58 Composed Magic Squares (14 x 14)
         Order 4 and 10 Magic Sub Squares

When the Magic Sum s14 is a multiple of 14 (e.g. 9660), order 14 (Semi) Magic Squares might be composed of:

  • One 10th order Simple Magic Corner Square with Magic Sum s10 = 10 * s14 / 14 = 6900 (top/left)
  • One 4th order Pan Magic Corner Square with Magic Sum s4 = 4 * s14 / 14 = 2760 (bottom/right)
  • Two Magic Rectangles order 4 x 10 with s4 = 2760 and s10 = 6900

An example is shown below for the consecutive prime numbers {89 ... 1367} with the related Magic Sums mentioned above.

Composed Simple Magic Square 1
163 257 1109 1231 733 593 739 857 503 937 541 569 809 619
1049 1291 103 317 823 433 971 1013 359 1031 383 397 991 499
271 149 1217 1123 557 947 409 367 1021 349 997 983 389 881
1277 1063 331 89 647 787 641 523 877 443 839 811 571 761
229 1151 859 521 1307 1361 1367 953 113 109 137 107 127 1319
1103 277 607 773 1297 797 1321 181 173 139 197 191 1303 1301
193 1187 1033 347 1051 1289 211 1327 241 1259 239 233 827 223
1201 179 311 1069 421 457 449 439 929 977 1061 1009 727 431
167 1213 1087 293 461 463 941 919 829 479 487 967 887 467
1223 157 283 1097 631 677 719 751 643 683 691 653 709 743
151 1229 1117 263 587 601 613 617 863 757 821 673 769 599
1249 131 251 1129 491 547 563 659 853 907 911 883 577 509
101 1279 1153 227 373 401 379 353 1019 419 1163 1093 661 1039
1283 97 199 1181 281 307 337 701 1237 1171 1193 1091 313 269

The Order 10 Simple Magic Sub Square (s10 = 6900), constructed with the Generator Principle as applied in previous sections, corresponds with 8 * 25/2 * (5!) = 8 * 1920 = 15360 Sub Squares.

The resulting square corresponds consequently with 384 * 15360 * (4! * 6!)2 = 1,761 1015 Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.59 Bordered Magic Squares (14 x 14)
         Order 12 Simple Magic Sub Squares

The consecutive prime numbers {89 ... 1367} - with the related Magic Sum s14 = 9660 - contain 52 regular pairs (pair sum = 1380), which enables the construction of Bordered Magic Squares as illustrated by following example:

Concentric Border(s)
1291 787 541 193 1201 167 1217 157 1229 149 1249 103 1279 97
359 311 367 1033 1049 317 499 947 997 349 503 937 971 1021
293 607 1307 1361 1367 953 113 109 137 107 127 1319 773 1087
283 859 1297 797 1321 181 173 139 197 191 1303 1301 521 1097
569 983 1051 1289 211 1327 241 1259 239 233 827 223 397 811
277 389 421 457 449 439 929 977 1061 1009 727 431 991 1103
1109 739 461 463 941 919 829 479 487 967 887 467 641 271
263 809 631 677 719 751 643 683 691 653 709 743 571 1117
1123 823 587 601 613 617 863 757 821 673 769 599 557 257
251 857 491 547 563 659 853 907 911 883 577 509 523 1129
1151 733 373 401 379 353 1019 419 1163 1093 661 1039 647 229
227 761 281 307 337 701 1237 1171 1193 1091 313 269 619 1153
1181 409 1013 347 331 1063 881 433 383 1031 877 443 1069 199
1283 593 839 1187 179 1213 163 1223 151 1231 131 1277 101 89

The Order 10 Simple Magic Sub Square (s10 = 6900), constructed with the Generator Principle as applied in Section 14.13.10, corresponds with 8 * 25/2 * (5!) = 8 * 1920 = 15360 Sub Squares.

The Order 14 Concentric Border corresponds with 8 * (12!)2 = 1,836 1018 borders for the applied corner pairs.

The Order 12 Concentric Border corresponds with 8 * (10!)2 = 1,053 1014 borders for the applied corner pairs.

The resulting square corresponds consequently with 15360 * 1,836 1018 * 1,053 1014 = 2,970 * 1036 Bordered Magic Squares with the same Magic Sum(s) and variable values.

The Concentric Bordered Magic Square shown above can be transformed into an Eccentric Magic Square with the order 10 embedded Simple Magic Square as Corner Square:

Eccentric Border(s)
1291 97 1279 541 193 1201 167 1217 157 1229 149 1249 103 787
1283 89 101 839 1187 179 1213 163 1223 151 1231 131 1277 593
359 1021 311 971 937 1033 1049 317 499 947 997 349 503 367
293 1087 409 1069 443 347 331 1063 881 433 383 1031 877 1013
283 1097 607 773 1307 1361 1367 953 113 109 137 107 127 1319
569 811 859 521 1297 797 1321 181 173 139 197 191 1303 1301
277 1103 983 397 1051 1289 211 1327 241 1259 239 233 827 223
1109 271 739 641 421 457 449 439 929 977 1061 1009 727 431
263 1117 809 571 461 463 941 919 829 479 487 967 887 467
1123 257 857 523 631 677 719 751 643 683 691 653 709 743
251 1129 733 647 587 601 613 617 863 757 821 673 769 599
1181 199 761 619 491 547 563 659 853 907 911 883 577 509
227 1153 389 991 373 401 379 353 1019 419 1163 1093 661 1039
1151 229 823 557 281 307 337 701 1237 1171 1193 1091 313 269

The Top/Right Diagonals have been corrected by means of permutation of the pairs in the four left and the four top border lines.

The Order 14 Eccentric Border corresponds with 2 * (10!)2 = 2,634 1013 borders for the applied corner pairs.

The Order 12 Eccentric Border corresponds with 2 * (8!)2 = 3,251 109 borders for the applied corner pairs.

The resulting square corresponds consequently with 15360 * 2,634 1013 * 3,251 109 = 1,315 * 1027 Eccentric Magic Squares with the same Magic Sum(s) and variable values.

14.13.60 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

-

-

-

-

14

Consecutive Primes, Simple Magic

CnstrGen14

Attachment 14.13.51

3

Square Inlays, Cons. Primes {89 ... 1367}

Prime1353

Attachment 14.13.53

4

Square Inlays, Cons. Primes {89 ... 1367}

Prime1354

Attachment 14.13.54

-

-

-

-

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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