Office Applications and Entertainment, Magic Squares

14.0   Special Magic Squares, Prime Numbers

Prime Number Magic squares can be generated with comparable routines as discussed in previous section, however based on prime variable values {a_i} with i = 1 ... n.

Prime Numbers can be separately generated and read by the guessing routine.

Attachment 14.0 shows the first 170 Prime Numbers, generated with routine Priem (based on a routine published in the Control Data Basic A Manual, 1979).

As the Magic Sum will depend from the selected Prime Numbers, the Magic Sum has to be made variable as well.

14.1   Magic Squares (3 x 3)

14.1.1 Simple Magic Squares (3 x 3)

The equations defining a Magic Square of the third order are:

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

with a(9) and a(8) the independent variables. Consequently the variable Magic Sum s1 should be divisible by 3.

For a certain range of Prime Numbers, the minimum (maximum) expected Magic Sum can be roughly estimated by one third of the sum of the first (last) nine prime numbers of the selected range.

Attachment 14.1 shows 184 Prime Number Magic Squares of order 3, based on the first 72 Prime Numbers, generated within 11.8 seconds (ref. Priem3).

Attachment 14.2 shows some Prime Number Magic Squares of order 3 with more interesting characteristics, which can be found within a wider range of Prime Numbers (2 ... 9923).

14.1.2 Semi Magic Squares (3 x 3), Seven Magic Lines

In Section 14.4.10 will be discussed how 6^th order Prime Number Simple Magic Squares can be constructed based on 3^th order Prime Number Magic Squares with only seven Magic Lines (Semi Magic).

The equations defining Semi Magic Squares with seven Magic Lines of the third order are:

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 a(9), a(8) and a(6) the independent variables.

Based on the equations shown above a routine can be written to generate Prime Number Semi Magic Squares of order 3 (ref. Priem3b).

Attachment 14.1.2 shows for each occurring Magic Sum one Prime Number Semi Magic Square of order 3, based on the first 563 Prime Numbers (1 ... 4079).

14.2   Magic Squares (4 x 4)

14.2.1 Simple Magic Squares (4 x 4)

Based on the equations defining a Magic Square of the fourth order:

a(13) = s1 - a(14) - a(15) - a(16)
a( 9) = s1 - a(10) - a(11) - a(12)
a( 7) =      a(8) - a(10) + a(12) - a(13) + a(16)
a( 6) = s1 - a(8) - a(11) - a(12) + a(13) - a(16)
a( 5) =    - a(8) + a(10) + a(11)
a( 4) = s1 - a(7) - a(10) - a(13)
a( 3) =-s1 - a(8) + a(9) + 2 * a(10) +         2 * a(13) + a(14)
a( 2) =      a(8) - a(9) - 2 * a(10) + a(15) + 2 * a(16)
a( 1) =      a(8) + a(12) - a(13)

a comparable routine can be written to generate Prime Number Magic Squares of order 4 (ref. Priem4).

As for certain Magic Sums the number of related Prime Number Magic Squares of order 4 might be quite high, the range of Prime Numbers considered has been limited to the first 30 (2 ... 113).

Attachment 14.2.0 shows one Prime Number Magic Square for each occurring Magic Sum.

Attachment 14.2.1 shows Prime Number Magic Squares with consecutive Prime Numbers for MC = 258 (32 ea) and MC = 276 (64 ea).

14.2.2 Pan Magic Squares (4 x 4)

Based on the equations defining a Pan Magic Square of the fourth order:

a(13) =        s1 - a(14) - a(15) - a(16)
a(11) =        s1 - a(12) - a(15) - a(16)
a(10) =             a(12) - a(14) + a(16)
a(9)  =           - a(12) + a(14) + a(15)
a(8)  =  0.5 * s1 - a(14)
a(7)  = -0.5 * s1 + a(14) + a(15) + a(16)
a(6)  =  0.5 * s1 - a(16)
a(5)  =  0.5 * s1 - a(15)
a(4)  =  0.5 * s1 - a(12) + a(14) - a(16)
a(3)  =  0.5 * s1 + a(12) - a(14) - a(15)
a(2)  =  0.5 * s1 - a(12)
a(1)  = -0.5 * s1 + a(12) + a(15) + a(16) 

a comparable routine can be written to generate Prime Number Pan Magic Squares of order 4 (ref. Priem4b).

