Office Applications and Entertainment, Magic Squares
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14.0     Special Magic Squares, Prime Numbers 14.13    Consecutive Primes (4) 14.13.31 Simple Magic Squares (12 x 12) Prime Number Simple Magic Squares of order 12 can be constructed with the Generator Principle, as applied in previous sections. Suitable Generators (12 Magic Series) can be constructed semi-automatically (ref. CnstrGen12). A possible Semi Magic Square and resulting Simple Magic Square are shown below, for the smallest consecutive prime numbers {89 ... 991} for which an Order 12 Simple Magic Square exists (MC12 = 6188):

Semi Magic Square 89 991 97 983 101 977 103 971 107 149 967 653 109 953 113 947 127 941 131 937 137 163 929 701 139 919 151 911 157 907 167 173 883 211 683 887 179 181 881 191 877 193 863 197 859 857 677 233 853 199 223 227 839 229 829 659 827 823 241 239 821 251 811 257 263 269 809 271 797 787 281 571 773 277 769 283 761 293 757 593 313 751 311 307 743 317 739 331 733 337 727 719 359 347 349 487 709 353 691 373 673 661 379 367 383 647 389 563 643 631 617 619 641 397 419 409 401 421 433 557 607 613 587 599 439 443 457 431 601 463 449 499 523 503 509 467 577 541 547 461 521 569 479 491

Simple Magic Square 89 97 983 101 967 971 107 149 103 653 977 991 139 151 911 157 683 173 883 211 167 887 907 919 179 881 191 877 677 197 859 857 863 233 193 181 853 223 227 839 241 659 827 823 829 239 229 199 773 769 283 761 311 593 313 751 757 307 293 277 743 739 331 733 349 719 359 347 727 487 337 317 607 587 599 439 449 431 601 463 457 499 443 613 643 617 619 641 433 409 401 421 419 557 397 631 821 811 257 263 281 271 797 787 809 571 269 251 709 691 373 673 389 367 383 647 379 563 661 353 523 509 467 577 479 461 521 569 547 491 541 503 109 113 947 127 929 937 137 163 131 701 941 953

The Generator Method has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.31. Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values. 14.13.32 Simple Magic Squares (12 x 12)          Single Square Inlays (Lower Order) Order 12 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 ... 991} with the related Magic Sum s12 = 6188 are shown below:

Order 3 Inlay, s3 = 807 97 983 673 991 103 971 127 977 89 967 101 109 163 389 191 227 911 181 773 179 887 461 907 919 883 107 269 431 881 197 821 193 863 499 877 167 173 311 347 149 857 211 827 199 839 563 853 859 829 229 823 233 811 241 691 239 797 263 809 223 647 733 307 727 313 719 293 317 353 709 331 739 641 397 631 401 619 613 601 617 383 457 409 419 503 571 467 541 487 523 577 521 479 509 547 463 701 683 349 677 359 373 643 661 659 379 367 337 787 257 769 271 277 757 251 761 283 751 281 743 607 599 421 593 439 449 433 587 443 491 569 557 157 929 941 947 131 953 151 937 113 139 137 653	Order 4 Inlay, Pan Magic, s4 = 1068 977 983 653 971 991 89 101 149 103 97 967 107 281 263 241 383 181 947 821 109 953 941 131 937 881 257 307 137 367 883 827 127 887 499 157 859 379 151 353 271 293 919 113 823 929 911 139 907 163 397 167 277 227 863 811 829 877 857 541 179 433 647 373 643 641 419 409 613 401 359 631 619 787 191 853 839 193 199 761 797 197 389 809 173 239 769 773 223 757 233 739 743 229 211 751 521 523 617 587 479 607 439 431 593 449 443 599 421 467 503 457 569 557 577 461 487 563 509 491 547 349 683 691 677 673 337 463 347 331 659 661 317 709 727 733 719 701 283 251 571 269 313 311 601
Order 5 Inlay, Associated, s5 = 1735 593 443 461 131 107 163 907 887 883 167 659 787 191 173 383 431 557 971 113 967 127 953 499 823 227 293 347 401 467 929 919 151 911 577 809 157 137 263 311 521 503 947 139 941 149 937 769 571 587 563 233 251 101 991 103 983 109 977 463 827 739 727 719 743 733 491 367 353 313 317 349 337 881 877 857 179 181 193 859 199 863 197 691 211 709 701 389 673 683 439 677 409 373 359 379 397 773 757 607 797 761 271 269 331 751 307 281 283 641 653 421 661 643 97 647 601 419 523 449 433 487 509 631 547 613 457 619 89 479 617 599 541 223 229 829 853 839 239 569 277 811 257 241 821	Order 5 with Diamond Inlay, s5 = 1735 131 251 557 569 227 163 907 179 883 887 821 613 587 101 431 173 443 947 139 937 941 149 769 571 383 401 347 293 311 991 103 109 983 977 463 827 167 521 263 593 191 971 113 967 127 953 499 823 467 461 137 107 563 929 151 911 919 157 809 577 617 653 677 673 643 641 449 457 97 433 439 409 761 757 787 751 773 271 661 277 317 283 269 281 839 233 853 811 229 239 797 829 257 241 313 547 733 727 743 359 739 331 719 349 491 353 307 337 193 877 181 863 881 197 859 199 211 857 647 223 691 683 709 397 701 419 659 367 421 389 373 379 619 523 503 599 487 89 631 607 541 509 479 601
Order 6 Concentric Inlay, s6 = 3312 113 953 941 937 241 127 839 229 829 239 277 463 823 881 227 907 193 281 857 199 211 257 499 853 821 487 613 461 647 283 179 181 883 233 859 541 421 197 911 223 877 683 137 919 173 887 191 569 157 643 457 617 491 947 109 967 131 929 601 139 977 151 163 167 863 991 103 983 107 971 149 563 593 479 599 587 89 547 557 607 509 523 577 521 313 769 311 317 773 331 761 337 757 349 751 419 653 101 659 571 449 439 431 467 673 443 641 661 353 97 347 743 359 367 727 631 719 373 739 733 263 811 251 269 827 809 797 271 787 307 503 293 701 619 709 389 379 383 691 397 409 677 401 433	Order 6, Concentric Pan Magic Inlay, s6 = 3378 953 317 617 263 659 569 251 191 907 181 397 883 179 587 491 881 293 947 409 929 151 157 937 227 353 593 797 479 383 773 229 421 919 163 167 911 887 419 827 149 857 239 941 439 131 967 193 139 449 653 137 743 719 677 197 443 109 977 971 113 557 809 509 863 467 173 127 499 103 991 107 983 563 101 571 541 577 641 599 631 643 619 613 89 751 359 307 311 337 331 739 347 727 757 461 761 199 859 223 839 877 211 829 277 547 241 853 233 257 271 821 313 269 823 503 787 811 283 281 769 661 487 521 97 401 431 673 523 683 463 601 647 379 733 367 709 349 373 691 701 457 389 607 433
Order 7 Concentric Inlay, s7 = 3801 103 937 929 907 677 139 109 173 823 163 401 827 859 967 317 367 181 883 227 911 131 881 193 271 853 307 863 439 829 277 233 137 839 151 877 383 643 577 563 433 619 523 443 953 113 941 283 97 199 607 313 757 229 809 887 991 127 971 107 191 167 257 659 719 857 223 919 811 197 211 347 821 977 149 157 179 409 947 983 101 797 239 787 463 769 773 269 241 263 251 281 353 761 751 733 743 449 431 487 613 467 617 491 461 457 499 647 569 701 331 709 293 739 311 349 337 691 359 641 727 379 373 419 683 397 661 673 389 653 421 631 509 89 479 503 557 521 547 593 571 599 601 541 587	Order 7 with Diamond Inlays, s7 = 3213 139 421 941 983 389 227 113 829 811 167 701 467 193 953 367 311 109 967 313 181 797 761 787 449 107 179 173 523 593 977 661 157 839 853 809 317 643 557 863 347 563 137 103 857 151 859 281 827 823 677 101 223 521 719 149 881 127 877 269 821 379 163 409 383 907 89 883 773 769 191 491 751 929 263 359 443 131 97 991 757 197 743 709 569 431 617 457 971 463 613 607 487 353 397 419 373 739 919 887 199 733 211 911 233 647 241 229 239 461 547 503 509 577 541 599 439 601 499 479 433 691 251 727 947 571 937 271 257 277 293 683 283 653 641 401 349 631 673 587 337 619 307 331 659

