Office Applications and Entertainment, Magic Squares

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 {b_i} = {a_i + 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 {b_i} for which also {b_i - 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 {a_i}/{b_i}, 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 {a_i}/{b_i}, 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 {a_i}/{b_i}, 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:

• Attachment 14.14.9  Prime Number Concentric Magic Twin Squares         (ref. Priem5c2)
  
  
• Attachment 14.14.13 Prime Number Magic Twin Squares with Square  Inlay (ref. Priem5g3)
  
  
• Attachment 14.14.14 Prime Number Magic Twin Squares with Diamond Inlay (ref. Priem5g2)

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {a_i}/{b_i}, 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:

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 {a_i}/{b_i}, 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):

• Attachment 14.14.81 Prime Number Concentric Magic Twin Squares
  
  
• Attachment 14.14.82 Prime Number Bordered   Magic Twin Squares, Center Square with Square  Inlay
  
  
• Attachment 14.14.83 Prime Number Bordered   Magic Twin Squares, Center Square with Diamond Inlay
  
  
• Attachment 14.14.84 Prime Number Bordered   Magic Twin Squares, Associated Center Square

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 {a_i}/{b_i}, 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 4^th 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:

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 {a_i}/{b_i}, 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 Composed   Magic Twin Squares with Associated Center Square
• 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 {a_i}/{b_i}, 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 14.14.12 Prime Number Concentric Magic Twin Squares
  
  
• Attachment 14.14.18 Prime Number Bordered   Magic Twin Squares, Composed Center Square

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 {a_i}/{b_i}, 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 {a_i}/{b_i}, 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 4^th 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 {a_i}/{b_i}, 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:

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