Office Applications and Entertainment, Magic Squares

14.0     Special Magic Squares, Prime Numbers

14.13    Consecutive Primes (2)

14.13.10 Simple Magic Squares (10 x 10)

Prime Number Simple Magic Squares of order 10 can be constructed with the Generator Principle, as applied in previous sections.

However, due to the vast amount of possible Magic Series for 100 Consecutive Prime Numbers, only partial collections can be considered.

Suitable Generators (10 Magic Series) can be constructed semi-automatically (ref. CnstrGen10a), although also Broken Series might be applied as illustrated in following example.

A possible Semi Magic Square and resulting Simple Magic Square are shown below, for the smallest consecutive prime numbers {23 ... 593} for which an Order 10 Simple Magic Square exists (MC10 = 2862):

Semi Magic Square 137 113 233 449 499 73 197 401 239 521 149 131 227 463 461 97 503 389 353 89 167 251 443 307 263 379 331 61 347 313 487 457 271 109 107 509 83 269 173 397 491 479 257 103 101 373 317 311 359 71 281 53 199 431 467 41 181 593 569 47 211 241 193 367 419 67 571 587 43 163 151 283 337 277 383 577 79 191 37 547 349 421 293 229 139 523 541 31 179 157 439 433 409 127 23 223 59 29 563 557

Simple Magic Square 137 113 233 449 499 521 239 197 73 401 149 131 227 463 461 89 353 503 97 389 167 251 443 307 263 313 347 331 379 61 487 457 271 109 107 397 173 83 509 269 491 479 257 103 101 71 359 317 373 311 151 283 337 277 383 547 37 79 577 191 281 53 199 431 467 47 569 181 41 593 211 241 193 367 419 163 43 571 67 587 439 433 409 127 23 557 563 59 223 29 349 421 293 229 139 157 179 541 523 31

The Generator Method has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.9.

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

14.13.11 Simple Magic Squares (10 x 10)
         Order 3 Square Inlay

Order 10 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 {41 ... 613} with the related Magic Sums
s10 = 3092 and s3 = 681 is shown below:

Generator, Order 3 Inlay 317 173 191 53 59 61 419 599 607 613 101 227 353 601 593 563 443 73 71 67 263 281 137 79 83 89 433 569 571 587 577 557 547 523 359 113 109 107 103 97 127 131 139 149 151 331 499 503 521 541 509 487 479 463 307 181 179 167 163 157 193 197 199 211 223 229 421 461 467 491 449 439 431 283 269 257 251 241 239 233 41 409 47 397 277 383 311 373 337 349 347 367 313 379 293 389 271 401 43 457

Semi Magic, Order 3 Inlay 317 173 191 53 59 599 419 607 613 61 101 227 353 601 593 443 71 67 563 73 263 281 137 79 83 569 433 571 89 587 577 557 547 523 359 109 113 103 97 107 499 127 139 149 503 331 151 131 521 541 179 479 487 463 307 157 509 167 163 181 491 211 199 197 229 223 461 421 193 467 251 449 431 283 257 239 233 241 439 269 47 311 337 397 409 379 313 401 41 457 367 277 271 347 293 43 389 383 373 349

Simple Magic, Order 3 Inlay 523 577 557 547 107 103 109 113 359 97 53 317 173 191 61 607 599 419 59 613 601 101 227 353 73 67 443 71 593 563 79 263 281 137 587 571 569 433 83 89 197 491 211 199 467 421 223 461 229 193 397 47 311 337 457 401 379 313 409 41 149 499 127 139 541 131 331 151 503 521 283 251 449 431 269 241 239 233 257 439 347 367 277 271 349 383 43 389 293 373 463 179 479 487 181 167 157 509 307 163

The Generator Method, as applied for Inlaid Magic Squares based on Consecutive Prime Numbers can be summarised as follows:

• The Generator (8 Magic Series, 2 Non Magic Series) can be constructed semi-automatically (ref, CnstrGen10b);
• The Non Magic Series have been corrected by permutating the numbers within the last two rows;
• The Semi Magic Square has been 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. CnstrSqrs10).

Potential order 3 Square Inlays (21 unique) have been constructed for the consecutive prime numbers (41 ... 613) with routine Prime1311 and are shown in Attachment 14.13.11.

The order 3 Square Inlay has been moved (one position) along the Main Diagonal (top/left to bottom/right) by means of row and column permutations.