Attachment 14.2.2 shows Prime Number Pan Magic Squares with MC = 240 (2 x 384 ea) and MC = 252 (384 ea), being the only Prime Number Pan Magic Squares occurring in the range (2 ... 113).

Attachment 14.2.3 shows for a wider range (5 ... 619) one Prime Number Pan Magic Square for each occurring Magic Sum.

Each square shown corresponds with 384 Prime Number Pan Magic Squares with the same Magic Sum and variable values {a_i}.

14.2.3 Associated Magic Squares (4 x 4)

Based on the equations defining an Associated Magic Square of the fourth order:

a(13) =       s1 - a(14) - a(15) - a(16)
a(11) =       s1 - a(12) - a(15) - a(16)
a(10) =       s1 - a(12) - a(14) - a(16)
a( 9) =       s1 - a(10) - a(11) - a(12)
a( 8) = 0.5 * s1 - a( 9)
a( 7) = 0.5 * s1 - a(10)
a( 6) = 0.5 * s1 - a(11)
a( 5) = 0.5 * s1 - a(12)
a( 4) = 0.5 * s1 - a(13)
a( 3) = 0.5 * s1 - a(14)
a( 2) = 0.5 * s1 - a(15)
a( 1) = 0.5 * s1 - a(16)

a comparable routine can be written to generate Prime Number Associated Magic Squares of order 4 (ref. Priem4d).

Attachment 14.2.4 shows for the same range one Prime Number Associated Magic Square for each occurring Magic Sum.

Each square shown corresponds with 384 Prime Number Associated Magic Squares with the same Magic Sum and variable values {a_i}.

14.3   Magic Squares (5 x 5)

A comparable routine can be written to generate Prime Number Magic Squares of order 5, however such a routine is not very feasible due to the high number of independent variables (14 ea).

In next sections solutions will be found for more strict defined Prime Number Magic Squares of the 5^th order.

14.3.1 Pan Magic Squares (5 x 5)

Based on the equations defining a Pan Magic Square of the fifth order:

a(21) =  s1 - a(22) - a(23) - a(24) - a(25)
a(16) =  s1 - a(17) - a(18) - a(19) - a(20)
a(15) =       a(17) + a(18) - a(25)
a(14) =  s1 - a(18) - a(19) - a(20) - a(24)
a(13) =  s1 - a(17) - a(18) - a(19) - a(23)
a(12) =       a(19) + a(20) - a(22)
a(11) =       a(18) + a(19) - a(21)
a(10) =  s1 - a(17) - a(18) - a(22) - a(23)
a( 9) =  s1 - a(16) - a(17) - a(21) - a(22)
a( 8) =  s1 - a(16) - a(20) - a(21) - a(25)
a( 7) =  s1 - a(19) - a(20) - a(24) - a(25)
a( 6) =  s1 - a(18) - a(19) - a(23) - a(24)
a( 5) =     - a(20) + a(22) + a(23)
a( 4) =     - a(19) + a(21) + a(22)
a( 3) =     - a(18) + a(21) + a(25)
a( 2) =     - a(17) + a(24) + a(25)
a( 1) =     - a(16) + a(23) + a(24) 

a comparable routine can be written to generate Prime Number Pan Magic Squares of order 5 (ref. Priem5b1).

The range of Prime Numbers considered has been limited to the first 72 (2 ... 353).

Attachment 14.3.1 shows one Prime Number Pan Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with 28800 Pan Magic Squares with the same Magic Sum and variable values {a_i}.

14.3.2 Concentric Magic Squares (5 x 5)

A 5^th order Prime Number Concentric Magic Square consists of a Prime Number Embedded Magic Square of the 3^th order with a border around it.

The Embedded Magic Square consists of a center element and 4 pairs. The border consists of 8 pairs. Consequently the variable values {a_i} on which a Prime Number Concentric Magic Square might be based should contain at least 12 pairs.

Based on the possible pairs for the first 170 Prime Numbers (2 ... 1013) the corresponding Magic Sums of the outer - and embedded squares (MC₅ and MC₃) can be determined.

