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14.0   Special Magic Squares, Prime Numbers

14.4   Magic Squares (6 x 6)

Comparable routines as discussed in previous sections can be written to generate Prime Number Magic Squares of order 6, however such routines are not very feasible due to the high number of independent variables, 23 ea for Magic Squares and 16 ea for Pan Magic Squares.

In next sections solutions will be found for more strict defined Prime Number (Pan) Magic Squares of the 6th order.

14.4.1 Concentric Pan Magic Squares (6 x 6)

The variable values {ai} on which a Prime Number Concentric Pan Magic Square with Embedded Magic Square might be based should contain at least 10 pairs (border).

Based on the possible pairs for the first 170 Prime Numbers (2 ... 1013) the corresponding Magic Sums of the Outer Pan Magic - and Embedded Magic Squares (MC6 and MC4) can be determined, as well as the corresponding variable values {ai}.

Based on the equations defining a Concentric Pan Magic Square (6 x 6) with Magic Center Square (4 x 4):

```a(31) =     s1   - a(32) - a(33) - a(34) - a(35) - a(36)
a(27) = 2 * s1/3 - a(28) - a(33) - a(34)
a(26) =          - a(29) + a(33) + a(34)
a(25) =     s1/3 - a(30)
a(23) = 5 * s1/3 - a(24) - a(28) - 2 * a(29) - a(30) - a(34) - a(35) - 2 * a(36)
a(22) =   - s1/3 + a(24) + a(34) + a(36)
a(21) =     s1   - a(24) - a(32) - a(34) - a(35) - a(36)
a(20) = 2 * s1/3 - a(21) - a(22) - a(23)
a(19) =     s1/3 - a(24)
a(18) =     s1/6 + a(29) - a(33)
a(17) =   - s1   - a(18) + a(28) + 2 * a(29) + a(30) + a(34) + a(35) + 2 * a(36)
a(16) =-2 * s1/3 + a(18) + a(32) + a(33) + a(35) + a(36)
a(15) =          - a(16) + a(32) + a(35)
a(14) = 2 * s1/3 - a(15) - a(16) - a(17)
a(13) =     s1/3 - a(18)
a(12) =     s1/3 + a(13) - a(24) - a(30) + a(31) - a(36)
a(11) =     s1/3 - a(13) + a(24) - a(29)
a(10) = 2 * s1/3 - a(11) - a(28) - a(29) + a(31) - a(36)
a( 9) =          - a(10) + a(33) + a(34)
a( 8) = 2 * s1/3 - a( 9) - a(10) - a(11)
a( 7) =     s1/3 - a(12)
a( 6) =     s1/3 - a(31)
a( 5) =     s1/3 - a(35)
a( 4) =     s1/3 - a(34)
a( 3) =     s1/3 - a(33)
a( 2) =     s1/3 - a(32)
a( 1) =     s1/3 - a(36)
```

a routine can be written to generate Prime Number Concentric Pan Magic Squares of order 6 (ref. Priem6a).

Attachment 14.4.1 shows Prime Number Concentric Pan Magic Squares for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {ai}.

