Office Applications and Entertainment, Magic Squares

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14.0    Special Magic Squares, Prime Numbers

14.14   Magic Twin Squares

Prime Number Magic Twin Squares are paired Magic Squares (A , B) with the property {bi} = {ai + d} for i = 1 ... n and d >= 2. This section will consider Twin Squares for d = 2.

Prime Number Magic Twin Squares can be generated with comparable routines as described in previous sections, however based on those prime variable values {bi} for which also {bi - 2} is prime for i = 1 ... n.

14.14.1 Magic Twin Squares (3 x 3)

The enumeration of order 3 Prime Number Magic Twin Squares, which can be found within prime number range (2 ... 9923) for d = 2 ... 9184, has been discussed in Attachment 14.2.

Attachment 14.14.1 shows for prime number range (17 ... 11779) and d = 2 the first occurring Prime Number Magic Twin Squares for miscellaneous Magic Sums.

Each pair shown corresponds with 8 pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 9.

14.14.2 Magic Twin Squares (4 x 4)

Attachment 14.14.2 shows for prime number range (7 ... 661) and d = 2 the first occurring Prime Number Magic Twin Squares for miscellaneous Magic Sums.

Each pair corresponds with 32 pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 16.

Attachment 14.14.3 shows for prime number range (13 ... 50551) and d = 2 the first occurring Prime Number Pan Magic Twin Squares for miscellaneous Magic Sums.

Each pair shown corresponds with 384 pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 16.

14.14.3 Magic Twin Squares (5 x 5)

Attachment 14.14.16 shows for prime number range (5 ... 1609) and d = 2 the first occurring Prime Number Simple Magic Twin Squares for miscellaneous Magic Sums.

Attachment 14.14.17 shows for prime number range (11 ... 9463) and d = 2 the first occurring Prime Number Pan Magic Twin Squares for miscellaneous Magic Sums, based on La Hirian Primaries as discussed in Section 14.12.2.

Attachment 14.14.6 shows for prime number range (13 ... 14629) and d = 2 the first occurring Prime Number Associated Magic Twin Squares for miscellaneous Magic Sums.

Based on the order 3 Prime Number Simple Magic Twin Squares as discussed in Section 14.4.1 above, following order 5 Prime Number Magic Twin Squares can be constructed:

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 25.

14.14.4 Magic Twin Squares (6 x 6)

Attachment 14.14.76 shows for prime number range (13 ... 3769) and d = 2 the first occurring Prime Number Simple Magic Twin Squares with Symmetric Main Diagonals for miscellaneous Magic Sums.

Attachment 14.14.72 shows for prime number range (19 ... 49921) and d = 2 the first occurring Prime Number Associated Magic Twin Squares for miscellaneous Magic Sums.

Attachment 14.14.75 shows the corresponding Prime Number Pan Magic and Complete Twin Squares (Eulers Transformation).

Based on order 4 Prime Number (Pan) Magic Twin Squares, order 6 Prime Number Concentric Magic Twin Squares can be constructed (ref. Priem6b2).

An example of a pair of Prime Number Concentric Magic Twin Squares, with Pan Magic Center Squares, is shown below:

MC6a = 18264
5741 569 1607 3119 5231 1997
431 1871 1667 3851 4787 5657
1451 2789 5849 809 2729 4637
2549 2237 1301 4217 4421 3539
4001 5279 3359 3299 239 2087
4091 5519 4481 2969 857 347
MC6b = 18276
5743 571 1609 3121 5233 1999
433 1873 1669 3853 4789 5659
1453 2791 5851 811 2731 4639
2551 2239 1303 4219 4423 3541
4003 5281 3361 3301 241 2089
4093 5521 4483 2971 859 349

Attachment 14.14.71 shows for prime number range (13 ... 17837) and d = 2 the first occurring Prime Number Concentric Magic Twin Squares for miscellaneous Magic Sums.

Following order 6 Prime Number Simple Magic Twin Squares can be constructed, based on combinations of order 3 Prime Number Semi Magic Twin Squares (6 Magic Lines):

  • Attachment 14.14.73 Prime Number Simple Magic Twin Squares, four order 3 Semi Magic Sub Squares

  • Attachment 14.14.74 Prime Number Simple Magic Twin Squares, two order 3 Semi Magic Bottom Squares

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 36.

