Office Applications and Entertainment, Magic Squares
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14.0     Special Magic Squares, Prime Numbers 14.13    Consecutive Primes (3) 14.13.21 Simple Magic Squares (11 x 11) Prime Number Simple Magic Squares of order 11 can be constructed with the Generator Principle, as applied in previous sections. Suitable Generators (11 Magic Series) can be constructed semi-automatically (ref. CnstrGen11). A possible Semi Magic Square and resulting Simple Magic Square are shown below, for the smallest consecutive prime numbers {67 ... 797} for which an Order 11 Simple Magic Square exists (MC11 = 4507):

Semi Magic Square 73 83 71 79 67 251 757 769 773 797 787 761 751 743 739 733 283 107 103 101 97 89 127 113 109 131 137 367 677 691 709 719 727 701 683 673 659 653 379 163 157 151 149 139 181 179 167 173 191 661 647 641 631 617 419 643 223 613 607 601 401 619 211 199 197 193 239 593 227 229 233 431 241 571 587 557 599 577 569 563 541 547 373 271 277 269 263 257 281 463 509 499 487 521 293 307 311 313 523 503 491 479 461 449 443 349 347 337 331 317 421 359 353 389 409 397 383 433 439 467 457

Simple Magic Square 751 743 739 107 97 761 89 103 283 101 733 113 109 131 677 719 127 727 691 367 709 137 683 673 659 163 149 701 139 157 379 151 653 223 613 607 619 197 643 193 211 401 199 601 593 227 229 241 557 239 599 571 431 587 233 83 71 79 757 797 73 787 769 251 773 67 569 563 541 271 263 577 257 277 373 269 547 359 353 389 383 467 421 457 433 397 439 409 463 509 499 293 313 281 523 307 521 311 487 491 479 461 349 331 503 317 347 443 337 449 179 167 173 647 617 181 419 641 661 631 191

The Generator Method has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.21. Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values. 14.13.22 Simple Magic Squares (11 x 11)          Order 3 Square Inlay Order 11 Simple Magic Squares with order 3 Square Inlay(s) can be constructed with the generator method. An example of the construction for the consecutive prime numbers {67 ... 797} with the related Magic Sums s11 = 4507 and s3 = 1011 is shown below:

Generator, Order 3 Inlay 71 73 79 83 89 229 757 769 773 787 797 101 577 127 307 113 131 157 739 743 751 761 103 67 337 607 137 139 257 691 709 727 733 107 367 547 97 149 151 313 673 683 701 719 109 163 167 173 179 419 647 653 659 661 677 181 191 193 197 199 401 613 617 631 641 643 211 223 227 233 239 397 571 587 599 601 619 241 251 263 269 271 389 541 557 563 569 593 277 281 283 293 331 487 499 503 509 521 523 311 317 347 349 373 449 461 463 467 479 491 353 359 379 383 409 421 431 433 439 443 457

Semi Magic, Order 3 Inlay 73 769 79 83 757 229 787 71 773 89 797 761 577 127 307 157 739 743 751 101 131 113 103 67 337 607 137 139 257 691 709 727 733 719 367 547 97 107 701 151 673 313 683 149 109 163 647 653 179 419 173 167 659 677 661 643 641 631 617 613 401 197 199 191 193 181 397 223 227 233 599 211 601 587 239 619 571 593 569 563 557 541 389 241 269 263 271 251 277 281 503 509 523 487 499 293 521 331 283 479 491 467 461 463 349 449 373 317 347 311 353 359 379 383 431 443 409 433 421 439 457

Simple Magic, Order 3 Inlay 613 641 631 617 401 643 199 193 191 181 197 157 577 127 307 739 761 751 131 101 113 743 137 67 337 607 139 103 691 727 709 733 257 107 367 547 97 701 719 673 683 313 149 151 541 569 563 557 389 593 269 271 263 251 241 757 769 79 83 229 73 71 89 773 797 787 463 491 467 461 349 479 373 347 317 311 449 523 281 503 509 487 277 293 331 521 283 499 179 163 647 653 419 109 167 677 659 661 173 431 359 379 383 443 353 433 439 421 457 409 599 223 227 233 211 397 587 619 239 571 601