14.4.2 Concentric Pan Magic Squares (6 x 6)
Associated Center Square

The equations defining a Concentric Pan Magic Square (6 x 6) with Associated Center Square (4 x 4) can be written as:

```a(36) =  5 * s1/6 - a(23) -     a(28) - 2 * a(29)
a(33) =  2 * s1/3 - a(34) -     a(27) -     a(28)
a(32) = -2 * s1/3 - a(35) + 2 * a(23) +     a(27) + a(28) + 2 * a(29)
a(31) =      s1/6 - a(23) + a(28)
a(30) =    - s1/6 - a(35) + a(23) + a(28) + a(29)
a(26) =  2 * s1/3 - a(27) - a(28) - a(29)
a(25) =      s1/2 + a(35) - a(23) - a(28) - a(29)
a(24) =      s1/6 - a(34) + a(29)
a(22) =  2 * s1/3 - a(23) - a(28) - a(29)
a(21) =  2 * s1/3 - a(23) - a(27) - a(29)
a(20) = -2 * s1/3 + a(23) + a(27) + a(28) + 2 * a(29)
a(19) =      s1/6 + a(34) - a(29)
a(18) =    - s1/2 + a(34) + a(27) + a(28) +     a(29)
a(17) =      s1   - a(23) - a(27) - a(28) - 2 * a(29)
a(16) =    - s1/3 + a(23) + a(27) + a(29)
a(15) =    - s1/3 + a(23) + a(28) + a(29)
a(14) =      s1/3 - a(23)
a(13) =  5 * s1/6 - a(34) - a(27) - a(28) - a(29)
a(12) =      s1/2 + a(35) - a(23) - a(27) - a(29)
a(11) =    - s1/3 + a(27) + a(28) + a(29)
a(10) =      s1/3 - a(27)
a( 9) =      s1/3 - a(28)
a( 8) =      s1/3 - a(29)
a( 7) =    - s1/6 - a(35) + a(23) + a(27) + a(29)
a( 6) =      s1/6 + a(23) - a(28)
a( 5) =      s1/3 - a(35)
a( 4) =      s1/3 - a(34)
a( 3) =    - s1/3 + a(34) +     a(27) +     a(28)
a( 2) =      s1   + a(35) - 2 * a(23) -     a(27) - a(28) - 2 * a(29)
a( 1) =    - s1/2 + a(23) +     a(28) + 2 * a(29)
```

with a(29), a(28) a(27) and a(23) the independent center square variables and a(35), a(34) the independent border variables.

An optimised routine can be written to generate Prime Number Concentric Pan Magic Squares with Associated Center Square (ref. Priem6g).

Attachment 14.4.8 shows one Prime Number Concentric Pan Magic Square as defined above for each of the occurring Magic Sums for the first 426 Prime Numbers (2 ... 2953).

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {ai}.

14.4.3 Concentric Magic Squares (6 x 6)

A 6th order Prime Number Concentric Magic Square consists of a Prime Number Embedded Magic Square of the 4th order with a border around it.

If the Embedded Magic Square is Pan Magic it consists of 8 pairs. The border consists of 10 pairs.

Consequently the variable values {ai} on which a Prime Number Concentric Magic Square with Pan Magic Embedded Square might be based should contain at least 18 pairs.

Based on the possible pairs for the first 170 Prime Numbers (2 ... 1013) the corresponding Magic Sums of the outer - and embedded squares (MC6 and MC4) can be determined.

• Attachment 14.4.1 page 1 shows these data for the occurring Magic Sums MC4 and npair >= 18;
• Attachment 14.4.1 page 2 shows for each of the listed Magic Sums MC6 the corresponding variable values {ai}.

Based on the equations defining a Concentric Magic Square (6 x 6) with Pan Magic Center Square (4 x 4):

```a(31) =      s1   - a(32) - a(33) - a(34) - a(35) - a(36)
a(26) =  2 * s1/3 - a(27) - a(28) - a(29)
a(25) =      s1/3 - a(30)
a(22) =  2 * s1/3 - a(23) - a(28) - a(29)
a(21) =             a(23) - a(27) + a(29)
a(20) =           - a(23) + a(27) + a(28)
a(19) =      s1/3 - a(24)
a(17) =      s1/3 - a(27)
a(16) =    - s1/3 + a(27) + a(28) + a(29)
a(15) =      s1/3 - a(29)
a(14) =      s1/3 - a(28)
a(13) =      s1/3 - a(18)
a(12) =  2 * s1/3 - a(18) - a(24) - a(30) + a(31) - a(36)
a(11) =      s1/3 - a(23) + a(27) - a(29)
a(10) =      s1/3 + a(23) - a(27) - a(28)
a( 9) =      s1/3 - a(23)
a( 8) =    - s1/3 + a(23) + a(28) + a(29)
a( 7) = -4 * s1/3 + a(18) + a(24) + a(30) + a(32) + a(33) + a(34) + a(35) + 2 * a(36)
a( 6) = -2 * s1/3 + a(32) + a(33) + a(34) + a(35) + a(36)
a( 5) =      s1/3 - a(35)
a( 4) =      s1/3 - a(34)
a( 3) =      s1/3 - a(33)
a( 2) =      s1/3 - a(32)
a( 1) =      s1/3 - a(36)
```

a routine can be written to generate Prime Number Concentric Magic Squares of order 6 (ref. Priem6b).