14.13.12 Simple Magic Squares (10 x 10)
         Order 4 Pan Magic Square Inlay

Order 10 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 {23 ... 593} with the related Magic Sums
s10 = 2862 and s4 = 1056 is shown below:

Generator, Order 4 Inlay 79 89 431 457 41 43 47 541 547 587 389 499 37 131 593 577 463 61 59 53 97 71 449 439 67 73 83 443 569 571 491 397 139 29 563 523 409 107 103 101 109 113 127 137 149 167 479 503 521 557 509 487 461 401 181 179 173 163 157 151 191 193 197 199 211 223 379 383 419 467 433 421 331 257 251 241 239 233 229 227 23 31 263 269 271 277 281 283 293 307 373 367 359 353 349 347 337 317 313 311

Semi Magic, Order 4 Inlay 79 89 431 457 47 41 547 43 541 587 389 499 37 131 593 577 463 59 61 53 97 71 449 439 571 67 83 569 73 443 491 397 139 29 101 523 409 563 103 107 127 137 167 149 113 503 109 479 557 521 487 461 509 401 179 157 173 163 181 151 383 191 223 193 379 467 211 199 419 197 433 421 229 331 251 227 257 239 241 233 353 283 311 373 347 31 293 271 337 263 23 313 367 359 281 269 317 277 349 307

Simple Magic, Order 4 Inlay 233 433 421 229 331 239 257 251 227 241 587 79 89 431 457 43 547 47 41 541 53 389 499 37 131 59 463 593 577 61 443 97 71 449 439 569 83 571 67 73 107 491 397 139 29 563 409 101 523 103 151 487 461 509 401 163 173 179 157 181 197 383 191 223 193 199 211 379 467 419 263 353 283 311 373 271 293 347 31 337 521 127 137 167 149 479 109 113 503 557 307 23 313 367 359 277 317 281 269 349

The Generator Method, as applied for Inlaid Magic Squares based on Consecutive Prime Numbers is described in Section 14.13.11 above.

Potential order 4 Pan Magic Square Inlays have been constructed for the consecutive prime numbers (23 ... 593) with routine Prime1312 and are shown in Attachment 14.13.12 (one square per occurring magic sum).

The order 4 Pan Magic Square Inlay has been moved (one position) along the Main Diagonal (top/left to bottom/right) by means of row and column permutations.

14.13.13 Simple Magic Squares (10 x 10)
         Order 5 Pan Magic Square Inlay

Order 10 Simple Magic Squares with order 5 Pan Magic Square Inlay(s) can be constructed with the generator method.

An example of the construction for the consecutive prime numbers {23 ... 593} with the related Magic Sums
s10 = 2862 and s5 = 1129 is shown below:

Generator, Order 5 Inlay 23 67 211 389 439 43 47 463 587 593 241 419 331 59 79 53 61 479 563 577 367 71 109 271 311 73 83 449 557 571 139 163 347 379 101 89 97 431 547 569 359 409 131 31 199 103 107 461 521 541 113 127 137 149 151 167 487 499 509 523 157 173 179 181 191 193 337 457 491 503 197 223 227 229 233 239 251 353 443 467 307 37 317 257 263 269 277 281 421 433 29 313 41 349 373 383 397 401 283 293

Semi Magic, Order 5 Inlay 23 67 211 389 439 43 47 463 587 593 241 419 331 59 79 61 563 577 53 479 367 71 109 271 311 449 73 83 557 571 139 163 347 379 101 547 569 431 89 97 359 409 131 31 199 541 103 461 521 107 523 509 499 487 167 137 151 37 113 239 157 181 191 337 503 491 197 179 349 277 443 467 353 251 233 283 281 229 173 149 317 263 307 257 433 41 421 373 227 223 293 313 383 401 397 269 457 29 193 127

Simple Magic, Order 5 Inlay 23 67 211 389 439 107 541 521 461 103 241 419 331 59 79 571 449 557 83 73 367 71 109 271 311 479 61 53 577 563 139 163 347 379 101 97 547 89 431 569 359 409 131 31 199 593 43 587 463 47 523 487 167 499 509 239 137 113 37 151 157 337 503 191 181 277 491 349 179 197 443 251 233 353 467 149 283 173 229 281 317 257 433 307 263 223 41 227 373 421 293 401 397 383 313 127 269 193 29 457

The construction method is as described in Section 14.13.11 above, with exception of the last step (main diagonals):

• The top/left  to bottom/right diagonal should be completed by permutating
  the columns of the right 5 x 10 rectangle
• 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 Pan Magic Square Inlays for the consecutive prime numbers (23 ... 593) have been based on La Hirian Primaries as described in Section 14.12.2.

A few of such Pan Magic Squares are shown in Attachment 14.13.13 (one square per occurring magic sum).

14.13.14 Simple Magic Squares (10 x 10)
         Order 5 Magic Square with Diamond Inlay

Alternatively it is possible to construct Order 10 Simple Magic Squares with order 5 Magic Squares with Diamond Inlay(s).