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 above 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. 14.13.33 Simple Magic Squares (12 x 12)          Order 3 Square Inlays (4 ea) Comparable with the method discussed in Section 14.13.27 it is possible to construct Order 12 Simple Magic Squares with four order 3 Simple Magic Square Inlays. Examples of subject Inlaid Magic Squares for the consecutive prime numbers {89 ... 991} with the related Magic Sum s12 = 6188 are shown below:

Simple Magic Type 1, Order 3 Inlays (4 ea) 389 167 233 577 211 313 823 431 991 967 977 109 107 263 419 103 367 631 953 547 127 919 811 941 293 359 137 421 523 157 971 457 113 947 983 827 761 317 449 751 463 409 149 907 911 151 727 193 197 509 821 199 541 883 937 131 163 929 139 739 569 701 257 673 619 331 227 881 887 173 179 691 281 283 773 769 743 757 397 311 307 487 347 733 719 677 709 433 647 353 461 683 373 401 349 383 863 859 877 181 191 857 239 479 853 229 337 223 839 829 251 809 797 269 439 277 379 787 241 271 607 653 661 659 443 587 503 467 491 97 499 521 563 571 601 613 643 641 89 617 593 101 599 557

Simple Magic Type 2, Order 3 Inlays (4 ea) 389 167 233 149 227 971 977 983 991 577 211 313 107 263 419 109 937 907 887 547 911 103 367 631 293 359 137 691 881 823 811 941 151 421 523 157 773 311 743 397 281 347 283 307 733 487 769 757 719 677 709 461 683 349 353 373 383 647 433 401 863 857 479 181 239 853 877 191 223 229 337 859 251 797 809 269 241 439 787 379 271 839 829 277 653 659 491 587 521 467 97 499 443 661 503 607 613 571 641 617 89 599 643 101 557 601 593 563 761 317 449 827 431 193 163 967 457 751 463 409 197 509 821 947 919 127 179 727 139 199 541 883 569 701 257 953 739 113 131 173 929 673 619 331

s3 789 1101 1527 1623

Miscellaneous (main) diagonal sets are possible for both square types. Type 1 (left), each resulting square corresponds with 84 = 4096 solutions, which can be obtained by selecting other aspects of the four inlays. Type 2 (right), 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 or 3 positions). Potential order 3 Simple Magic Square Inlays (unique) for the consecutive prime numbers (89 ... 991), resulting in 12928 valid sets of four, are shown in Attachment 14.13.33. 14.13.34 Simple Magic Squares (12 x 12)          Order 4 Pan Magic Square Inlays (4 ea) Comparable with the method discussed in Section 14.13.27 it is possible to construct Order 12 Simple Magic Squares with four order 4 Pan Magic Square Inlays. Examples of subject Inlaid Magic Squares for the consecutive prime numbers {89 ... 991} with the related Magic Sum s12 = 6188 are shown below:

Simple Magic Type 1, Order 4 Inlays (4 ea) 293 283 569 787 379 409 463 829 137 977 863 199 499 857 223 353 433 859 349 439 127 983 239 827 397 179 673 683 577 211 661 631 113 991 251 821 743 613 467 109 691 601 607 181 139 967 313 757 311 457 509 823 149 401 617 953 947 151 677 193 479 853 281 487 599 971 131 419 941 157 523 347 541 227 739 593 443 107 911 659 937 167 701 163 769 563 571 197 929 641 461 89 173 229 919 647 421 761 359 719 367 373 751 733 727 491 103 383 557 389 101 449 503 547 643 709 587 431 619 653 271 773 809 797 877 811 331 277 521 307 97 317 907 233 887 191 241 257 263 269 839 337 883 881