Attachment 14.4.3 shows one Prime Number Concentric Magic Square for 48 of the occurring Magic Sums.

Each square shown corresponds with multiples of 645120 for the same Magic Sum, depending from the selected variable values {ai} and the related number of possible Embedded Pan Magic Squares (multiples of 384).

14.4.4 Concentric Magic Squares (6 x 6)
Non Overlapping Sub Squares (2 x 2)

Based on the equations defining a Concentric Magic Square (6 x 6), composed out of 9 Non Overlapping Sub Squares (2 x 2), with Pan Magic Center Square (4 x 4):

```a(31) =     s1   - a(32) - a(33) - a(34) - a(35) - a(36)
a(29) = 2 * s1/3 - a(30) - a(35) - a(36)
a(27) = 2 * s1/3 - a(28) - a(33) - a(34)
a(26) =     s1/3 + a(30) - a(31) - a(32)
a(25) =     s1/3 - a(30)
a(22) = 2 * s1/3 - a(23) - a(28) - a(29)
a(21) =            a(23) - a(27) + a(29)
a(20) = 2 * s1/3 - a(23) - a(33) - a(34)
a(19) =     s1/3 - a(24)
a(18) =     s1/3 - a(23) - a(24) + a(27)
a(17) =   - s1/3 + a(28) + a(33) + a(34)
a(16) =          - a(30) + a(31) + a(32)
a(15) =   - s1/3 + a(30) + a(35) + a(36)
a(14) =     s1/3 - a(28)
a(13) =            a(23) + a(24) - a(27)
a(12) = 2 * s1/3 + a(23) + a(28) - a(30) - a(32) - a(35) - 2 * a(36)
a(11) = 2 * s1/3 - a(17) - a(23) - a(29)
a(10) =   - s1/3 + a(23) + a(33) + a(34)
a( 9) =     s1/3 - a(23)
a( 8) = 2 * s1/3 - a(11) - a(33) - a(34)
a( 7) =     s1/3 - a(12)
a( 6) =     s1/3 - a(31)
a( 5) =     s1/3 - a(35)
a( 4) =     s1/3 - a(34)
a( 3) =     s1/3 - a(33)
a( 2) =     s1/3 - a(32)
a( 1) =     s1/3 - a(36)
```

a comparable routine can be written to generate Prime Number Concentric Magic Squares of order 6, composed out of 2 x 2 Non Overlapping Sub Squares (ref. Priem6c).

Attachment 14.4.4 shows one Prime Number Concentric Magic Square as defined above for 48 of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum, depending from the selected variable values {ai} and the related number of possible Embedded Pan Magic Squares.

14.4.5 Eccentric Magic Squares (6 x 6)

Based on the equations defining an Eccentric Magic Square (6 x 6) with Pan Magic Border Square (4 x 4):

```a(33) =  2 * s1/3 - a(34) - a(35) - a(36)
a(31) =      s1/3 - a(32)
a(29) =  2 * s1/3 - a(30) - a(35) - a(36)
a(28) =             a(30) - a(34) + a(36)
a(27) =           - a(30) + a(34) + a(35)
a(25) =      s1/3 - a(26)
a(24) =      s1/3 - a(34)
a(23) =    - s1/3 + a(34) + a(35) + a(36)
a(22) =      s1/3 - a(36)
a(21) =      s1/3 - a(35)
a(19) =      s1/3 - a(20)
a(18) =      s1/3 - a(30) + a(34) -a(36)
a(17) =      s1/3 + a(30) - a(34) -a(35)
a(16) =      s1/3 - a(30)
a(15) =    - s1/3 + a(30) + a(35) + a(36)
a(13) =      s1/3 - a(14)
a(11) =  2 * s1/3 + a(12) - a(16) - a(21) - a(26) - a(31)
a( 8) =     (s1/3 - a( 9) - a(10) - a(11) - a(12) + a(13) + a(19) + a(25) + a(31))/2
a( 7) =      s1   - a( 8) - a( 9) - a(10) - a(11) - a(12)
a( 6) =      s1/3 - a(12)
a( 5) =      s1/3 - a(11)
a( 4) =      s1/3 - a(10)
a( 3) =      s1/3 - a( 9)
a( 2) =      s1   - a( 8) - a(14) - a(20) - a(26) - a(32)
a( 1) =      s1/3 - a( 8)
```