The Generator Method, as applied for Inlaid Magic Squares based on Consecutive Prime Numbers can be summarised as follows: The Generator (11 Magic Series) can be constructed semi-automatically (ref, CnstrGen11); The Semi Magic Square can be constructed by permutating the numbers within the rows, while leaving the order 3 Inlay as constructed; The Magic Square can be obtained semi automatically by permutating the rows and columns within the Semi Magic Square (ref. CnstrSqrs11). Potential order 3 Square Inlays (34 unique) have been constructed for the consecutive prime numbers (67 ... 797) with routine Prime1322 and are shown in Attachment 14.13.22. The order 3 Square Inlay might be moved along the Main Diagonal (top/left to bottom/right) by means of row and column permutations (-1, 1, 5, 6 or 7 positions). 14.13.23 Simple Magic Squares (11 x 11)          Order 4 Pan Magic Square Inlay Order 11 Simple Magic Squares with order 4 Pan Magic Square Inlay(s) can be constructed with the generator method. An example of the construction for the consecutive prime numbers {67 ... 797} with the related Magic Sums s11 = 4507 and s4 = 2340 is shown below:

Generator, Order 4 Inlay 67 71 73 79 83 311 743 751 769 773 787 89 479 487 647 727 109 113 127 233 739 757 97 577 797 409 557 131 137 139 211 719 733 101 523 443 691 683 149 151 157 199 701 709 103 761 613 593 373 163 167 173 223 661 677 107 179 181 191 193 397 631 643 653 659 673 197 227 229 239 241 307 599 601 607 619 641 251 257 263 269 271 313 547 563 569 587 617 277 281 283 293 317 421 499 503 521 541 571 331 337 347 349 353 431 439 457 463 491 509 359 367 379 383 389 401 419 433 449 461 467

Semi Magic, Order 4 Inlay 83 71 73 67 79 311 773 769 751 787 743 757 479 487 647 727 739 233 127 113 109 89 137 577 797 409 557 131 139 733 97 211 719 701 523 443 691 683 709 199 157 149 151 101 167 761 613 593 373 103 163 173 661 223 677 673 193 179 181 191 397 643 659 631 653 107 197 227 229 239 241 619 607 307 599 641 601 617 587 563 547 569 269 271 313 257 263 251 421 283 281 293 317 277 521 503 541 571 499 353 439 463 457 337 491 509 347 349 431 331 401 367 379 383 433 461 449 419 359 467 389

Simple Magic, Order 4 Inlay 83 71 67 73 79 311 751 787 773 769 743 757 479 487 647 727 739 113 109 233 127 89 137 577 797 409 557 131 97 211 139 733 719 701 523 443 691 683 709 149 151 199 157 101 167 761 613 593 373 103 661 223 163 173 677 401 367 383 379 433 461 359 467 449 419 389 617 587 547 563 569 269 257 263 271 313 251 353 439 457 463 337 491 349 431 509 347 331 421 283 293 281 317 277 541 571 521 503 499 197 227 239 229 241 619 599 641 607 307 601 673 193 181 179 191 397 631 653 643 659 107

The Generator Method, as applied for Inlaid Magic Squares based on Consecutive Prime Numbers is described in Section 14.13.22 above. Potential order 4 Pan Magic Square Inlays have been constructed for the consecutive prime numbers (67 ... 797) with routine Prime1323 and are shown in Attachment 14.13.23 (one square per occurring magic sum). The order 4 Pan Magic Square Inlay might be moved along the Main Diagonal (top/left to bottom/right) by means of row and column permutations (-1, 5 or 6 positions). 14.13.24a Simple Magic Squares (11 x 11)           Order 5 Associated Magic Square Inlay Order 11 Simple Magic Squares with order 5 Associated Square Inlay(s) can be constructed with the generator method. An example of the construction for the consecutive prime numbers {67 ... 797} with the related Magic Sums s11 = 4507 and s5 = 1985 is shown below:

Generator, Order 5 Inlay 727 571 151 163 373 127 131 673 733 761 97 337 331 103 613 601 137 139 683 719 743 101 307 271 397 523 487 149 157 619 739 751 107 193 181 691 463 457 167 199 661 677 709 109 421 631 643 223 67 173 241 641 653 701 113 191 197 211 227 379 593 607 617 647 659 179 239 251 257 263 389 557 563 569 587 599 233 277 281 283 293 409 509 521 541 547 577 269 313 317 347 349 431 467 479 491 499 503 311 359 367 383 401 419 433 439 443 449 461 353 73 79 83 89 229 757 769 773 787 797 71