• Attachment 14.3.2 page 1 shows these data for the occurring Magic Sums MC₃ and n_pair >= 12;
• Attachment 14.3.2 page 2 shows for each of the listed Magic Sums MC₅ the corresponding variable values {a_i}.

Based on the equations defining a Concentric Magic Square of the fifth order:

a(21) =        s1 - a(22) - a(23) - a(24) - a(25) 
a(17) =  0.6 * s1 - a(18) - a(19) 
a(16) =  0.4 * s1 - a(20) 
a(14) =  0.8 * s1 - a(18) - 2 * a(19) 
a(13) =  0.2 * s1 
a(12) =  0.4 * s1 - a(14) 
a(11) =  0.4 * s1 - a(15) 
a(10) =  0.6 * s1 - a(15) - a(20) + a(21) - a(25) 
a( 9) =  0.4 * s1 - a(17) 
a( 8) =  0.4 * s1 - a(18) 
a( 7) =  0.4 * s1 - a(19) 
a( 6) =  0.4 * s1 - a(10) 
a( 5) =  0.4 * s1 - a(21) 
a( 4) =  0.4 * s1 - a(24) 
a( 3) =  0.4 * s1 - a(23) 
a( 2) =  0.4 * s1 - a(22) 
a( 1) =  0.4 * s1 - a(25) 

a comparable routine can be written to generate Prime Number Concentric Magic Squares of order 5 (ref. Priem5c).

Attachment 14.3.3 shows one Prime Number Concentric Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with multiples of 2880 for the same Magic Sum, depending from the selected variable values {a_i} and the related number of possible Embedded Magic Squares (n3).

14.3.3 Eccentric Magic Squares (5 x 5)

A 5^th order Prime Number Eccentric Magic Square consists of one Prime Number Magic Corner Square of the 3^th order, supplemented with two rows and two columns.

Based on the method discussed above and the equations defining an Eccentric Magic Square of the fifth order:

a(23) =  0.6 * s1 - a(24) - a(25) 
a(21) =  0.4 * s1 - a(22) 
a(20) =  0.8 * s1 - a(24) - 2 * a(25) 
a(19) =  0.2 * s1 
a(18) =  0.4 * s1 - a(20) 
a(16) =  0.4 * s1 - a(17) 
a(15) =  0.4 * s1 - a(23) 
a(14) =  0.4 * s1 - a(24) 
a(13) =  0.4 * s1 - a(25) 
a(11) =  0.4 * s1 - a(12) 
a( 9) =  0.6 * s1 + a(10) - a(13) - a(17) - a(21) 
a( 7) =  0.4 * s1 -(a( 8) + a( 9) + a(10) + a(12) - a(16) - a(21))/2
a( 6) =        s1 - a( 7) - a( 8) - a( 9) - a(10) 
a( 5) =  0.4 * s1 - a(10) 
a( 4) =  0.4 * s1 - a( 9) 
a( 3) =  0.4 * s1 - a( 8) 
a( 2) =  0.4 * s1 - a( 6) 
a( 1) =  0.4 * s1 - a( 7) 

a comparable routine can be written to generate Prime Number Eccentric Magic Squares of order 5 (ref. Priem5d).

Attachment 14.3.4 shows one Prime Number Eccentric Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum, depending from the selected variable values {a_i}, the related number of possible Magic Corner Squares (n3) and the key variable a(13).

14.3.4 Center Symmetric Magic Squares (5 x 5)

A 5^th order Prime Number Center Symmetric Magic Square is a Prime Number Magic Square where the two numbers of each of the 12 Center Symmetric Pairs are Complementary.

