Office Applications and Entertainment, Magic Squares

14.0   Special Magic Squares, Prime Numbers

14.5   Magic Squares (7 x 7)

Comparable routines as discussed in previous sections can be written to generate Prime Number Magic Squares of order 7, however such routines are not very feasible due to the high number of independent variables, e.g. 24 ea for Pan Magic Squares and 12 ea for Ultra Magic Squares.

In next sections solutions will be found for some more strict defined Prime Number Magic Squares of the 7^th order.

14.5.1 Concentric Magic Squares (7 x 7)

A 7^th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 5^th order, as discussed in Section 14.3.2, with a border around it.

The variable values {a_i} on which a Prime Number Concentric Magic Square of the 7^th order might be based should contain at least 24 pairs.

Based on the possible pairs for a certain amount of Prime Numbers, the corresponding Magic Sums of the outer - and embedded squares (MC₇ and MC₅) can be determined.

Based on the equations defining the border of a Concentric Magic Square (7 x 7):

a(43) =      s1   - a(44) - a(45) - a(46) - a(47) - a(48) - a(49)
a(36) =  2 * s1/7 - a(42)
a(29) =  2 * s1/7 - a(35)  
a(22) =  2 * s1/7 - a(28) 
a(15) =  2 * s1/7 - a(21)
a(14) = -3 * s1/7 + a(15) + a(22) + a(29) + a(36) + a(43) - a(49)
a( 8) =  2 * s1/7 - a(14) 
a( 7) =  2 * s1/7 - a(43)
a( 6) =  2 * s1/7 - a(48)
a( 5) =  2 * s1/7 - a(47)
a( 4) =  2 * s1/7 - a(46)
a( 3) =  2 * s1/7 - a(45)
a( 2) =  2 * s1/7 - a(44)
a( 1) =  2 * s1/7 - a(49) 

a routine can be written to generate Prime Number Concentric Magic Squares of order 7 (ref. Priem7a1).

Attachment 14.6.1 shows one Prime Number Concentric Magic Square for some of the occurring Magic Sums.

It should be noted that:

• the routine Priem7a1 searched only those solutions per Magic Sum which could be found within 10 seconds and
  
  
• although not usual, one (1) has not been excluded from the collection of defining prime numbers.

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

Note:

For Prime Number Concentric Magic Squares of order 7 with Magic Sum s7, it is convenient to split the supplementary rows and columns into parts summing to s4 = 4 * s7 / 7 and s3 = 3 * s7 / 7:

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)	a(37)	a(38)	a(39)	a(40)	a(41)	a(42)
a(43)	a(44)	a(45)	a(46)	a(47)	a(48)	a(49)

This results in following alternative border equations:

a( 4) = s4 - a( 5) - a( 6) - a(7) a(28) = s4 - a(21) - a(14) - a(7) a(43) = s4/2 - a( 7) a(46) = s4/2 - a( 4) a(47) = s4/2 - a( 5) a(48) = s4/2 - a( 6) a( 8) = s4/2 - a(14) a(15) = s4/2 - a(21) a(22) = s4/2 - a(28)

a( 1) = s3 - a( 2) - a( 3) a(35) = s3 - a(42) - a(49) a(44) = s4/2 - a( 2) a(45) = s4/2 - a( 3) a(49) = s4/2 - a( 1) a(36) = s4/2 - a(42) a(29) = s4/2 - a(35)

which enable the development of a much faster routine to generate Prime Number Concentric Magic Squares of order 7 (ref. Priem7a2).

Subject routine produced, based on 1689 previously generated Concentric Magic Squares of order 5, 1689 Prime Number Concentric Magic Square of order 7 within 272 seconds (one square per Magic Sum).