a comparable routine can be written to generate Prime Number Eccentric Magic Squares of order 6 (ref. Priem6d1).

Attachment 14.4.5 shows one Prime Number Eccentric Magic Square for 48 of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum, depending from the selected variable values {ai}, the related number of possible Pan Magic Corner Squares and the key variables.

Note:

Although the results obtained above are satisfactorily, a much faster routine (ref. Priem6d2) can be developed based on following principles:

• to read the previously generated (Pan Magic) Corner Squares of order 4;
• to complete the Main Diagonal and determine the related Border Pairs;
• to complete the Eccentric Magic Square of order 6 with the remaining Border Pairs.

Subject routine produced, based on 2236 previously generated Pan Magic Squares of order 4, 2236 Prime Number Eccentric Magic Square of order 6 within 340 seconds (one square per Magic Sum).

14.4.6 Associated Pan Magic Squares (6 x 6)

Based on the equations defining an Associated Pan Magic Square (Ultra Magic) of the sixth order:

```a(31) =     s1 - a(32) - a(33) - a(34) - a(35) - a(36)
a(27) = 2 * s1 - a(28) - 2 * a(29) - 2 * a(30) + a(32) - 2 * a(34) - 3 * a(35) - 2 * a(36)
a(26) =     s1 - 2 * a(27) - a(29) - 2 * a(30)
a(25) =              a(27) - a(28) + a(30)
a(24) = 3 * s1/2 - a(29) - 2 * a(30) - a(34) - 2 * a(35) - 2 * a(36)
a(23) =     s1/2 - a(25) - a(28) - a(29) + a(32)
a(22) = 3 * s1/2 - 2 * a(28) - a(29) - 2 * a(34) - 2 * a(35) - a(36)
a(21) =     s1 - a(24) - a(32) - a(33) - a(35) - a(36)
a(20) =          a(22) + a(28) - a(30) + a(32) - a(36)
a(19) =        - a(20) + a(28) - a(30) + a(32) + a(33)
```
 a(18) = s1/3 - a(19) a(17) = s1/3 - a(20) a(16) = s1/3 - a(21) a(15) = s1/3 - a(22) a(14) = s1/3 - a(23) a(13) = s1/3 - a(24) a(12) = s1/3 - a(25) a(11) = s1/3 - a(26) a(10) = s1/3 - a(27) a( 9) = s1/3 - a(28) a(8) = s1/3 - a(29) a(7) = s1/3 - a(30) a(6) = s1/3 - a(31) a(5) = s1/3 - a(32) a(4) = s1/3 - a(33) a(3) = s1/3 - a(34) a(2) = s1/3 - a(35) a(1) = s1/3 - a(36)

a comparable routine can be written to generate Associated Pan Magic Squares of order 6 (ref. Priem6i).

Subject routine produced, for the smallest possible Magic Sum (s1 = 990), 1408 ( = 11 * 128) solutions which are shown in Attachment 14.4.60 and includes the one recently published by Max Alekseyev (2015).