An example of the construction for the consecutive prime numbers {23 ... 593} with the related Magic Sums s10 = 2862 and s5 = 1255 is shown below:

Semi Magic, Order 5 Inlay 311 47 461 173 263 43 61 347 563 593 257 23 449 59 467 587 569 313 71 67 269 389 251 113 233 73 79 307 571 577 179 443 53 479 101 557 547 331 89 83 239 353 41 431 191 97 103 367 499 541 107 521 137 487 151 439 433 277 281 29 109 491 139 421 157 199 223 373 349 401 359 337 409 383 379 241 293 37 227 197 509 131 419 167 457 397 271 317 31 163 523 127 503 149 463 229 283 193 181 211

Simple Magic, Order 5 Inlay 311 47 461 173 263 103 541 499 367 97 257 23 449 59 467 79 577 571 307 73 269 389 251 113 233 61 593 563 347 43 179 443 53 479 101 569 67 71 313 587 239 353 41 431 191 547 83 89 331 557 137 521 487 151 107 433 29 281 277 439 139 491 421 157 109 223 401 349 373 199 409 337 383 379 359 293 197 227 37 241 419 131 167 457 509 271 163 31 317 397 503 127 149 463 523 283 211 181 193 229

Potential order 5 Magic Squares with Diamond Inlays have been constructed for the consecutive prime numbers (23 ... 593) with routine Prime1313 and are shown in Attachment 14.13.15 (one square per occurring magic sum).

14.13.15 Simple Magic Squares (10 x 10)
         Order 6 Ultra Magic Square Inlay

Order 10 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 {23 ... 593} with the related Magic Sums
s10 = 2862 and s6 = 1638 is shown below:

Semi Magic, Order 6 Inlay 353 419 263 229 307 67 23 31 577 593 149 179 89 349 463 409 587 571 37 29 173 449 383 157 277 199 41 73 541 569 347 269 389 163 97 373 563 557 61 43 137 83 197 457 367 397 47 107 523 547 479 239 317 283 127 193 521 499 151 53 59 509 71 461 101 379 401 251 337 293 211 503 223 443 271 433 227 181 113 257 467 109 439 241 431 131 313 233 331 167 487 103 491 79 421 281 139 359 191 311

Simple Magic, Order 6 Inlay 353 419 263 229 307 67 23 31 577 593 149 179 89 349 463 409 587 571 37 29 173 449 383 157 277 199 563 557 61 43 347 269 389 163 97 373 41 73 541 569 137 83 197 457 367 397 47 107 523 547 479 239 317 283 127 193 521 499 151 53 59 509 461 71 101 379 401 251 337 293 211 503 443 223 271 433 227 181 113 257 467 109 241 439 431 131 313 233 331 167 487 103 79 491 421 281 139 359 191 311

The construction method is as described in Section 14.13.11 above, with exception of the last step (main diagonals):

• The top/left  to bottom/right diagonal should be completed by permutating
  the columns of the right 4 x 10 rectangle (if required)
• The top/right to bottom/left  diagonal should be completed by permutating
  the rows    of the top/right   4 x 6 corner and
  the columns of the bottom/left 6 x 4 corner

Potential order 6 Ultra Magic Square Inlays have been constructed for the consecutive prime numbers (23 ... 593) with routine Prime1314 and are shown in Attachment 14.13.14 (one square per occurring magic sum).

14.13.16 Simple Magic Squares (10 x 10)
         Order 3 Square Inlays (4 ea)

Comparable with the above it is possible to construct Order 10 Simple Magic Squares with four order 3 Simple Magic Square Inlays.

An example of subject construction for the consecutive prime numbers {23 ... 593} with the related Magic Sums s10 = 2862 is shown below:

Semi Magic, Order 3 Inlays (4 ea) 313 109 157 439 163 211 23 283 577 587 37 193 349 43 271 499 571 569 263 67 229 277 73 331 379 103 449 257 367 397 491 83 269 557 173 311 223 167 199 389 59 281 503 101 347 593 181 233 337 227 293 479 71 383 521 137 197 139 241 401 29 547 31 487 47 463 373 251 191 443 307 523 433 419 79 359 149 353 127 113 541 317 467 61 421 97 239 179 409 131 563 53 509 41 461 89 457 431 151 107

Simple Magic 1, Order 3 Inlays (4 ea) 373 251 29 547 31 487 47 463 191 443 149 353 307 523 433 419 79 359 127 113 23 283 313 109 157 439 163 211 577 587 571 569 37 193 349 43 271 499 263 67 449 257 229 277 73 331 379 103 367 397 223 167 491 83 269 557 173 311 199 389 181 233 59 281 503 101 347 593 337 227 197 139 293 479 71 383 521 137 241 401 239 179 541 317 467 61 421 97 409 131 457 431 563 53 509 41 461 89 151 107

s3 579 813 843 1041

Simple Magic 2, Order 3 Inlays (4 ea) 373 29 547 31 251 191 487 47 463 443 23 313 109 157 283 577 439 163 211 587 571 37 193 349 569 263 43 271 499 67 449 229 277 73 257 367 331 379 103 397 149 307 523 433 353 127 419 79 359 113 239 541 317 467 179 409 61 421 97 131 223 491 83 269 167 199 557 173 311 389 181 59 281 503 233 337 101 347 593 227 197 293 479 71 139 241 383 521 137 401 457 563 53 509 431 151 41 461 89 107

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 (23 ... 593), resulting in 82 valid sets of four, are shown in Attachment 14.13.16.