14.5.2 Bordered Magic Squares (7 x 7)
       Miscellaneous Inlays

Based on the collections of 5^th order Ultra Magic and Inlaid Magic Squares, as deducted in Section 14.3.5 thru 14.3.9, also following Bordered Magic Squares can be generated with routine Priem7a1:

• Ultra Magic Center Square                   (ref. Attachment 14.6.2);
  
  
• Associated Center Square with Diamond Inlay (ref. Attachment 14.6.3);
  
  
• Concentric Center Square with Diamond Inlay (ref. Attachment 14.6.4);
  
  
• Center Square with Square Inlay             (ref. Attachment 14.6.5);
  
  
• Center Square with Square and Diamond Inlay (ref. Attachment 14.6.6).

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

14.5.3 Eccentric Magic Squares (7 x 7)

Based on the equations defining the supplementary rows and columns of an Eccentric Magic Square (7 x 7):

a(43) =  2 * s1/7 - a(44) 
a(36) =  2 * s1/7 - a(37) 
a(29) =  2 * s1/7 - a(30) 
a(22) =  2 * s1/7 - a(23) 
a(15) =  2 * s1/7 - a(16) 
a(13) =  5 * s1/7 + a(14) - a(19) - a(25) - a(31) - a(37) - a(43)  
a( 8) = -5 * s1/7 + a( 9) + a(16) + a(23) + a(30) + a(37) + a(44) 
a( 9) =  6 * s1/7 - 0.5 * (a(10) + a(11) + a(12) + a(13) + a(14) + a(16) + a(23) + a(30) + a(37) + a(44)) 
a( 7) =  2 * s1/7 - a(14) 
a( 6) =  2 * s1/7 - a(13) 
a( 5) =  2 * s1/7 - a(12) 
a( 4) =  2 * s1/7 - a(11) 
a( 3) =  2 * s1/7 - a(10) 
a( 2) =  2 * s1/7 - a( 8) 
a( 1) =  2 * s1/7 - a( 9) 

a routine can be written to generate Prime Number Eccentric Magic Squares of order 7 (ref. Priem7b).

Attachment 14.6.7 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} and the related number of possible Magic Corner Squares.

Note:

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

• to read the previously generated Magic Corner Squares of order 5;
• to determine, by means of transformation of an order 3 Magic Square, the top left Corner Pairs (4 ea);
• to complete the Main Diagonal and determine the related Border Pairs (4 ea);
• to complete the Eccentric Magic Square of order 7 with the remainder of the Border Pairs (4 ea).

These principles have been applied successfully in Section 14.8.5, Composed Magic Squares of order 13, for the completion of one of the Embedded Eccentric Magic Squares of order 7 (ref. PriemI7).

14.5.4 Eccentric Magic Squares, Overlapping Sub Squares

Prime Number Eccentric Magic Squares of order 7 with a Magic Sum s1 might contain:

• One 5^th order Magic Corner Square with Magic Sum s5 = 5 * s1 / 7 (bottom/right)
• One 3^th order Semi Magic Corner Square with Magic Sum s3 = 3 * s1 / 7 (top/left)

as shown 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)	a(37)	a(38)	a(39)	a(40)	a(41)	a(42)
a(43)	a(44)	a(45)	a(46)	a(47)	a(48)	a(49)

A dedicated procedure (Priem7e) can be used:

• to read a square from a collection of previously generated 5^th order Ultra Magic Squares (ref. Attachment 14.3.6);
• to transform the Ultra Magic Square to a suitable Corner Square;
• to calculate the 3^th order Semi Magic Corner Square and
• to complete the 7 x 7 Eccentric Magic Square.

Attachment 14.6.10 shows for miscellaneous Magic Sums the first occurring order 7 Prime Number Eccentric Magic Square with Overlapping Sub Squares.