Based on the method discussed above and the equations defining a Center Symmetric Magic Square of the fifth order:

a(21) =     s1     - a(22) - a(23) - a(24) - a(25)
a(16) =     s1     - a(17) - a(18) - a(19) - a(20)
a(15) =     s1 / 5 + a(16) - a(20) + a(21) - a(25)
a(14) =     s1 / 5 + a(17) - a(19) + a(22) - a(24)
a(13) =     s1 / 5
a(12) = 2 * s1 / 5 - a(14)
a(11) = 2 * s1 / 5 - a(15)
a(10) = 2 * s1 / 5 - a(16)
a( 9) = 2 * s1 / 5 - a(17)
a( 8) = 2 * s1 / 5 - a(18)
a( 7) = 2 * s1 / 5 - a(19)
a( 6) = 2 * s1 / 5 - a(20)
a( 5) = 2 * s1 / 5 - a(21)
a( 4) = 2 * s1 / 5 - a(22)
a( 3) = 2 * s1 / 5 - a(23)
a( 2) = 2 * s1 / 5 - a(24)
a( 1) = 2 * s1 / 5 - a(25)

a comparable routine can be written to generate Prime Number Center Symmetric Magic Squares of order 5 (ref. Priem5e).

Attachment 14.3.5 shows one Prime Number Center Symmetric Magic Square for some of the occurring Magic Sums.

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

14.3.5 Ultra Magic Squares (5 x 5)

A 5^th order Prime Number Ultra Magic Square is a Prime Number Center Symmetric Pan Magic Square.

Based on the method discussed above and the equations defining an Ultra Magic Square of the fifth order:

a(21) =        s1 - a(22) - a(23) - a(24) - a(25)
a(20) =  0.6 * s1 - a(24) - a(25)
a(19) =  0.6 * s1 + a(22) - a(23) - a(24) - a(25)
a(18) =  0.6 * s1 - a(22) - a(24)
a(17) = -0.4 * s1 + 2 * a(24) + a(25)
a(16) = -0.4 * s1 + a(23) + a(24) + a(25)
a(15) =  0.2 * s1 - a(22) + a(24)
a(14) = -0.8 * s1 + a(23) + 2 * a(24) + 2 * a(25)
a(13) =        s1 / 5
a(12) =    2 * s1 / 5 - a(14)
a(11) =    2 * s1 / 5 - a(15)
a(10) =    2 * s1 / 5 - a(16)
a( 9) =    2 * s1 / 5 - a(17)
a( 8) =    2 * s1 / 5 - a(18)
a( 7) =    2 * s1 / 5 - a(19)
a( 6) =    2 * s1 / 5 - a(20)
a( 5) =    2 * s1 / 5 - a(21)
a( 4) =    2 * s1 / 5 - a(22)
a( 3) =    2 * s1 / 5 - a(23)
a( 2) =    2 * s1 / 5 - a(24)
a( 1) =    2 * s1 / 5 - a(25)

a comparable routine can be written to generate Prime Number Ultra Magic Squares of order 5 (ref. Priem5f).

Attachment 14.3.6 shows one Prime Number Ultra Magic Square for 48 of the occurring Magic Sums.

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

14.3.6 Diamond Inlay, Associated Magic Squares (5 x 5)

When an embedded Magic Square is rotated 45 degrees, the embedded square is referred to as a Diamond Inlay.
Based on the equations defining a 5^th order Associated Magic Square with a 3^th order Diamond Inlay:

a(21) =        s1 - a(22) - a(23) - a(24) - a(25)
a(18) =  0.8 * s1 + a(19) - 2 * a(20) - a(22) + 2 * a(23) - a(24) - 2 * a(25)
a(17) =  0.8 * s1 - a(19) - 2 * a(23)
a(16) = -0.6 * s1 - a(19) + a(20) + a(22) + a(24) + 2 * a(25)
a(15) =  0.6 * s1 - a(19) - a(23)
a(14) =        s1 - 2 * a(19) + a(22) - 2 * a(23) - a(24)
a(13) =  0.2 * s1
a(12) =  0.4 * s1 - a(14)
a(11) =  0.4 * s1 - a(15)
a(10) =  0.4 * s1 - a(16)
a( 9) =  0.4 * s1 - a(17)
a( 8) =  0.4 * s1 - a(18)
a( 7) =  0.4 * s1 - a(19)
a( 6) =  0.4 * s1 - a(20)
a( 5) =  0.4 * s1 - a(21)
a( 4) =  0.4 * s1 - a(22)
a( 3) =  0.4 * s1 - a(23)
a( 2) =  0.4 * s1 - a(24)
a( 1) =  0.4 * s1 - a(25)

a comparable routine can be written to generate Associated Magic Squares with Diamond Inlays (ref. Priem5g1).