14.14.5 Magic Twin Squares (7 x 7)

Based on the order 5 Prime Number Magic Twin Squares found in Section 14.4.3 above, following order 7 Prime Number Magic Twin Squares can be constructed (ref. Section 14.5.1):

Based on the order 3 and 4 Prime Number Magic Twin Squares found in Section 14.4.1 and 2 above, following order 7 Prime Number Magic Twin Squares can be constructed (ref. Section 14.5.7):

  • Attachment 14.14.85 Prime Number Composed   Magic Twin Squares, Simple Magic Corner Squares

  • Attachment 14.14.86 Prime Number Composed   Magic Twin Squares, Simple Embedded Magic Squares

Attachment 14.14.87 contains a few examples of sets of Prime Number Pan Magic Twin Squares, based on La Hirian Primaries as discussed in Section 14.12.4.

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 49.

14.14.6 Magic Twin Squares, Composed (8 x 8)

In Section 14.6.1 was discussed how Prime Number Magic Squares of order 8 - with Magic Sum 2 * s1 - can be composed out of 4th order Prime Number Magic Squares with Magic Sum s1.

An example of a Magic Sum for which a set of 4 Prime Number Magic Twin Squares can be found is s1 = 77620:

MC8a = 155224
11057 7559 31247 27749 10709 1787 37019 28097
2339 29567 9239 36467 7487 32609 6197 31319
25469 1877 36929 13337 20747 4649 34157 18059
38747 38609 197 59 38669 38567 239 137
18917 10499 28307 19889 29387 23669 15137 9419
4049 25409 13397 34757 3299 13007 25799 35507
16187 5867 32939 22619 6689 4157 34649 32117
38459 35837 2969 347 38237 36779 2027 569
MC8b = 155240
11059 7561 31249 27751 10711 1789 37021 28099
2341 29569 9241 36469 7489 32611 6199 31321
25471 1879 36931 13339 20749 4651 34159 18061
38749 38611 199 61 38671 38569 241 139
18919 10501 28309 19891 29389 23671 15139 9421
4051 25411 13399 34759 3301 13009 25801 35509
16189 5869 32941 22621 6691 4159 34651 32119
38461 35839 2971 349 38239 36781 2029 571

Attachment 14.14.4 contains a few more sets of Prime Number Magic Twin Squares, which can be used to construct Composed Magic Twin Squares of order 8 (ref. Priem4c2).

Attachment 14.14.7 shows for prime number range (59 ... 98929) and d = 2 the first occurring Prime Number Composed Magic Twin Squares of order 8 for a few Magic Sums.

Attachment 14.14.5 shows for prime number range (13 ... 50593) and d = 2 the first occurring Prime Number Concentric Magic Twin Squares for a few Magic Sums.

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 64.

14.14.7 Magic Twin Squares (9 x 9)

Composed Magic Twin Squares of order 9 can be constructed based on a combination of one order 3 Magic Twin Square with 8 Semi Magic Twin Squares (6 Magic Lines).

Attachment 14.14.20 shows for prime number range (5 ... 45343) and d = 2 the first occurring Prime Number Composed Magic Twin Squares for a few Magic Sums (ref. Priem9b).

Based on the order 7 Prime Number Magic Twin Squares found in Section 14.4.5 above, following order 9 Prime Number Magic Twin Squares can be constructed (ref. Section 14.7.4):

  • Prime Number Concentric Magic Twin Squares
  • Prime Number Bordered   Magic Twin Squares, Center Square with Square Inlay
  • Prime Number Bordered   Magic Twin Squares, Center Square with Diamond Inlay
  • Prime Number Bordered   Magic Twin Squares, Associated Center Square

of which a few examples are shown in Attachment 14.14.10.

Order 9 Associated Magic Squares with order 4 and 5 Square Inlays, can be obtained by means of a transformation of order 9 Composed Magic Squares (ref. Section 14.7.11).

An Example, based on order 4 and 5 Prime Number Magic Twin Squares found in Section 14.4.2 and 3 above, is enclosed in Attachment 14.14.95 which contains:

  • Prime Number Composed   Magic Twin Squares with Associated Corner Squares and Rectangles
  • Prime Number Associated Magic Twin Squares with Associated Embedded Magic Squares

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 81.

14.14.8 Magic Twin Squares (10 x 10)

Based on the order 8 Prime Number Concentric Magic Twin Squares found in Section 14.4.6 above, order 10 Prime Number Magic Twin Squares can be constructed (ref. Section 14.8.3):

Attachment 14.14.30 shows for prime number range (29 ... 42703) and d = 2 the first occurring Prime Number Composed Magic Twin Squares, with order 4 Associated Center Square, for a few Magic Sums.