14.4.7 Associated Pan Magic Squares (6 x 6)
Non Overlapping Sub Squares (2 x 2)

Based on the equations defining an Associated Pan Magic Square (Ultra Magic) of the sixth order, composed out of 9 Non Overlapping Sub Squares (2 x 2):

```a(32) =             - a(33) + a(34) + a(35)
a(31) =      s1 - 2 * a(34) - 2 * a(35) - a(36)
a(29) =  2 * s1 / 3 - a(30) - a(35) - a(36)
a(27) =  2 * s1 / 3 - a(28) - a(33) - a(34)
a(26) =    - s1 + 2 * a(28) - a(30) + 2 * a(33) + 2 * a(34) + a(35) + a(36)
a(25) =  2 * s1 / 3 - 2 * a(28) + a(30) - a(33) - a(34)
a(24) =  5 * s1 / 6 - a(30) - a(34) - a(35) - a(36)
a(23) = -5 * s1 / 6 + a(28) + 2 * a(34) + 2 * a(35) + a(36)
a(22) =  5 * s1 / 6 - 2 * a(28) + a(30) - 2 * a(34) - a(35)
a(21) =      s1 / 6 + a(30) - a(35)
a(20) =  5 * s1 / 6 - a(28) - a(33) - a(34) - a(36)
a(19) = -5 * s1 / 6 + 2 * a(28) - a(30) + a(33) + 2 * a(34) + a(35) + a(36)
```
 a(18) = s1/3 - a(19) a(17) = s1/3 - a(20) a(16) = s1/3 - a(21) a(15) = s1/3 - a(22) a(14) = s1/3 - a(23) a(13) = s1/3 - a(24) a(12) = s1/3 - a(25) a(11) = s1/3 - a(26) a(10) = s1/3 - a(27) a( 9) = s1/3 - a(28) a(8) = s1/3 - a(29) a(7) = s1/3 - a(30) a(6) = s1/3 - a(31) a(5) = s1/3 - a(32) a(4) = s1/3 - a(33) a(3) = s1/3 - a(34) a(2) = s1/3 - a(35) a(1) = s1/3 - a(36)

a comparable routine can be written to generate subject Associated Pan Magic Squares of order 6 (ref. Priem6f).

Attachment 14.4.7 shows one Prime Number Associated Pan Magic Square, composed out of 2 x 2 Non Overlapping Sub Squares, for a number of the occurring Magic Sums for the first 398 Prime Numbers (2 ... 2719).

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {ai}.

14.4.8 Associated Pan Magic Squares (6 x 6)
Compact (2 x 2)

Based on the equations defining a Compact Ultra Magic Square of the sixth order:

```a(33) =      s1   - a(34) - 2 * a(35) - 2 * a(36)
a(32) =    - s1   + 2 * a(34) + 3 * a(35) + 2 * a(36)
a(31) =      s1   - 2 * a(34) - 2 * a(35) - a(36)
a(29) =  2 * s1/3 - a(30) - a(35) - a(36)
a(28) =             a(30) - a(34) + a(36)
a(27) =    - s1/3 - a(30) + a(34) + 2 * a(35) + a(36)
a(26) =      s1   + a(30) - 2 * a(34) - 3 * a(35) - a(36)
a(25) =    - s1/3 - a(30) + 2 * a(34) + 2 * a(35)
a(24) =  5 * s1/6 - a(30) - a(34) - a(35) - a(36)
a(23) = -5 * s1/6 + a(30) + a(34) + 2 * a(35) + 2 * a(36)
a(22) =  5 * s1/6 - a(30) - a(35) - 2 * a(36)
a(21) =      s1/6 + a(30) - a(35)
a(20) =    - s1/6 - a(30) + a(34) + 2 * a(35)
a(19) =      s1/6 + a(30) - a(34) - a(35) + a(36)
```
 a(18) = s1/3 - a(19) a(17) = s1/3 - a(20) a(16) = s1/3 - a(21) a(15) = s1/3 - a(22) a(14) = s1/3 - a(23) a(13) = s1/3 - a(24) a(12) = s1/3 - a(25) a(11) = s1/3 - a(26) a(10) = s1/3 - a(27) a( 9) = s1/3 - a(28) a(8) = s1/3 - a(29) a(7) = s1/3 - a(30) a(6) = s1/3 - a(31) a(5) = s1/3 - a(32) a(4) = s1/3 - a(33) a(3) = s1/3 - a(34) a(2) = s1/3 - a(35) a(1) = s1/3 - a(36)

a comparable routine can be written to generate Compact Ultra Magic Squares of order 6 (ref. Priem6h).