Simple Magic Type 2, Order 4 Inlays (4 ea) 227 647 307 787 911 181 433 631 263 523 439 839 211 883 131 743 277 907 659 313 349 929 173 613 677 197 757 337 881 89 389 797 593 193 769 509 853 241 773 101 571 829 557 199 859 419 683 103 109 113 127 137 251 331 607 733 967 953 941 919 257 383 709 353 601 863 373 727 691 317 271 643 823 701 367 719 409 487 569 421 239 617 673 163 991 983 977 971 491 229 347 449 139 149 151 311 457 443 499 641 461 191 809 599 167 751 283 887 401 739 359 541 653 397 547 463 223 947 107 811 521 379 563 577 179 827 467 587 761 157 877 293 661 479 619 281 503 857 431 269 937 233 821 97

s4 1932 2080 2100 2120

s4 1968 2064 2040 2088

Miscellaneous (main) diagonal sets are possible for both square types. Type 1 (left), each resulting square corresponds with 3844 = 21.743.271.936 solutions, which can be obtained by selecting other aspects of the four inlays. Type 2 (right), 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 ... 991), resulting in numerous valid sets of four, are shown in Attachment 14.13.34. 14.13.35 Simple Magic Squares (12 x 12)          Order 3 Semi Magic Square Inlays (10 ea) It has been investigated whether Prime Number Magic Squares composed of 16 order 3 (Semi) Magic Squares with different Magic Sums, as discussed in Section 14.21.5, can be constructed for Consecutive Prime Numbers. Up to now the best result is a Prime Number Magic Square with ten order 3 Semi Magic Square Inlays as shown below for the consecutive prime numbers {89 ... 991} with the related Magic Sum s12 = 6188.

Simple Magic, Order 3 Inlays (10 ea) 653 701 199 269 389 439 577 733 683 103 613 829 281 409 863 317 239 241 823 709 761 919 193 433 619 443 491 953 911 859 151 109 107 523 739 283 113 127 157 593 821 149 101 677 757 719 983 991 137 163 673 353 383 827 967 311 257 947 607 563 277 809 787 617 359 587 467 547 521 307 743 167 751 977 601 97 641 811 661 479 401 197 349 223 857 773 211 853 139 557 419 431 691 569 179 509 941 227 647 599 769 181 461 631 449 373 251 659 89 541 929 907 397 233 727 643 191 839 463 229 971 131 457 337 263 937 331 347 883 379 271 881 499 887 173 293 877 367 503 571 487 313 797 421

s3 1553 s12 s13 1545 s21 1563 1535 s24 s31 1549 1541 s34 1559 1537 1561 1531

The applied order 3 Semi Magic Square Inlays (7 Magic Lines) have been selected from a collection of potential order 3 Semi Magic Squares containing: 66530 Semi Magic Sqauers (unique) with  1068 different Magic Sums as generated with routine SemiMgc37, based on the consecutive prime numbers (89 ... 991) and summarised in the graph below: The resulting Square shown above corresponds with 1210 * (3!)4 * (3!)3 = 1,733 1016 solutions, which can be obtained by selecting other aspects of the ten inlays and variation of the remaining parts. 14.13.36 Composed Magic Squares (12 x 12)          Order 6 Semi Magic Sub Squares (4 ea) Order 12 (Semi) Magic Squares might be composed of order 6 (Semi) Magic Sub Squares when the Magic Sum s12 is a multiple of 2. An example is shown below for the consecutive prime numbers {107 ... 1019} with the related Magic Sums s12 = 6492 and s6 = 3246.

Composed Semi Magic Square 331 643 173 937 149 1013 607 397 193 883 157 1009 181 223 947 179 857 859 971 853 163 151 977 131 137 139 863 983 997 127 1019 991 907 113 109 107 367 569 461 709 313 827 601 389 359 701 373 823 347 433 503 487 719 757 739 691 463 317 743 293 311 353 727 751 821 283 881 811 733 281 277 263 587 509 431 563 479 677 419 547 499 599 523 659 541 443 593 467 571 631 577 617 421 521 619 491 439 457 641 647 653 409 683 673 661 449 401 379 307 557 337 839 239 967 613 383 271 809 241 929 269 349 829 257 769 773 877 787 251 233 887 211 227 229 761 911 919 199 953 941 797 197 191 167