14.5.5 Associated Magic Squares, Overlapping Sub Squares

Prime Number Magic Squares of order 7 with a Magic Sum s1 can also be composed out of:

• Two each 4^th order Magic Corner Squares A and D (s4 = 4 * s1 / 7), with the center element in common, and
• Two each 3^th order Semi Magic Corner Squares B and C (s3 = 3 * s1 / 7).

as shown below.

a1	a2	a3	a4	b1	b2	b3
a5	a6	a7	a8	b4	b5	b6
a9	a10	a11	a12	b7	b8	b9
a13	a14	a15	a16/d1	d2	d3	d4
c1	c2	c3	d5	d6	d7	d8
c4	c5	c6	d9	d10	d11	d12
c7	c8	c9	d13	d14	d15	d16

Associated Magic Squares of order 7 can be constructed based on:

• One Complementary Pair of order 4 Magic Anti Symmetric Corner Squares and
• One Complementary Pair of order 3 Semi Magic Anti Symmetric Corner Squares (7 Magic Lines).

A (Semi) Magic Anti Symmetric Square of order n is a (Semi) Magic Square for which a_i + a_j ≠ 2 * s_n / n for any i and j (i,j = 1 ... n²; i ≠ j).

Anti Symmetric Magic Squares of order 4 can be generated, based on the equations deducted in Section 14.2.1, with the slightly addapted guessing routine Priem4g.

Attachment 14.6.9a shows for miscellaneous Magic Sums the first occurring 4^th order Anti Symmetric Magic Square.

A dedicated procedure (Priem3g) can be used:

• to read the Anti Symmetric Magic Square A;
• to calculate the Complementary Magic Square D;
• to generate the Complementary Semi Magic Squares B and C and
• to construct the 7 x 7 Associated Magic Square.

Attachment 14.6.9b shows for miscellaneous Magic Sums the first occurring order 7 Prime Number Associated Magic Square with Overlapping Sub Squares.

14.5.6 Associated Magic Squares, Diamond Inlays Order 3 and 4

Based on the equations defining Associated Magic Squares (Diamond Inlays) of the seventh order:

a(46) =   4 * s1/7 - a(40) - a(34) - a(28)
a(45) =       s1/7 + a(47) - 2 * a(27) + a(26) - a(20) + a(46) + a(40) - a(28)
a(43) =       s1   - a(44) - a(45) - a(47) - a(48) - a(49) - a(46)
a(39) =   3 * s1/7 - a(33) - a(27)
a(38) =              a(20) - a(46) + a(28)
a(37) = - 3 * s1/7 + a(41) - a(44) + a(48) + a(27) + a(20) + a(34)
a(36) =       s1   - 2 * a(41) - a(42) + a(44) - a(48) + a(33) - 2 * a(20) + a(46) - a(40) - a(34) - a(28)
a(35) = (17 * s1/7 - 2 * a(41) - 2*a(42) - 2*a(47) - 2 * a(48) - 2 * a(49) - a(33) + 2 * a(20) - 2 * a(46) +
                                                                                   - 2 * a(40) - 2 * a(34)) / 2
a(32) =   4 * s1/7 - a(26) - 2 * a(20) + a(46) - a(28)
a(29) =   3 * s1/7 - a(35) - 2 * a(33) - 2 * a(27) + a(26) + 3 * a(20) - a(46) - a(34) + a(28)
a(26) =   4 * s1/7 - a(20) -     a(34) -     a(28)
a(25) =       s1/7
a(19) =   4 * s1/7 - a(33) - 2 * a(27)

a routine can be written to generate subject Associated Magic Squares of order 7 (ref. Priem7e1).

Attachment 14.6.12 shows for miscellaneous Magic Sums the first occurring Prime Number Associated Magic Square.

