Office Applications and Entertainment, Magic Squares Index About the Author

 14.0    Special Magic Squares, Prime Numbers 14.22   Magic Squares, Higher Order, Inlaid (2) 14.22.1 Introduction Following sections will describe how Prime Number Inlaid Magic Squares with Concentric Main and/or Sub Squares can be generated with comparable routines as described in previous sections. The advantage of applying Concentric Magic Squares either as Main Square or as Inlays is the relative high generation speed compared with other Magic Squares. 14.22.2 Magic Squares, Composed (15 x 15) Examples of 15 x 15 Prime Number Inlaid Magic Squares with Concentric Sub Squares have been provided in Attachment 14.6.53. The 15th order Inlaid Magic Square shown below (s15 = 110535), is composed out of twenty five each 3th order Simple Magic Squares with different Magic Sums s(1) ... s(25).
 17239 1579 8329 139 9049 17959 9769 16519 859
 12511 601 6151 61 6421 12781 6691 12241 331
 12517 643 6121 31 6427 12823 6733 12211 337
 16921 1009 8263 73 8731 17389 9199 16453 541
 11887 937 5827 157 6217 12277 6607 11497 547
 11467 1483 5881 691 6277 11863 6673 11071 1087
 15727 1021 7753 193 8167 16141 8581 15313 607
 12541 2389 5683 13 6871 13729 8059 11353 1201
 13399 1129 6679 349 7069 13789 7459 13009 739
 15601 2371 7411 271 8461 16651 9511 14551 1321
 11239 3163 4951 163 6451 12739 7951 9739 1663
 11731 1423 5659 199 6271 12343 6883 11119 811
 13831 1327 6949 487 7369 14251 7789 13411 907
 15307 2503 7591 751 8467 16183 9343 14431 1627
 14797 2767 7297 787 8287 15787 9277 13807 1777
 10957 2617 6067 1657 6547 11437 7027 10477 2137
 13879 1999 7129 919 7669 14419 8209 13339 1459
 14683 1861 7057 241 7867 15493 8677 13873 1051
 12619 733 6361 313 6571 12829 6781 12409 523
 15319 1093 8161 1033 8191 15349 8221 15289 1063
 15241 2221 8101 1381 8521 15661 8941 14821 1801
 13903 3121 7927 2341 8317 14293 8707 13513 2731
 13159 4933 6841 1993 8311 14629 9781 11689 3463
 11287 883 5851 571 6007 11443 6163 11131 727
 9679 1831 5557 1567 5689 9811 5821 9547 1699
 The Magic Sums s(1) thru s(25) of the Sub Squares comply with the equations defining a Concentric Magic Square of order 5 (ref. Section 14.3.2). The Prime Number Inlaid Magic Square shown above can be constructed by selecting: An order 5 Concentric Magic Main Square containing possible Center Elements for the Sub Squares Twenty five order 3 Simple Magic Squares based on the corresponding Magic Sums and the equations deducted in Section 14.1.1 Attachment 14.6.58 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 15. Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or rotation/reflection of the twenty five inlays. 14.22.3 Magic Squares, Composed (18 x 18) The 18th order Inlaid Magic Square shown below (s18 = 126234), is composed out of thirty six each 3th order Simple Magic Squares with different Magic Sums s(1) ... s(36).