Composed Simple Magic Square 331 643 173 937 149 1013 607 397 193 883 157 1009 181 223 947 179 857 859 971 853 163 151 977 131 137 139 863 983 997 127 1019 991 907 113 109 107 367 569 461 709 313 827 601 389 359 701 373 823 347 433 503 487 719 757 739 691 463 317 743 293 311 353 727 751 821 283 881 811 733 281 277 263 677 431 563 587 509 479 659 499 599 419 547 523 631 593 467 541 443 571 491 421 521 577 617 619 409 641 647 439 457 653 379 661 449 683 673 401 557 839 307 967 239 337 383 809 613 929 241 271 349 257 269 773 769 829 787 233 877 211 887 251 229 911 227 199 919 761 941 197 953 167 191 797

The square left is a Semi Magic Square composed of Semi Magic Sub Squares. The Simple Magic Square right - also composed of Semi Magic Sub Squsres - is obtained by row and column permutations within the sub squares. The Composed Simple Magic Square (right) corresponds with 4 * (6!)4 = 1,075 1012 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:

Composed Simple Magic Square 331 643 173 937 149 1013 367 569 461 709 313 827 607 397 193 883 157 1009 601 389 359 701 373 823 181 223 947 179 857 859 347 433 503 487 719 757 971 853 163 151 977 131 739 691 463 317 743 293 137 139 863 983 997 127 311 353 727 751 821 283 1019 991 907 113 109 107 881 811 733 281 277 263 677 431 563 587 509 479 557 839 307 967 239 337 659 499 599 419 547 523 383 809 613 929 241 271 631 593 467 541 443 571 349 257 269 773 769 829 491 421 521 577 617 619 787 233 877 211 887 251 409 641 647 439 457 653 229 911 227 199 919 761 379 661 449 683 673 401 941 197 953 167 191 797

Four Way V Type ZigZag Magic Square 331 367 643 569 173 461 937 709 149 313 1013 827 677 557 431 839 563 307 587 967 509 239 479 337 607 601 397 389 193 359 883 701 157 373 1009 823 659 383 499 809 599 613 419 929 547 241 523 271 181 347 223 433 947 503 179 487 857 719 859 757 631 349 593 257 467 269 541 773 443 769 571 829 971 739 853 691 163 463 151 317 977 743 131 293 491 787 421 233 521 877 577 211 617 887 619 251 137 311 139 353 863 727 983 751 997 821 127 283 409 229 641 911 647 227 439 199 457 919 653 761 1019 881 991 811 907 733 113 281 109 277 107 263 379 941 661 197 449 953 683 167 673 191 401 797

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values. 14.13.37 Composed Magic Squares (12 x 12)          Order 5 and 7 Magic Sub Squares When the Magic Sum s12 is a multiple of 12 (e.g. 6492), order 12 (Semi) Magic Squares might be composed of: One 7th order Simple Magic Corner Square with Magic Sum s7 = 7 * s12 / 12 = 3787 (top/left) One 5th order Simple Magic Corner Square with Magic Sum s5 = 5 * s12 / 12 = 2705 (bottom/right) Two Magic Rectangles order 5 x 7 with s5 = 2705 and s7 = 3787 An example is shown below for the consecutive prime numbers {107 ... 1019} with the related Magic Sums mentioned above.

Composed Semi Magic Square 179 947 607 431 317 797 509 313 353 827 229 983 859 563 569 173 419 227 977 373 359 811 233 929 113 151 881 1019 887 599 137 293 257 691 557 907 1013 503 107 541 109 523 991 331 727 421 829 397 653 449 521 997 571 163 433 757 647 251 839 211 269 401 953 277 661 769 457 809 587 239 877 193 701 773 149 349 823 709 283 911 857 547 223 167 383 367 461 683 641 491 761 1009 181 241 307 967 379 499 677 389 487 617 739 439 971 197 941 157 659 643 337 347 577 593 631 199 743 467 443 853 463 409 479 673 719 733 311 919 191 863 131 601 821 787 751 613 281 271 263 139 619 937 883 127