Attachment 14.4.9 shows for miscellaneous Magic Sums the first occurring Prime Number Compact Ultra Magic Square.

Each square shown corresponds with 288 squares for the applicable Magic Sum and variable values {ai}.

14.4.9 Most Perfect Pan Magic Squares (6 x 6)

Based on the equations defining a Most Perfect Pan Magic Square (6 x 6):

```a(33) =      s1 - 2 * a(34) - 2 * a(35) - a(36)
a(32) =    - s1 + 2 * a(34) + 3 * a(35) + 2 * a(36)
a(31) =      s1 - a(34) - 2 * a(35) - 2 * a(36)
a(29) =  4 * s1/6 - a(30) - a(35) - a(36)
a(28) =             a(30) - a(34) + a(36)
a(27) = -2 * s1/6 - a(30) + 2 * a(34) + 2 * a(35)
a(26) =      s1   + a(30) - 2 * a(34) - 3 * a(35) - a(36)
a(25) = -2 * s1/6 - a(30) + a(34) + 2 * a(35) + a(36)
a(24) =  5 * s1/6 - a(30) - a(34) - a(35) - a(36)
a(23) = -5 * s1/6 + a(30) + a(34) + 2 * a(35) + 2 * a(36)
a(22) =  5 * s1/6 - a(30) - a(35) - 2 * a(36)
a(21) =      s1/6 + a(30) - a(34) - a(35) + a(36)
a(20) =    - s1/6 - a(30) + a(34) + 2 * a(35)
a(19) =      s1/6 + a(30) - a(35)
```
 a(18) = s1/3 - a(33) a(17) = s1/3 - a(32) a(16) = s1/3 - a(31) a(15) = s1/3 - a(36) a(14) = s1/3 - a(35) a(13) = s1/3 - a(34) a(12) = s1/3 - a(27) a(11) = s1/3 - a(26) a(10) = s1/3 - a(25) a( 9) = s1/3 - a(30) a(8) = s1/3 - a(29) a(7) = s1/3 - a(28) a(6) = s1/3 - a(21) a(5) = s1/3 - a(20) a(4) = s1/3 - a(19) a(3) = s1/3 - a(24) a(2) = s1/3 - a(23) a(1) = s1/3 - a(22)

a comparable routine can be written to generate Prime Number Most Perfect Pan Magic Squares of order 6 (ref. Priem6e).

The first Most Perfect Pan Magic Square of Prime Numbers occurs for MC = 29790, which confirms Natalia Makarova's findings (June 01, 2015).

Attachment 14.4.6 shows for miscellaneous Magic Sums the first occurring Prime Number Most Perfect Pan Magic Square.

Each square shown corresponds with 288 squares for the applicable Magic Sum and variable values {ai}, which are shown in Attachment 14.4.6a.

14.4.10 Simple Magic Squares (6 x 6) composed of Semi Magic Sub Squares (3 x 3)

Prime Number Magic Squares of order 6 - with Magic Sum 2 * s1 - can be composed out of Prime Number Semi Magic Squares of order 3 with Magic Sum s1.

In section 14.1.2, a procedure was developed to generate 3th order Prime Number Semi Magic Squares (Seven Magic Lines) with Magic Sum s1, based on the equations:

```a(7) =      s1            -     a(8) -     a(9)
a(5) = -    s1 +     a(6) +     a(8) + 2 * a(9)
a(4) =  2 * s1 - 2 * a(6) -     a(8) - 2 * a(9)
a(3) =      s1 -     a(6)            -     a(9)
a(2) =  2 * s1 -     a(6) - 2 * a(8) - 2 * a(9)
a(1) = -2 * s1 + 2 * a(6) + 2 * a(8) + 3 * a(9)
```

With some minor modifications subject procedure can be used to find a set of 4 (or more) Prime Number Semi Magic Squares with Magic Sum s1 - each containing 9 different Prime Numbers.