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

14.5.7 Associated Magic Squares, Square Inlays Order 3 and 4

Based on the equations defining Associated Magic Squares (Square Inlays) of the seventh order:

a(44) =     s1   - a(46) - a(48) - a(43) - a(45) - a(47) - a(49)
a(43) = 4 * s1/7 - a(45) - a(47) - a(49)
a(37) = 3 * s1/7 - a(39) - a(41)
a(36) = 4 * s1/7 - a(38) - a(40) - a(42)
a(33) = 2 * s1/7 - a(35) + 0.5 * a(43) + 0.5 * a(45) - 0.5 * a(47) - 0.5 * a(49)
a(32) = 6 * s1/7 - 2 * a(34) - a(46) - 2 * a(48)
a(31) =            a(33) - a(45) + a(47)
a(30) = 3 * s1/7 - a(32) - a(34)
a(29) = 4 * s1/7 - 2 * a(33) - a(35) + a(45) - a(47)
a(28) = 5 * s1/7 - a(38) - a(40) - 2 * a(42)
a(27) = 4 * s1/7 - a(39) - 2 * a(41)
a(26) =     s1/7 + a(38) - a(40)
a(25) =     s1/7

a routine can be written to generate subject Associated Magic Squares of order 7 (ref. Priem7e2).

Attachment 14.6.13 shows for miscellaneous Magic Sums the first occurring Prime Number Associated Magic Square.

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

Note:

The order 7 Associated Magic Squares as described above can also be obtained by means of a transformation of order 7 Composed Magic Squares as illustrated below:

The Magic Square shown at the left side above is composed out of:

• One 3^th order Simple Magic Corner Square with Magic Sum s3 = 3 * s1 / 7 (top/left)
• One 4^th order Associated Magic Corner Square with Magic Sum s4 = 4 * s1 / 7 (bottom/right)
• Two Associated Magic Rectangles order 3 x 4 with s3 = 3 * s1 / 7 and s4 = 4 * s1 / 7

Based on this definition a routine can be developed to generate subject Composed Magic Squares (ref. Priem7e3).

Attachment 14.6.30 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square.

It should be noted that the reversed transformation is not necessarily possible because of the bottom-left / top-right Main Diagonal.

14.5.8 Ultra Magic Squares (7 x 7)

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

a(43) =       s1   - a(44) - a(45) - a(46) - a(47) - a(48) - a(49)
a(36) =       s1   - a(37) - a(38) - a(39) - a(40) - a(41) - a(42)
a(35) =   6 * s1/7 - a(41) - a(42) - a(47) - a(48) - a(49)
a(34) =              a(35) + a(37) - a(40) + a(44) + a(45) - a(46) - a(48)
a(33) =   6 * s1/7 + a(38) - a(39) - a(40) - a(41) + a(44) - 2 * a(47) - a(48) - a(49)
a(32) =   6 * s1/7 - a(38) - a(40) - a(44) - a(46) - a(48)
a(31) =  12 * s1/7 - a(33) - a(37) - 2 * a(39) - a(41) - a(43) - 2 * a(45) - 2 * a(47) - a(49)
a(30) = - 8 * s1/7 + a(39) + a(40) + 2 * a(41) + a(42) - a(44) + 2 * a(47) + 2 * a(48) + a(49)
a(29) = - 8 * s1/7 + a(38) + a(39) + a(40) + a(41) + a(42) + a(46) + a(47) + a(48) + a(49)
a(28) =       s1/7 - a(37) + a(41) - a(44) - a(45) + a(47) + a(48)
a(27) = -13 * s1/7 + a(39) + 2*a(40) + 2*a(41) + 2*a(42) - a(44) - a(45) + a(46) + 3*a(47) + 3*a(48) + 2*a(49)
a(26) =              a(27) + a(36) - a(42) + a(45) - a(47)
a(25) =       s1/7

a routine can be written to generate Ultra Magic Squares of order 7 (ref. Priem7c).

Attachment 14.6.11 shows for miscellaneous Magic Sums the first occurring Prime Number Ultra Magic Square.