Each pair corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 100.

14.14.9 Magic Twin Squares (11 x 11)

Based on the order 9 Prime Number Magic Twin Squares found in Section 14.4.7 above, following order 11 Prime Number Magic Twin Squares can be constructed (ref. Section 14.9.1):

  • Prime Number Concentric Magic Twin Squares
  • Prime Number Bordered   Magic Twin Squares, Center Square with Square Inlay
  • Prime Number Bordered   Magic Twin Squares, Center Square with Diamond Inlay

of which a few examples are shown in Attachment 14.14.11.

Each pair corresponds with numerous pairs with the same Magic Sum and variable values {ai}/{bi}, i = 1 ... 121.

14.14.10 Magic Twin Squares (12 x 12)

Based on the order 10 Prime Number Concentric Magic Twin Squares found in Section 14.4.8 above, order 12 Prime Number Concentric Magic Twin Squares can be constructed (ref. Priem12a).

Attachment 14.14.15 shows for prime number range (13 ... 98809) and d = 2 the first occurring Prime Number Concentric Magic Twin Squares for a few Magic Sums.

Based on the order 6 Prime Number Magic Twin Squares as discussed in Section 14.4.4 above, following order 12 Composed Prime Number Magic Twin Squares can be constructed (ref. Priem12b):

  • Composed, 16 Semi Magic Sub Squares (3 x 3)
  • Composed, Associated Center Square  (6 x 6), 12 Semi Magic Border Squares (3 x 3)
  • Composed, Concentric Center Square  (6 x 6), 12 Semi Magic Border Squares (3 x 3)

of which a few examples are shown in Attachment 14.14.22.

Prime Number Magic Squares of order 12 - with Magic Sum 3 * s1 - can be composed out of 4th order Prime Number Magic Squares with Magic Sum s1.

Attachment 14.14.23 shows for prime number range (19 ... 49939) and d = 2 the first occurring Prime Number Composed Magic Twin Squares of order 12, for a few Magic Sums.

Each pair corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 144.

14.14.11 Summary

The obtained results regarding the miscellaneous types of Prime Number Magic Twin Squares as deducted and discussed in previous sections are summarized in following table:

Order

Main Characteristics

Subroutine

Results

3

Simple Magic

Priem3

Attachment 14.14.1

4

Simple Magic

Priem4

Attachment 14.14.2

Pan Magic

Priem4b

Attachment 14.14.3

5

Simple Magic

Priem5a2

Attachment 14.14.16

Pan Magic

La Hirian

Attachment 14.14.17

Associated

Priem5e

Attachment 14.14.6

Concentric

Priem5c2

Attachment 14.14.9

Magic, Square  Inlay

Priem5g3

Attachment 14.14.13

Magic, Diamond Inlay

Priem5g2

Attachment 14.14.14

6

Concentric

Priem6b2

Attachment 14.14.71

Associated

Priem6i2

Attachment 14.14.72

Pan Magic and Complete

Euler

Attachment 14.14.75

Simple, Symmetric Main Diagonals

Priem6a2

Attachment 14.14.76

Simple, Four Semi Magic Sub    Squares

Priem6c2

Attachment 14.14.73

Simple, Two  Semi Magic Bottom Squares

Priem6c4

Attachment 14.14.74

7

Bordered

Priem7a2

Ref. Sect. 14.14.5

Composed, Simple Magic Corner Squares

Priem7f2

Attachment 14.14.85

Pan Magic

La Hirian

Attachment 14.14.87

8

Composed, Simple Magic Sub    Squares

Priem4c2

Attachment 14.14.7

Concentric

Priem8a2

Attachment 14.14.5

9

Composed, Semi Magic Sub Squares

Priem9b

Attachment 14.14.20

Concentric

Priem9a2

Attachment 14.14.10

10

Concentric

Priem10c

Attachment 14.14.12

Bordered, Composed   Center Square

Priem10c

Attachment 14.14.18

Composed, Associated Center Square

Priem10b

Attachment 14.14.30

11

Concentric

Priem11a

Attachment 14.14.11

12

Concentric

Priem12a

Attachment 14.14.15

Composed, Miscellaneous Types

Priem12b

Attachment 14.14.22

Composed, Simple Magic Sub Squares

Priem4c2

Attachment 14.14.23

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-

-

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Following sections will provide miscellaneous construction methods for paired squares based on Sophie Germain Primes.


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