Semi Magic, Order 5 Inlay 727 571 151 163 373 127 673 97 131 733 761 337 331 103 613 601 137 139 719 743 683 101 307 271 397 523 487 739 619 157 149 107 751 193 181 691 463 457 167 661 199 709 677 109 421 631 643 223 67 653 173 641 241 113 701 191 659 211 647 617 227 379 607 197 593 179 587 569 557 563 257 263 389 251 239 233 599 281 277 269 409 509 577 521 293 547 541 283 347 499 349 431 491 467 479 313 317 311 503 359 439 367 383 419 353 401 443 461 433 449 757 79 769 89 229 797 73 787 773 83 71

Simple Magic, Order 5 Inlay 727 571 151 163 373 653 173 641 241 113 701 337 331 103 613 601 167 661 199 709 677 109 307 271 397 523 487 739 619 157 149 107 751 193 181 691 463 457 137 139 719 743 683 101 421 631 643 223 67 127 673 97 131 733 761 587 557 257 563 569 263 389 251 239 233 599 281 269 509 409 277 577 521 293 547 541 283 757 769 229 89 79 797 73 787 773 83 71 359 367 419 383 439 353 401 443 461 433 449 347 349 491 431 499 467 479 313 317 311 503 191 211 617 647 659 227 379 607 197 593 179

The construction method is as described in Section 14.13.22 above, with exception of the last step (main diagonals): The top/left  to bottom/right diagonal should be completed by permutating the six bottom rows The top/right to bottom/left  diagonal should be completed by permutating the rows    of the top/right   5 x 5 corner and the columns of the bottom/left 5 x 5 corner Potential order 5 Associated Square Inlays have been constructed for the consecutive prime numbers (67 ... 797) with routine Prime1324 and are shown in Attachment 14.13.24 (one square per occurring magic sum). 14.13.24b Simple Magic Squares (11 x 11)           Order 5 Magic Square with Diamond Inlay Alternatively it is possible to construct Order 11 Simple Magic Squares with order 5 Magic Squares with Diamond Inlay(s). An example of the construction for the consecutive prime numbers {67 ... 797} with the related Magic Sums s11 = 4507 and s5 = 1985 is shown below:

Semi Magic, Order 5 Inlay 487 97 631 619 151 89 101 641 107 787 797 439 223 613 103 607 769 757 127 647 109 113 337 373 397 421 457 131 137 139 599 743 773 79 691 181 571 463 761 719 569 167 157 149 643 601 163 271 307 739 191 179 557 683 173 331 193 199 211 229 227 709 727 733 197 751 701 673 677 661 311 233 251 239 257 241 263 283 269 281 313 653 293 353 509 617 659 277 593 317 461 383 389 379 367 359 347 349 563 73 523 401 433 449 419 479 587 67 499 577 541 547 503 521 491 467 443 431 409 83 71

Simple Magic, Order 5 Inlay 487 97 631 619 151 761 719 569 167 157 149 439 223 613 103 607 739 191 179 557 683 173 337 373 397 421 457 131 137 139 599 743 773 79 691 181 571 463 769 757 127 647 109 113 643 601 163 271 307 89 101 641 107 787 797 211 229 199 193 331 227 709 727 733 197 751 521 491 503 547 541 467 443 431 409 83 71 383 389 461 317 593 379 367 359 347 349 563 661 311 677 673 701 233 251 239 257 241 263 313 653 281 269 283 293 353 509 617 659 277 433 449 401 523 73 419 479 587 67 499 577

Potential order 5 Magic Squares with Diamond Inlays have been constructed for the consecutive prime numbers (67 ... 797) with routine Prime1325 and are shown in Attachment 14.13.25 (one square per occurring magic sum). 14.13.25a Simple Magic Squares (11 x 11)           Order 7 Bordered Magic Square Inlay Order 11 Simple Magic Squares with order 7 Borderd Magic Square Inlay(s) can be constructed with the generator method. An example of the construction for the consecutive prime numbers {67 ... 797} with the related Magic Sums s11 = 4507 and s7 = 2793 is shown below:

Semi Magic, Order 7 Inlay 71 701 691 641 521 89 79 101 113 739 761 647 593 353 131 499 419 151 103 127 733 751 631 613 643 229 283 227 167 149 617 769 179 449 181 109 619 487 599 349 163 797 191 563 139 367 433 439 73 683 659 557 787 173 197 137 241 457 577 653 67 661 757 541 223 193 719 97 107 157 277 709 727 773 199 233 509 211 251 263 269 317 607 673 743 257 677 239 571 601 443 569 547 313 293 281 271 307 311 523 359 587 397 383 331 347 491 337 379 373 409 503 421 479 467 463 401 389 461 83 431

Simple Magic, Order 7 Inlay 71 701 691 641 521 89 79 769 179 617 149 647 593 353 131 499 419 151 223 193 541 757 631 613 643 229 283 227 167 739 761 113 101 449 181 109 619 487 599 349 733 751 127 103 139 367 433 439 73 683 659 191 563 797 163 137 241 457 577 653 67 661 173 197 787 557 719 97 107 157 277 709 727 233 509 199 773 211 673 607 251 263 269 317 677 239 257 743 571 293 313 601 443 569 547 307 311 271 281 523 347 331 359 587 397 383 379 373 337 491 409 401 463 503 421 479 467 83 431 461 389

The construction method is as described in Section 14.13.22 above, with exception of the last step (main diagonals): The top/left  to bottom/right diagonal should be completed by permutating the four right colomns The top/right to bottom/left  diagonal should be completed by permutating the rows    of the top/right   7 x 4 rectangle and the columns of the bottom/left 4 x 7 rectangle The square(s) shown above correspond with numerous Prime Number Magic Squares with the same Magic Sum and variable values. Potential order 7 Border Inlays have been constructed for the consecutive prime numbers (67 ... 797) with routine Prime1325b and are shown in Attachment 14.13.25b (one border per occurring magic sum). 14.13.25b Simple Magic Squares (11 x 11)           Order 7 Magic Square with Diamond Inlays Alternatively it is possible to construct Order 11 Simple Magic Squares with order 7 Magic Squares with Diamond Inlay(s). An example of the construction for the consecutive prime numbers {67 ... 797} with the related Magic Sums s11 = 4507 and s7 = 2793 is shown below:

Semi Magic, Order 7 Inlay 67 647 313 571 409 449 337 101 107 733 773 379 97 241 569 659 307 541 109 127 751 727 769 401 467 131 167 71 787 113 139 701 761 251 293 397 443 653 593 163 137 683 151 743 577 277 719 283 419 439 79 677 709 149 179 487 557 457 317 383 503 89 757 181 619 157 263 521 199 479 103 431 797 673 641 173 227 191 193 197 211 229 607 643 739 661 613 223 461 601 587 691 617 269 271 257 239 233 281 563 373 331 349 359 347 367 523 631 311 353 499 547 599 463 509 491 433 421 389 73 83

Simple Magic, Order 7 Inlay 67 647 313 571 409 449 337 113 701 761 139 379 97 241 569 659 307 541 101 733 773 107 769 401 467 131 167 71 787 109 751 727 127 251 293 397 443 653 593 163 137 151 743 683 577 277 719 283 419 439 79 677 149 179 709 487 557 457 317 383 503 89 757 619 157 181 263 521 199 479 103 431 797 673 173 227 641 193 191 607 643 229 197 211 739 613 223 661 601 461 269 271 617 587 691 257 233 281 239 373 563 347 367 359 331 349 523 311 353 631 547 499 491 433 509 599 463 421 73 83 389

The order 7 Magic Square Inlay contains: One 3th order Magic Diamond Inlay with Magic Sum s3 = 1329, One 4th order Magic Diamond Inlay with Magic Sum s4 = 1464 The square(s) shown above correspond with numerous Prime Number Magic Squares with the same Magic Sum and variable values. 14.13.26 Simple Magic Squares (11 x 11)          Order 6 Ultra Magic Square Inlay Order 11 Simple Magic Squares with order 6 Ultra Magic Square Inlay(s) can be constructed with the generator method. An example of the construction for the consecutive prime numbers {67 ... 797} with the related Magic Sums s11 = 4507 and s6 = 2250 is shown below:

Semi Magic, Order 6 Inlay 677 383 137 149 311 593 101 97 503 769 787 229 193 619 571 457 181 107 103 797 709 541 467 443 227 263 251 599 109 547 727 761 113 151 499 487 523 307 283 461 757 127 139 773 569 293 179 131 557 521 167 433 743 163 751 157 439 601 613 367 73 733 739 173 421 191 197 199 211 223 233 719 701 683 653 449 239 277 587 647 673 257 269 691 313 271 241 281 661 349 331 353 617 317 463 337 359 347 373 479 491 409 401 641 389 397 431 71 419 379 643 631 659 607 509 563 577 67 83 89 79

Simple Magic, Order 6 Inlay 677 383 137 149 311 593 109 547 727 761 113 229 193 619 571 457 181 167 433 743 163 751 467 443 227 263 251 599 107 103 797 709 541 151 499 487 523 307 283 461 757 127 139 773 569 293 179 131 557 521 101 97 503 769 787 157 439 601 613 367 73 733 739 173 421 191 509 631 659 607 643 563 577 67 83 89 79 233 199 211 223 197 719 701 683 653 449 239 257 587 647 673 277 269 691 313 271 241 281 617 349 331 353 661 317 463 337 359 347 373 641 491 409 401 479 389 397 431 71 419 379

The construction method is as described in Section 14.13.22 above, with exception of the last step (main diagonals): The top/left  to bottom/right diagonal should be completed by permutating the five bottom rows (if required) The top/right to bottom/left  diagonal should be completed by permutating the rows    of the top/right   5 x 5 corner and the columns of the bottom/left 5 x 5 corner Potential order 6 Ultra Magic Square Inlays have been constructed for the consecutive prime numbers (67 ... 797) with routine Prime1326 and are shown in Attachment 14.13.26 (one square per occurring magic sum). 14.13.27 Simple Magic Squares (11 x 11)          Order 3 Square Inlays (4 ea) Comparable with the above it is possible to construct Order 11 Simple Magic Squares with four order 3 Simple Magic Square Inlays. An example of subject construction for the consecutive prime numbers {67 ... 797} with the related Magic Sum s11 = 5707 is shown below:

Semi Magic, Order 3 Inlays (4 ea) 389 167 233 461 131 251 103 433 769 773 797 107 263 419 71 281 491 109 479 743 757 787 293 359 137 311 431 101 113 563 709 751 739 577 127 307 719 641 617 139 149 151 347 733 67 337 607 557 659 761 673 163 157 173 353 367 547 97 701 677 599 727 179 241 191 181 569 653 661 211 223 227 683 691 199 197 193 601 443 631 269 277 229 643 271 647 257 239 409 619 373 283 317 331 379 613 349 521 313 541 421 449 401 383 397 439 457 463 467 89 587 571 593 523 487 503 499 509 79 73 83

Simple Magic, Order 3 Inlays (4 ea) 389 167 233 461 131 251 797 773 769 433 103 107 263 419 71 281 491 787 757 743 479 109 293 359 137 311 431 101 739 751 709 563 113 577 127 307 719 641 617 733 347 151 149 139 67 337 607 557 659 761 353 173 157 163 673 367 547 97 701 677 599 181 191 241 179 727 443 631 601 269 277 229 239 257 647 271 643 653 661 569 211 223 227 193 197 199 691 683 619 373 409 283 317 331 313 521 349 613 379 421 449 541 401 383 397 89 467 463 457 439 571 593 587 523 487 503 83 73 79 509 499

s3 789 843 1011 1977

Miscellaneous (main) diagonal sets are possible. Each resulting square corresponds with 2 * 83 = 1024 solutions, which can be obtained by selecting other aspects of the four inlays. An other example of subject construction - for the same prime numbers and Magic Sums - results in considerable more Inlaid Magic Squares:

Semi Magic, Order 3 Inlays (4 ea) 389 167 233 461 131 251 773 769 751 479 103 107 263 419 71 281 491 109 727 743 563 733 293 359 137 311 431 101 739 757 673 593 113 577 127 307 719 641 617 683 149 151 139 397 67 337 607 557 659 761 163 317 173 157 709 367 547 97 701 677 599 691 181 277 191 179 193 643 521 503 227 463 239 241 271 409 797 569 619 509 331 223 449 421 197 313 787 89 631 601 523 443 283 467 347 83 349 379 401 653 257 541 211 383 229 269 647 373 457 487 661 587 613 199 571 79 73 439 433 353 499	Simple Magic 1, Order 3 Inlays (4 ea) 389 167 233 773 769 751 479 103 461 131 251 107 263 419 109 727 743 563 733 71 281 491 293 359 137 739 757 673 593 113 311 431 101 193 643 521 239 241 271 409 797 503 227 463 569 619 509 421 197 313 787 89 331 223 449 631 601 523 347 83 349 379 401 443 283 467 653 257 541 269 647 373 457 487 211 383 229 661 587 613 73 439 433 353 499 199 571 79 577 127 307 683 149 151 139 397 719 641 617 67 337 607 163 317 173 157 709 557 659 761 367 547 97 691 181 277 191 179 701 677 599	s3 789 843 1011 1977
Simple Magic 2, Order 3 Inlays (4 ea) 239 193 643 521 241 271 409 503 227 463 797 773 389 167 233 769 751 479 461 131 251 103 109 107 263 419 727 743 563 71 281 491 733 739 293 359 137 757 673 593 311 431 101 113 421 569 619 509 197 313 787 331 223 449 89 347 631 601 523 83 349 379 443 283 467 401 269 653 257 541 647 373 457 211 383 229 487 683 577 127 307 149 151 139 719 641 617 397 163 67 337 607 317 173 157 557 659 761 709 691 367 547 97 181 277 191 701 677 599 179 73 661 587 613 439 433 353 199 571 79 499	Simple Magic 3, Order 3 Inlays (4 ea) 239 241 193 643 521 271 503 227 463 409 797 421 197 569 619 509 313 331 223 449 787 89 773 769 389 167 233 751 461 131 251 479 103 109 727 107 263 419 743 71 281 491 563 733 739 757 293 359 137 673 311 431 101 593 113 347 83 631 601 523 349 443 283 467 379 401 683 149 577 127 307 151 719 641 617 139 397 163 317 67 337 607 173 557 659 761 157 709 691 181 367 547 97 277 701 677 599 191 179 269 647 653 257 541 373 211 383 229 457 487 73 439 661 587 613 433 199 571 79 353 499

Each square shown above 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). Potential order 3 Simple Magic Square Inlays (unique) for the consecutive prime numbers (67 ... 797), resulting in 456 valid sets of four, are shown in Attachment 14.13.27. 14.13.28a Composed Magic Squares (11 x 11)           Order 5 and 6 Simple Magic Sub Squares When the Magic Sum s11 is a multiple of 11 (e.g. 9823), order 11 (Semi) Magic Squares might be composed of: One 6th order Magic Corner Square with Magic Sum s6 = 6 * s11 / 11 = 5358 (top/left) One 5th order Magic Corner Square with Magic Sum s5 = 5 * s11 / 11 = 4465 (bottom/right) Two Magic Rectangles order 5 x 6 with s5 = 4465 and s6 = 5358 An example is shown below for the consecutive prime numbers {487 ... 1303} with the related Magic Sums mentioned above.

Composed Semi Magic Square 1229 631 541 547 1223 1187 739 797 827 1009 1093 587 1217 661 1213 1109 571 929 937 787 751 1061 607 769 1193 947 1201 641 863 953 757 859 1033 1249 773 839 593 733 1171 761 877 1039 967 821 1087 677 887 907 569 1231 997 743 971 1063 691 599 1291 1237 1151 523 557 1069 1051 977 709 659 857 811 829 1031 701 1129 1303 617 727 521 1297 683 853 823 1021 881 1097 809 1301 563 1283 509 983 919 673 647 1117 1019 577 1103 883 911 991 653 719 1091 1123 1153 619 1277 503 1013 491 1181 1289 1163 1049 643 613 601 499 941 1279 1259 487