Composed Simple Magic Square 179 947 607 431 317 797 509 313 353 827 229 983 859 563 569 173 419 227 977 373 359 811 233 929 113 151 881 1019 887 599 137 293 257 691 557 907 1013 503 107 541 109 523 991 331 727 421 829 397 653 449 521 997 571 163 433 757 647 251 839 211 269 401 953 277 661 769 457 809 587 239 877 193 701 773 149 349 823 709 283 911 857 547 223 167 491 383 683 641 367 761 461 1009 181 241 307 967 617 379 389 487 499 739 677 439 971 197 941 157 593 659 347 577 643 631 337 199 743 467 443 853 733 463 673 719 409 311 479 919 191 863 131 601 271 821 613 281 787 263 751 139 619 937 883 127

The order 7 Simple Magic Sub Square applied above is composed of: Two each order 4 Simple Magic Squares with Magic Sum s4 = 2164 and Two each order 3 Semi   Magic Squares with Magic Sum s3 = 1623 Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values. 14.13.38 Composed Magic Squares (12 x 12)          Order 4 and 8 Magic Sub Squares When the Magic Sum s12 is a multiple of 12 (e.g. 6492), order 12 (Semi) Magic Squares might be composed of: One 8th order Simple Magic Corner Square with Magic Sum s8 = 8 * s12 / 12 = 4328 (top/left) One 4th order Simple Magic Corner Square with Magic Sum s4 = 4 * s12 / 12 = 2164 (bottom/right) Two Magic Rectangles order 4 x 8 with s4 = 2164 and s8 = 4328 An example is shown below for the consecutive prime numbers {107 ... 1019} with the related Magic Sums mentioned above.

Composed Semi Magic Square 337 619 421 787 137 881 193 953 367 313 757 727 241 811 229 883 149 907 131 977 863 503 419 379 709 271 853 331 911 113 1013 127 277 347 701 839 877 463 661 163 967 263 827 107 467 719 479 499 577 433 631 523 311 739 257 857 569 599 509 487 439 673 643 409 197 797 151 1019 571 587 617 389 457 307 541 859 769 269 947 179 601 653 449 461 691 751 349 373 887 359 809 109 613 607 397 547 191 1009 167 181 211 659 919 991 443 677 283 761 173 683 227 937 929 733 353 293 157 641 383 983 971 239 997 223 593 251 491 563 743 199 941 281 829 233 773 823 431 521 401 317 821 647 557 139

Composed Simple Magic Square 337 619 421 787 137 881 193 953 367 313 757 727 241 811 229 883 149 907 131 977 863 503 419 379 709 271 853 331 911 113 1013 127 277 347 701 839 877 463 661 163 967 263 827 107 467 719 479 499 577 433 631 523 311 739 257 857 569 599 509 487 439 673 643 409 197 797 151 1019 571 587 617 389 457 307 541 859 769 269 947 179 601 653 449 461 691 751 349 373 887 359 809 109 613 607 397 547 181 211 991 659 919 191 1009 167 443 677 283 761 937 929 293 733 353 173 683 227 157 641 383 983 223 593 563 251 491 971 239 997 743 199 941 281 823 431 317 521 401 829 233 773 821 647 557 139

The resulting Composed Magic Squares can be transformed to: Order 12 Bordered Magic Squares with order 4 Magic Center Square, or Order 12 Bordered Magic Squares with order 8 Magic Center Square as illustrated below:

Bordered Magic Square, Type 1 337 619 421 787 367 313 757 727 137 881 193 953 241 811 229 883 863 503 419 379 149 907 131 977 709 271 853 331 277 347 701 839 911 113 1013 127 877 463 661 163 467 719 479 499 967 263 827 107 191 1009 167 181 443 677 283 761 211 659 919 991 173 683 227 937 157 641 383 983 929 733 353 293 971 239 997 223 743 199 941 281 593 251 491 563 829 233 773 823 821 647 557 139 431 521 401 317 577 433 631 523 569 599 509 487 311 739 257 857 439 673 643 409 571 587 617 389 197 797 151 1019 457 307 541 859 601 653 449 461 769 269 947 179 691 751 349 373 613 607 397 547 887 359 809 109