Attachment 14.8.1 contains such sets for MC = 2439 , MC = 2475 and MC = 2655 which were found by means of procedure Priem3c for the first 274 Prime Numbers (2 ... 1747).

Each set of four squares can be arranged in 24 ways into a 6th order Prime Number Magic Square with Magic Sum 2 * s1, providing that the proper orientation of the magic diagonal has been taken into account (ref. Attachment 14.8.2).

Further it should be realized that each Semi Magic Square of the 3th order, is a member of a collection of 12 Semi Magic Squares of the 3th order with the same magic diagonal (ref. Attachment 14.8.3).

Consequently, based on one single set of 4 Prime Number Semi Magic Squares of the 3th order as shown in Attachment 14.8.1, 24 * 124 = 497664 Prime Number Magic Squares of the 6th order can be constructed.

Attachment 14.8.4 shows for miscellaneous Magic Sums (48 ea) the first occurring Prime Number Simple Magic Square composed of Semi Magic Sub Squares, generated with procedure Priem3e1 in 180 seconds.

14.4.11 Associated Magic Squares (6 x 6) composed of Semi Magic Anti Symmetric Sub Squares (3 x 3)

Comparable as in Section 14.4.10 above, Prime Number Associated Magic Squares of order 6 - with Magic Sum 2 * s1 - can be composed out of Prime Number Semi Magic Anti Symmetric Squares of order 3 with Magic Sum s1.

A Semi Magic Anti Symmetric Square of order 3 is a Semi Magic Square for which ai + aj ≠ 2 * s1 / 3 for any i and j (i,j = 1 ... 9; i ≠ j).

Semi Magic Anti Symmetric Squares occur in Complementary Pairs. Associated Magic Squares of order 6 can be constructed based on two such pairs.

Attachment 14.8.5 shows for miscellaneous Magic Sums (48 ea) the first occurring Prime Number Associated Magic Square composed of Semi Magic Anti Symmetric Sub Squares, generated with procedure Priem3e2 in 175 seconds.

14.4.12 Pan Magic Squares (6 x 6) composed of Semi Magic Sub Squares (3 x 3)

Based on the equations defining order 6 Pan Magic Squares composed of order 3 Semi Magic Sub Squares (Six Magic Lines):

```a(34) =      s1 / 2 - a(35) - a(36)
a(15) =     -s1 / 6 + a(34) + a(35)
a(14) =  5 * s1 / 6 - a(34) - 2 * a(35) - a(36)
a(13) =     -s1 / 6 + a(35) + a(36)
a(18) =  5 * s1 / 6 - a(33) - a(34) - a(35) - a(36)
a(31) =      s1 / 2 - a(32) - a(33)
a(17) =     -s1 / 6 - a(32) + a(34) + a(35) + a(36)
a(16) =     -s1 / 6 + a(32) + a(33)
a(24) =      s1 / 2 - a(30) - a(36)
a( 9) =      s1 / 3 - a(30)
a( 3) =      s1 / 3 + a(30) - a(34) - a(35)
a(28) =      s1 / 2 - a(29) - a(30)
a(23) =      s1 / 2 - a(29) - a(35)
a(22) =               a(29) + a(30) - a(34)
a( 8) =      s1 / 3 - a(29)
a( 7) =     -s1 / 6 + a(29) + a(30)
a( 2) = -2 * s1 / 3 + a(29) + a(34) + 2 * a(35) + a(36)
a( 1) =  5 * s1 / 6 - a(29) - a(30) - a(35) - a(36)
a(21) =      s1 / 2 - a(27) - a(33)
a(12) =      s1 / 3 - a(27)
a( 6) = -2 * s1 / 3 + a(27) + a(33) + a(34) + a(35) + a(36)
a(25) =      s1 / 2 - a(26) - a(27)
a(20) =      s1 / 2 - a(26) - a(32)
a(19) =      s1 / 2 - a(25) - a(31)
a(11) =      s1 / 3 - a(26)
a(10) =     -s1 / 6 + a(26) + a(27)
a( 5) =      s1 / 3 + a(26) + a(32) - a(34) - a(35) - a(36)
a( 4) =  5 * s1 / 6 - a(26) - a(27) - a(32) - a(33)
```