Composed Simple Magic Square 1229 631 541 547 1223 1187 929 937 787 751 1061 587 1217 661 1213 1109 571 761 877 1039 967 821 607 769 1193 947 1201 641 863 953 757 859 1033 1249 773 839 593 733 1171 997 743 971 1063 691 1087 677 887 907 569 1231 739 797 827 1009 1093 599 1291 1237 1151 523 557 1069 1051 977 709 659 983 919 673 647 1117 1019 1303 617 727 521 1297 653 719 1091 1123 1153 619 809 1301 563 1283 509 1289 1163 1049 643 613 601 577 1103 883 911 991 683 853 823 1021 881 1097 1277 503 1013 491 1181 857 811 829 1031 701 1129 499 941 1279 1259 487

The resulting Composed Magic Squares can be transformed to: Order 11 Bordered Magic Squares with order 5 Magic Center Square, or Order 11 Inlaid Magic squares with order 5 and 6 Magic Square Inlays as illustrated below:

Bordered Magic Square 1229 631 541 929 937 787 751 1061 547 1223 1187 587 1217 661 761 877 1039 967 821 1213 1109 571 607 769 1193 863 953 757 859 1033 947 1201 641 983 919 673 1303 617 727 521 1297 647 1117 1019 653 719 1091 809 1301 563 1283 509 1123 1153 619 1289 1163 1049 577 1103 883 911 991 643 613 601 683 853 823 1277 503 1013 491 1181 1021 881 1097 857 811 829 499 941 1279 1259 487 1031 701 1129 1249 773 839 997 743 971 1063 691 593 733 1171 1087 677 887 739 797 827 1009 1093 907 569 1231 599 1291 1237 1069 1051 977 709 659 1151 523 557

Inlaid Magic Square 1229 929 631 937 541 787 547 751 1223 1061 1187 983 1303 919 617 673 727 647 521 1117 1297 1019 587 761 1217 877 661 1039 1213 967 1109 821 571 653 809 719 1301 1091 563 1123 1283 1153 509 619 607 863 769 953 1193 757 947 859 1201 1033 641 1289 577 1163 1103 1049 883 643 911 613 991 601 1249 997 773 743 839 971 593 1063 733 691 1171 683 1277 853 503 823 1013 1021 491 881 1181 1097 1087 739 677 797 887 827 907 1009 569 1093 1231 857 499 811 941 829 1279 1031 1259 701 487 1129 599 1069 1291 1051 1237 977 1151 709 523 659 557

It should be noted that for Bordered Magic Squares the reversed transformation is not necessarily possible because of the bottom-left / top-right Main Diagonal. Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values. 14.13.28b Composed Magic Squares (11 x 11)           Order 4 and 7 Simple Magic Sub Squares When the Magic Sum s11 is a multiple of 11 (e.g. 9823), order 11 (Semi) Magic Squares might be composed of: One 7th order Bordered Magic Corner Square with Magic Sum s7 = 7 * s11 / 11 = 6251 (top/left) One 4th order Simple Magic Corner Square with Magic Sum s4 = 4 * s11 / 11 = 3572 (bottom/right) Two Magic Rectangles order 4 x 7 with s4 = 3572 and s7 = 6251 An example is shown below for the consecutive prime numbers {487 ... 1303} with the related Magic Sums mentioned above.

Composed Semi Magic Square 503 1277 1193 1109 1013 557 599 853 827 691 1201 1223 1031 761 997 643 1033 563 971 797 641 1163 1103 751 1039 887 719 1069 683 743 821 911 1097 857 907 769 757 1049 983 929 863 701 1061 947 809 839 1019 883 991 733 977 811 1051 1091 619 569 937 877 941 1063 647 1217 859 967 1129 617 1187 509 593 677 773 1229 1283 1151 1087 727 607 577 631 541 829 1181 1213 1279 739 659 1021 1153 1171 1117 919 613 601 571 1259 787 1009 653 1123 521 523 823 1249 1291 1297 547 953 673 1237 709 1303 1301 1289 881 499 491 487 1093 1231 661 587