 16223 389 8117 137 8243 16349 8369 16097 263
 11243 2267 5099 59 6203 12347 7307 10139 1163
 11867 1811 5669 251 6449 12647 7229 11087 1031
 13757 881 6689 41 7109 14177 7529 13337 461
 16067 113 8081 101 8087 16073 8093 16061 107
 11717 317 5927 197 5987 11777 6047 11657 257
 9923 3209 5387 1637 6173 10709 6959 9137 2423
 11897 1109 6053 509 6353 12197 6653 11597 809
 11069 3671 4967 467 6569 12671 8171 9467 2069
 12497 3023 6221 971 7247 13523 8273 11471 1997
 14813 1889 6947 17 7883 15749 8819 13877 953
 14657 1619 7283 479 7853 15227 8423 14087 1049
 11447 1613 5531 281 6197 12113 6863 10781 947
 12743 1847 6359 599 6983 13367 7607 12119 1223
 15077 2243 7121 191 8147 16103 9173 14051 1217
 10889 1721 5657 857 6089 11321 6521 10457 1289
 11933 3413 5153 53 6833 13613 8513 10253 1733
 13109 4397 5981 701 7829 14957 9677 11261 2549
 12011 617 6269 557 6299 12041 6329 11981 587
 11987 2711 5639 431 6779 13127 7919 10847 1571
 11093 2027 5309 359 6143 11927 6977 10259 1193
 14243 1187 7589 1019 7673 14327 7757 14159 1103
 14867 71 7433 23 7457 14891 7481 14843 47
 14771 863 7547 503 7727 14951 7907 14591 683
 10391 5573 5417 2153 7127 12101 8837 8681 3863
 14447 1877 7487 977 7937 14897 8387 13997 1427
 11483 3359 6737 2447 7193 11939 7649 11027 2903
 11003 4793 5333 1373 7043 12713 8753 9293 3083
 10499 1697 5441 821 5879 10937 6317 10061 1259
 10799 5849 4049 149 6899 13649 9749 7949 2999
 14639 2609 6869 269 8039 15809 9209 13469 1439
 12953 5039 5477 347 7823 15299 10169 10607 2693
 10337 4943 7451 4691 7577 10463 7703 10211 4817
 11057 5171 4523 383 6917 13451 9311 8663 2777
 10559 2459 4799 179 5939 11699 7079 9419 1319
 9227 3119 5003 1559 5783 10007 6563 8447 2339
 The Magic Sums s(1) thru s(36) of the Simple Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 6 (ref. Section 14.4.3). The Prime Number Inlaid Magic Squares described above can be constructed by selecting: An order 6 Concentric Magic Main Square containing possible Center Elements for the Sub Squares Thirty six order 3 Simple Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.1.1 Attachment 14.6.59 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 18, based on Concentric Magic Main Squares. Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or rotation/reflection of the thirty six inlays. Alternatively Order 18 Prime Number Inlaid Magic Squares can be constructed by selecting: An order 3 Simple Magic Main Square containing possible Order 6 Magic Sums Nine order 6 Concentric Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.4.4 Attachment 14.6.60 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 18. Each square shown corresponds with numerous solutions, which can be obtained by rotation/reflection of the main square or selecting other aspects of the nine inlays. 14.22.4 Magic Squares, Composed (20 x 20) Examples of 20th order Inlaid Magic Squares, composed out of twenty five each 4th order Pan Magic Squares with different Magic Sums s(1) ... s(25), are shown in Attachment 14.6.83. The Magic Sums s(1) thru s(25) of the Pan Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 5 (ref. Section 14.3.2). The Prime Number Inlaid Magic Squares described above can be constructed by selecting: An order 5 Concentric Magic Main Square containing possible Order 4 Magic Sums Twenty five order 4 Pan Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.2.2 Attachment 14.6.83 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 20, based on Concentric Magic Main Squares. Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the twenty five inlays. Alternatively Order 20 Prime Number Inlaid Magic Squares can be constructed by selecting: An order 4 Pan Magic Main Square containing possible Center Elements for the Sub Squares Sixteen order 5 Concentric Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.3.2 Attachment 14.6.84 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 20. Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the sixteen inlays. 14.22.5 Magic Squares, Composed (21 x 21) Examples of 21th order Inlaid Magic Squares, composed out of forty nine each 3th order Simple Magic Squares with different Magic Sums s(1) ... s(49), are shown in Attachment 14.6.85. The Magic Sums s(1) thru s(49) of the Simple Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 7 (ref. Section 14.5.1). The Prime Number Inlaid Magic Squares described above can be constructed by selecting: An order 7 Concentric Magic Main Square containing possible Center Elements for the Sub Squares Forty nine order 3 Simple Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.1.1 Attachment 14.6.85 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 21, based on Concentric Magic Main Squares. Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or rotation/reflection of the forty nine inlays. Alternatively Order 21 Prime Number Inlaid Magic Squares can be constructed by selecting: An order 3 Simple Magic Main Square containing possible Center Elements for the Sub Squares Nine order 7 Concentric Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.5.1 Attachment 14.6.86 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 21. Each square shown corresponds with numerous solutions, which can be obtained by rotation/reflection of the main square or selecting other aspects of the nine inlays. 14.22.6 Magic Squares, Composed (24 x 24) Attachment 14.6.87 shows an example of a 24th order Inlaid Magic Square, composed out of thirty six each 4th order Pan Magic Squares with different Magic Sums s(1) ... s(36). The Magic Sums s(1) thru s(36) of the Pan Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 6 (ref. Section 14.4.3). The Prime Number Inlaid Magic Square described above can be constructed by selecting: An order 6 Concentric Magic Main Square containing possible Order 4 Magic Sums Thirty six order 4 Pan Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.2.2 The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the thirty six inlays. Attachment 14.6.88 shows an example of a 24th order Inlaid Magic Square, composed out of sixteen each 6th order Concentric Magic Squares with different Magic Sums s(1) ... s(16). The Magic Sums s(1) thru s(16) of the Concentric Magic Sub Squares comply with the equations defining a Pan Magic Square of order 4 (ref. Section 14.2.2). The Prime Number Inlaid Magic Square described above can be constructed by selecting: An order 4 Pan Magic Main Square containing possible Order 6 Magic Sums Sixteen order 6 Concentric Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.4.3 The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the sixteen inlays. 14.22.7 Magic Squares, Composed (25 x 25) Attachment 14.6.92 shows an example of a 25th order Inlaid Magic Square, composed out of twenty five each 5th order Magic Squares with different Magic Sums s(1) ... s(25). The Magic Sums s(1) thru s(25) of the Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 5 (ref. Section 14.3.2). The Prime Number Inlaid Magic Square described above can be constructed by selecting: An order 5 Concentric Magic Main Square containing possible Center Elements for the Sub Squares Twenty five order 5 Concentric Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.3.2 The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the twenty five inlays. 14.22.8 Magic Squares, Composed (28 x 28) Attachment 14.6.93 shows an example of a 28th order Inlaid Magic Square, composed out of forty nine each 4th order Pan Magic Squares with different Magic Sums s(1) ... s(49). The Magic Sums s(1) thru s(49) of the Pan Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 7 (ref. Section 14.5.1). The Prime Number Inlaid Magic Square described above can be constructed by selecting: An order 7 Concentric Magic Main Square containing possible Order 4 Magic Sums Forty nine order 4 Pan Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.2.2 The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the forty nine inlays. Attachment 14.6.94 shows an example of a 28th order Inlaid Magic Square, composed out of sixteen each 7th order Concentric Magic Squares with different Magic Sums s(1) ... s(16). The Magic Sums s(1) thru s(16) of the Concentric Magic Sub Squares comply with the equations defining a Pan Magic Square of order 4 (ref. Section 14.2.2). The Prime Number Inlaid Magic Square described above can be constructed by selecting: An order 4 Pan Magic Main Square containing possible Center Elements for the Sub Squares Sixteen order 7 Concentric Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.5.1 The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the sixteen inlays. 14.22.9 Summary The obtained results regarding miscellaneous types of higher order Prime Number Inlaid Magic Squares as deducted and discussed in previous sections are summarized in following table:
 Type Characteristics Orders Subroutine Results Order 15 Composed Concentric Main Square 5 x 3 Order 18 Composed Concentric Main Square Concentric Sub  Squares 6 x 3 3 x 6 Order 20 Composed Concentric Main Square Concentric Sub  Squares 5 x 4 4 x 5 Order 21 Composed Concentric Main Square Concentric Sub  Squares 7 x 3 3 x 7 Order 24 Composed Concentric Main Square Concentric Sub  Squares 6 x 4 4 x 6 Order 25 Composed Concentric Main and Sub Squares 5 x 5 Order 28 Composed Concentric Main Square Concentric Sub  Squares 7 x 4 4 x 7 - - - - - -
 Following sections will describe how Prime Number Magic Squares with Consecutive Primes can be found with comparable routines as described in previous chapters.