Attachment 14.3.7 shows one Prime Number Associated Magic Square with Diamond Inlay for 48 of the occurring Magic Sums.

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

14.3.7 Diamond Inlay, Concentric Magic Squares (5 x 5)

Based on the equations defining a fifth order Concentric Magic Square with a third order Diamond Inlay:

a(21) =        s1 - a(22) - a(23) - a(24) - a(25)
a(18) = -0.2 * s1 + 2 * a(23)
a(17) =  0.8 * s1 - a(19) - 2 * a(23)
a(16) =  0.4 * s1 - a(20)
a(15) =  0.6 * s1 - a(19) - a(23)
a(14) =        s1 - 2 * a(19) - 2 * a(23)
a(13) =  0.2 * s1
a(12) =  0.4 * s1 - a(14)
a(11) =  0.4 * s1 - a(15)
a(10) =        s1 + a(19) - a(20) - a(22) - a(24) - 2 * a(25)
a( 9) =  0.4 * s1 - a(17)
a( 8) =  0.4 * s1 - a(18)
a( 7) =  0.4 * s1 - a(19)
a( 6) =  0.4 * s1 - a(10)
a( 5) =  0.4 * s1 - a(21)
a( 4) =  0.4 * s1 - a(24)
a( 3) =  0.4 * s1 - a(23)
a( 2) =  0.4 * s1 - a(22)
a( 1) =  0.4 * s1 - a(25)

a comparable routine can be written to generate Concentric Magic Squares with Diamond Inlays (ref. Priem5g2).

Attachment 14.3.8 shows one Prime Number Concentric Magic Square with Diamond Inlay for each of the occurring Magic Sums for the first 1224 Prime Numbers (2 ... 9923).

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

14.3.8 Square Inlay, General (5 x 5)

Based on the equations defining a fifth order Magic Square with a third order Square Inlay:

a(22) =  0.4 * s1 - a(24)
a(21) =        s1 - a(22) - a(23) - a(24) - a(25)
a(16) =        s1 - a(17) - a(18) - a(19) - a(20)
a(15) = -0.4 * s1 + 2 * a(21) + a(23)
a(13) =  0.2 * s1
a(12) =  0.4 * s1 - a(14)
a(11) =             a(13) - a(21) + a(25)
a(10) =  0.4 * s1 - a(20)
a( 9) =  0.4 * s1 - a(17)
a( 8) =  0.4 * s1 - a(18)
a( 7) =  0.4 * s1 - a(19)
a( 6) =  0.4 * s1 - a(16)
a( 5) =        s1 - a(10) - a(15) - a(20) - a(25)
a( 4) =        s1 - a( 9) - a(14) - a(19) - a(24)
a( 3) =        s1 - a( 8) - a(13) - a(18) - a(23)
a( 2) =        s1 - a( 7) - a(12) - a(17) - a(22)
a( 1) =        s1 - a( 6) - a(11) - a(16) - a(21)

a comparable routine can be written to generate Magic Squares with Square Inlays (ref. Priem5g3).

Attachment 14.3.9 shows one Prime Number Magic Square with third order Square Inlay for 48 of the occurring Magic Sums.

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

14.3.9 Square and Diamond Inlay (5 x 5)

Based on the equations defining a fifth order Magic Square with third order Square and Diamond Inlay:

a(22) =  0.4 * s1 - a(24)
a(21) =  0.6 * s1 - a(23) - a(25)
a(19) = -0.2 * s1 + 2 * a(25)
a(17) =        s1 - 2 * a(23) - 2 * a(25)
a(16) =  0.2 * s1 - a(18) - a(20) + 2 * a(23)
a(15) =  0.8 * s1 - a(23) - 2 * a(25)
a(13) =  0.2 * s1
a(12) =  0.4 * s1 - a(14)
a(11) = -0.4 * s1 + a(23) + 2 * a(25)
a(10) =  0.4 * s1 - a(20)
a( 9) = -0.6 * s1 + 2 * a(23) + 2 * a(25)
a( 8) =  0.4 * s1 - a(18)
a( 7) =  0.6 * s1 - 2 * a(25)
a( 6) =  0.2 * s1 + a(18) + a(20) - 2 * a(23)
a( 5) = -0.2 * s1 + a(23) + a(25)
a( 4) =  1.8 * s1 - a(14) - 2 * a(23) - a(24) - 4 * a(25)
a( 3) =  0.4 * s1 - a(23)
a( 2) = -1.4 * s1 + a(14) + 2 * a(23) + a(24) + 4 * a(25)
a( 1) =  0.4 * s1 - a(25)