Composed Simple Magic Square 503 1277 1193 1109 1013 557 599 853 827 691 1201 1223 1031 761 997 643 1033 563 971 797 641 1163 1103 751 1039 887 719 1069 683 743 821 911 1097 857 907 769 757 1049 983 929 863 701 1061 947 809 839 1019 883 991 733 977 811 1051 1091 619 569 937 877 941 1063 647 1217 859 967 1129 617 1187 509 593 677 773 1229 1283 1151 1087 727 607 541 631 1279 829 1181 1213 577 739 659 1021 1153 919 1117 1259 613 601 571 1171 787 1009 653 1123 823 523 547 1249 1291 1297 521 953 673 1237 709 1289 1301 487 881 499 491 1303 1093 1231 661 587

The resulting Composed Magic Squares can be transformed to order 11 Bordered Magic Squares with order 7 Bordered Magic Center Square, as illustrated below:

Bordered Magic Square 1237 709 823 523 547 1249 1291 1297 521 953 673 661 587 1289 1301 487 881 499 491 1303 1093 1231 691 1201 503 1277 1193 1109 1013 557 599 853 827 641 1163 1223 1031 761 997 643 1033 563 971 797 911 1097 1103 751 1039 887 719 1069 683 743 821 1061 947 857 907 769 757 1049 983 929 863 701 1091 619 809 839 1019 883 991 733 977 811 1051 1129 617 569 937 877 941 1063 647 1217 859 967 727 607 1187 509 593 677 773 1229 1283 1151 1087 1021 1153 541 631 1279 829 1181 1213 577 739 659 653 1123 919 1117 1259 613 601 571 1171 787 1009

It should be noted that for Bordered Magic Squares the reversed transformation is not necessarily possible because of the bottom-left / top-right Main Diagonal. Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values. Potential order 7 Border Inlays (unique) have been constructed, based on the regular pairs contained in the consecutive prime numbers {487 ... 1303} , with routine Prime1325b and are shown in Attachment 14.13.28b. 14.13.29 Inlaid Magic Squares (11 x 11)          Diamond Inlays Order 5 and 6 Prime Number Inlaid Magic Squares of order 11 with Diamond Inlays can be constructed when the Magic Sum s11 is a multiple of 11. An example is shown below, for the consecutive prime numbers {487 ... 1303} with the related Magic Sum s11 = 9823. Inlaid Magic Square 839 757 911 743 751 1103 887 1229 1063 821 719 947 691 1009 613 1259 1163 659 1123 811 787 761 919 1033 727 1237 1291 773 587 523 1049 827 857 1129 991 619 571 673 823 953 1231 499 1153 1181 1289 631 503 733 1223 599 541 941 997 1297 1069 509 937 1217 1171 977 677 809 1193 569 487 1277 653 1303 661 1213 907 1187 1117 739 617 863 563 929 797 1093 1109 643 601 1249 877 1301 647 577 709 769 1039 1279 557 1013 1283 593 859 1021 701 1019 883 983 607 491 1201 641 853 1087 1091 967 881 1031 1061 547 1051 683 1097 521 971 829 1151 The Inlaid Magic Square shown above contains: One 5th order Magic Diamond Inlay with Magic Sum s5 = 5 * s11 / 11 = 4465 One 6th order Magic Diamond Inlay with Magic Sum s6 = 6 * s11 / 11 = 5358 and corresponds with miscellaneous Prime Number Magic Squares with the same Magic Sum(s) and variable values. 14.13.30 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
-	-	-	-
11	Consecutive Primes, Simple Magic	CnstrSqrs11	Attachment 14.13.21
3	Square Inlays, Consecutive Primes (67 ... 797)	Prime1322	Attachment 14.13.22 Attachment 14.13.27
4	Square Inlays, Consecutive Primes (67 ... 797)	Prime1323	Attachment 14.13.23
5	Square Inlays, Consecutive Primes (67 ... 797)	Prime1324	Attachment 14.13.24
		Prime1325	Attachment 14.13.25
6	Square Inlays, Consecutive Primes (67 ... 797)	Prime1326	Attachment 14.13.26
7	Border Inlays, Consecutive Primes (67 ... 797) Border Inlays, Consecutive Primes (487 ... 1303}	Prime1325b	Attachment 14.13.25b Attachment 14.13.28b
-	-	-	-

Following sections will describe how Order 12 Prime Number Magic Squares with Consecutive Primes can be found with comparable routines as described in previous sections.

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