Bordered Magic Square, Type 2 941 281 223 593 563 251 491 971 239 997 743 199 557 139 823 431 317 521 401 829 233 773 821 647 757 727 337 619 421 787 137 881 193 953 367 313 419 379 241 811 229 883 149 907 131 977 863 503 701 839 709 271 853 331 911 113 1013 127 277 347 479 499 877 463 661 163 967 263 827 107 467 719 509 487 577 433 631 523 311 739 257 857 569 599 617 389 439 673 643 409 197 797 151 1019 571 587 449 461 457 307 541 859 769 269 947 179 601 653 397 547 691 751 349 373 887 359 809 109 613 607 283 761 181 211 991 659 919 191 1009 167 443 677 383 983 937 929 293 733 353 173 683 227 157 641

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. The order 8 Simple Magic Sub Square applied above is composed of four each order 4 Simple Magic Squares with Magic Sum s4 = 2164. Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values. 14.13.39 Composed Magic Squares (12 x 12)          Order 3 and 9 Magic Sub Squares When the Magic Sum s12 is a multiple of 12 (e.g. 6492), order 12 (Semi) Magic Squares might be composed of: One 9th order Simple Magic Corner Square with Magic Sum s9 = 9 * s12 / 12 = 4869 (top/left) One 3th order Simple Magic Corner Square with Magic Sum s3 = 3 * s12 / 12 = 1623 (bottom/right) Two Magic Rectangles order 3 x 9 with s3 = 1623 and s9 = 4869 An example is shown below for the consecutive prime numbers {107 ... 1019} with the related Magic Sums mentioned above.

Composed Semi Magic Square 947 907 929 179 167 631 829 173 107 109 991 523 421 659 431 617 457 439 641 557 647 1019 113 491 349 313 359 733 739 569 691 373 743 127 1009 487 449 593 479 587 499 563 577 521 601 131 809 683 863 227 857 239 811 571 229 839 233 137 977 509 419 709 383 701 467 389 677 397 727 353 317 953 293 311 283 769 761 719 307 653 773 983 197 443 241 269 271 821 787 797 257 599 827 997 193 433 887 881 877 223 181 191 661 757 211 1013 263 347 139 971 151 157 163 461 919 941 967 751 199 673 937 149 859 823 853 251 337 281 379 463 541 619 547 503 613 643 607 911 367 401 277 409 883 331

Composed Simple Magic Square 947 907 929 179 167 631 829 173 107 109 991 523 421 659 431 617 457 439 641 557 647 1019 113 491 349 313 359 733 739 569 691 373 743 137 977 509 449 593 479 587 499 563 577 521 601 127 1009 487 863 227 857 239 811 571 229 839 233 131 809 683 419 709 383 701 467 389 677 397 727 353 317 953 293 311 283 769 761 719 307 653 773 983 197 443 241 269 271 821 787 797 257 599 827 997 193 433 887 881 877 223 181 191 661 757 211 1013 263 347 461 157 139 971 151 163 919 941 967 751 199 673 251 823 937 149 859 853 337 281 379 463 541 619 911 643 547 503 613 607 367 401 277 409 883 331

The Order 9 Simple Magic Sub Square (s9 = 4869), constructed with the Generator Principle as applied in previous sections, corresponds with 8 * 24/2 * (4!) = 8 * 192 = 1536 Sub Squares. The resulting square corresponds consequently with 8 * 1536 * (3! * 6!)2 = 2,293 1011 Prime Number Magic Squares with the same Magic Sum(s) and variable values. 14.13.40 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
-	-	-	-
12	Consecutive Primes, Simple Magic	CnstrSqrs12	Attachment 14.13.31
3	Square Inlays, Consecutive Primes (89 ... 991)	Prime1333	Attachment 14.13.33
3	Square Inlays, Consecutive Primes (89 ... 991)	SemiMgc37	66530
4	Square Inlays, Consecutive Primes (89 ... 991)	Prime1334	Attachment 14.13.34
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Following sections will describe how Order 13 Prime Number Magic Squares with Consecutive Primes can be found with comparable routines as described in previous sections.

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