a comparable routine can be written to generate Magic Squares with Square and Diamond Inlays (ref. Priem5g4).

Attachment 14.3.10 shows one Prime Number Magic Square with third order Square and Diamond Inlay for each of the occurring Magic Sums for the first 1224 Prime Numbers (2 ... 9923).

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

14.3.10 Anti Symmetric Magic Squares (5 x 5)

Anti Symmetric Magic Squares of order 5 are Magic Squares for which a_i + a_j ≠ 0.4 * s1 for any i and j (i,j = 1 ... 25; i ≠ j). Anti Symmetric Magic Squares occur in complementary pairs.

Anti Symmetric Magic Squares can be used for the construction of:

1. Partly Symmetric Composed Magic Squares of order 9 (ref. Section 14.7.2)
2. Associated Composed Magic Squares of order 9       (ref. Section 14.7.3)
3. Associated Composed Magic Squares of order 10      (ref. Section 14.8.13)
4. Associated Composed Magic Squares of order 15      (ref. Section 14.11.15)
5. Concentric Magic Cubes of order 5                  (ref. Section 7.3.3)

Anti Symmetric Magic Squares, suitable for the construction of ad. a and b above, must have the number s1/5 in one of the corners, say the bottom/right corner.

Heuristically I worked out following procedure:

1. Identify the Magic Sums for which Ultra Magic Squares exist;
2. Generate, based on the formula's for Pan Magic Squares and the Magic Sums found under ad. 1, Pan Magic Squares with Corner Element s1/5 (ref. AntSym1a);
3. Generate, based on the formula's of Simple Magic Squares, starting with the values of the Pan Magic Squares found under ad. 2, Simple Anti Magic Squares, with Corner Element s1/5 (ref. AntSym1b).

Above described procedure generated 184 Prime Number Anti Symmetric Magic Squares - with Corner Element s1/5 - per two hours (average), of which 48 are shown in Attachment 14.3.11.

Note: Occasionally Anti Symmetric Pan Magic Squares were found.

For Anti Symmetric Magic Squares, suitable for the construction of ad. c, d and e above, the number s1/5 should be eliminated.

This can be achieved with a procedure, based on the formula's of Simple Magic Squares:

a(21) =  s1 - a(22) - a(23) - a(24) - a(25)
a(16) =  s1 - a(17) - a(18) - a(19) - a(20)
a(11) =  s1 - a(12) - a(13) - a(14) - a(15)
a( 9) =       a(10) - a(13) + a(15) - a(17) + a(20) - a(21) + a(25)
a( 7) = (s1 - a( 8) - a( 9) - a(10) + a(11) - a(13) + a(16) - a(19) + a(21) - a(25)) / 2
a( 6) =  s1 - a( 7) - a( 8) - a( 9) - a(10)
a( 5) =  s1 - a( 9) - a(13) - a(17) - a(21)
a( 4) =  s1 - a( 9) - a(14) - a(19) - a(24)
a( 3) =  s1 - a( 8) - a(13) - a(18) - a(23)
a( 2) =  s1 - a( 7) - a(12) - a(17) - a(22)
a( 1) =  s1 - a( 2) - a( 3) - a( 4) - a( 5)

but starting with the values of the Anti Symmetric Magic Squares listed in Attachment 14.3.11 (ref. AntSym2).

Procedure AntSym2 transferred 184 Prime Number Anti Symmetric Magic Squares per hour (average), of which 48 are shown in Attachment 14.3.12.

14.3.11 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:

Comparable routines as listed above, can be used to generate Prime Number Magic Squares of order 6, which will be described in following sections.