14.13.17 Composed Magic Squares (10 x 10)
         Order 4 and 6 Simple Magic Sub Squares

When the Magic Sum s10 is a multiple of 10 (e.g. 3500), order 10 (Semi) Magic Squares might be composed of:

• One 4^th order Magic Corner Square with Magic Sum s4 = 4 * s10 / 10 = 1400 (top/left)
• One 6^th order Magic Corner Square with Magic Sum s6 = 6 * s10 / 10 = 2100 (bottom/right)
• Two Magic Rectangles order 4 x 6 with s4 = 1400 and s6 = 2100

An example is shown below (left) for the consecutive prime numbers {71 ... 653} with the related Magic Sums mentioned above.

Subject (Semi) Magic Squares can be transformed to Order 10 Bordered Magic Squares with order 6 Magic Center Square, as illustrated below (right):

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

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

It should be noted that 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 and variable values.

14.13.18 Composed Magic Squares (10 x 10)
         Order 5 Semi Magic Sub Squares

Order 10 (Semi) Magic Squares might be composed of order 5 (Semi) Magic Sub Squares when the Magic Sum s10 is a multiple of 2.

An example is shown below for the consecutive prime numbers {131 ... 739} with the related Magic Sums s10 = 4222 and s5 = 2111.

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

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

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 * (5!)⁴ = 829.440.000 squares for the applied diagonal elements (highlighted).

Order 10 Simple Magic Squares composed of (Semi) Magic Sub Squares can be transformed into Four Way V type ZigZag Magic Squares of order 10 as illustrated below:

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

Four Way V Type ZigZag Magic Square 167 193 739 661 179 199 317 367 709 691 373 541 647 419 229 307 599 593 263 251 181 239 701 641 163 241 389 337 677 653 281 349 613 557 293 257 571 569 353 379 311 347 401 461 359 397 601 463 439 443 499 331 431 277 409 547 503 433 269 523 719 659 139 191 683 631 421 457 149 173 491 313 223 587 563 479 227 283 607 449 733 673 131 157 727 643 383 487 137 151 467 577 197 271 617 521 211 233 619 509

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

14.13.19 Bordered Magic Squares (10 x 10)
         Order 8 Simple Magic Sub Squares

The consecutive prime numbers {71 ... 653} - with the related Magic Sum s10 = 3500 - contain 18 regular pairs (pair sum = 700), which enables the construction of Bordered Magic Squares as illustrated by following examples:

Concentric Border 599 461 419 317 257 197 113 269 251 617 191 607 619 97 293 367 401 139 277 509 353 359 79 433 313 457 271 491 397 347 107 223 631 227 193 487 229 157 653 593 179 211 71 643 601 89 499 613 73 521 563 337 283 163 523 571 557 199 167 137 569 263 641 439 241 331 127 181 577 131 389 421 373 151 173 349 307 479 547 311 467 379 103 647 463 149 409 541 109 233 83 239 281 383 443 503 587 431 449 101

Eccentric Border 599 617 461 419 317 257 197 113 269 251 83 101 239 281 383 443 503 587 431 449 191 509 227 631 487 223 653 193 157 229 353 347 433 79 457 359 397 313 491 271 107 593 163 283 571 337 167 523 199 557 179 521 97 619 367 607 277 293 139 401 563 137 647 103 149 379 109 463 541 409 569 131 643 71 89 211 73 601 613 499 389 311 151 373 349 421 547 173 479 307 467 233 439 641 331 263 577 241 181 127

The Order 8 Simple Magic Sub Square (s8 = 2800), constructed with the Generator Principle as applied in previous sections, corresponds with 8 * 2⁴/2 * (4!) = 8 * 192 = 1536 Sub Squares.

The Concentric Border (left) corresponds with 8 * (8!)² = 13.005.619.200 borders for the applied corner pairs.

The Eccentric Border (right) corresponds with 2 * (6!)² = 1.036.800 borders for the applied (aspect of the) Sub Square and corner pairs.

14.13.20 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:

Following sections will describe how Order 11 Prime Number Magic Squares with Consecutive Primes can be found with comparable routines as described in previous sections.