a routine can be written to generate subject Prime Number Pan Magic Squares of order 6 (ref. Priem6e3).

The consequential symmetry (complete) of the defining properties is worth to be noticed and allows for a further deduction to:

```a(34) =      s1 / 2 - a(35) - a(36)
a(31) =      s1 / 2 - a(32) - a(33)
a(28) =      s1 / 2 - a(29) - a(30)
a(25) =      s1 / 2 - a(26) - a(27)
a(24) =      s1 / 2 - a(30) - a(36)
a(23) =      s1 / 2 - a(29) - a(35)
a(22) =               a(29) + a(30) - a(34)
a(21) =      s1 / 2 - a(27) - a(33)
a(20) =      s1 / 2 - a(26) - a(32)
a(19) =      s1 / 2 - a(25) - a(31)
```
 a(18) = s1 / 3 - a(33) a(17) = s1 / 3 - a(32) a(16) = s1 / 3 - a(31) a(15) = s1 / 3 - a(36) a(14) = s1 / 3 - a(35) a(13) = s1 / 3 - a(34) a(12) = s1 / 3 - a(27) a(11) = s1 / 3 - a(26) a(10) = s1 / 3 - a(25) a( 9) = s1 / 3 - a(30) a( 8) = s1 / 3 - a(29) a( 7) = s1 / 3 - a(28) a(6) = s1 / 3 - a(21) a(5) = s1 / 3 - a(20) a(4) = s1 / 3 - a(19) a(3) = s1 / 3 - a(24) a(2) = s1 / 3 - a(23) a(1) = s1 / 3 - a(22)

Attachment 14.8.6 shows for miscellaneous Magic Sums (48 ea) the first occurring Prime Number Pan Magic Square composed of Semi Magic Sub Squares, generated with procedure Priem6e3 in 56 seconds.

14.4.13 Inlaid Magic Squares (6 x 6)

An order 6 Magic Square might be composed out of:

• One 4th order Pan Magic Corner Square D with Magic Sum s4 (top/left);
• One 3th order Simple Magic Corner Square A with Magic Sum s31 (bottom/right);
• Two 2 x 3 Magic Rectangles B/C with Pair Sum Pr3 and Magic Sum s32.

As illustrated below:

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32) a(33) a(34) a(35) a(36)

Based on the defining equations of an order 4 Pan Magic Square, as deducted in Section 14.2.2, a dedicated procedure can be developed (ref. MgcSqr413):

• to read previously selected suitable order 3 Simple Magic Squares A;
• to generate the order 4 Pan Magic Squares D;
• to complete the 6 x 6 Inlaid Magic Squares with the 2 x 3 Magic Rectangles B/C.

Attachment 14.4.13 shows for miscellaneous Magic Sums (48 ea) the first occurring Prime Number Inlaid Magic Square. Each square shown corresponds with miscellaneous Inlaid Magic Squares.

14.4.14 Summary

The obtained results regarding the miscellaneous types of order 6 Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table:

 Type Characteristics Subroutine Results Concentric Pan Magic General Associated Center Square Concentric General Sub Squares 2 x 2 Eccentric - Associated Pan Magic Sub Squares 2 x 2 Compact Most Perfect Compact and Complete Composed Semi Magic Sub Squares 3 x 3 Composed, Associated Composed, Pan Magic Inlaid Overlapping Sub Squares
 Comparable routines as listed above, can be used to generate less conventional Prime Number Magic Squares of order 6, which will be described in following sections.