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

14.5.9 Ultra Magic Squares, Order 3 Concentric Square and Square Inlay (a)

Based on the equations defining seventh order Ultra Magic Squares with Order 3 Concentric Square and Square Inlay (a):

a(45) = (    s1   - 2 * a(48) - a(32) - 2 * a(33)) / 2
a(44) =  4 * s1/7 - a(45) - a(47) - a(48)
a(43) =  3 * s1/7 - a(46) - a(49)
a(40) = (9 * s1/7 - 2 * a(48) - 2 * a(32) - 2 * a(33) - a(46)) / 2
a(39) =  3 * s1/7 - a(41) - 2 * a(47) + a(32) + a(33) - a(49)
a(38) = -    s1   + a(40) + a(45) + a(47) + 2 * a(48) + a(32) + 2 * a(33)
a(37) = -    s1   + a(41) + 2 * a(47) + 2 * a(48) + a(46) + 2 * a(49)
a(36) =  9 * s1/7 - a(41) - a(42) - a(45) - a(47) - 2 * a(48) - a(33) - a(49)
a(35) =  6 * s1/7 - a(41) - a(42) - a(47) - a(48) - a(49)
a(34) =  3 * s1/7 - a(40) - a(42) - a(48) + a(49)
a(31) =  3 * s1/7 - a(32) - a(33)
a(30) = -9 * s1/7 + a(40) + a(41) + a(42) + a(45) + a(47) + 3 * a(48) + a(32) + a(33)
a(29) = -3 * s1/7 + a(42) + a(45) + a(48) + a(33)
a(28) =  4 * s1/7 - a(46) - 2 * a(49)
a(27) = -5 * s1/7 + a(41) + 2 * a(42) + 2 * a(47) + 2 * a(48) - a(32) - a(33) + a(49)
a(26) =  4 * s1/7 - a(32) - 2 * a(33)
a(25) =      s1/7

a routine can be written to generate subject Ultra Magic Squares of order 7 (ref. Priem7f1).

Attachment 14.6.14 shows for miscellaneous Magic Sums the first occurring Prime Number Ultra Magic Square.

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

14.5.10 Ultra Magic Squares, Order 3 Concentric Square and Square Inlay (b)

Based on the equations defining seventh order Ultra Magic Squares with Order 3 Concentric Square and Square Inlay (b):

a(46) =       s1   - 2 * a(48) - a(49) - a(32) - a(33) - a(41)
a(44) =  -6 * s1/7 + a(45) + a(47) + a(48) + a(49) + a(33) + a(39) + a(41)
a(43) =   6 * s1/7 - 2 * a(45) - 2 * a(47) - a(49) + a(32) - a(39)
a(42) = (-    s1/7 + 2 * a(45) + a(32) + 2 * a(33) - 2 * a(41))/2
a(40) = ( 5 * s1/7 - 2 * a(47) - a(39))/2
a(38) =            a(40) - a(45) + a(47)
a(37) =   3 * s1/7 - a(39) - a(41)
a(36) =  -    s1/7 - a(42) + a(45) + a(47) + a(39)
a(35) =   6 * s1/7 - a(42) - a(47) - a(48) - a(49) - a(41)
a(34) =   3 * s1/7 - a(35) - a(40) - a(47) + a(41)
a(31) =   3 * s1/7 - a(32) - a(33)
a(30) =  -3 * s1/7 - a(36) + a(38) + a(45) + a(47) + a(48) - a(33) + a(39) + a(41)
a(29) =   4 * s1/7 - a(30) - a(34) - a(35)
a(28) =   4 * s1/7 - 2 * a(45) - a(49) - a(33) + a(41)
a(27) =   4 * s1/7 - a(39) - 2 * a(41)
a(26) =   4 * s1/7 - a(32) - 2 * a(33)
a(25) =       s1/7

a routine can be written to generate subject Ultra Magic Squares of order 7 (ref. Priem7f2).

Attachment 14.6.15 shows for miscellaneous Magic Sums the first occurring Prime Number Ultra Magic Square.

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

14.5.11 Ultra Magic Squares, Order 3 Square Inlays

Based on the equations defining seventh order Ultra Magic Squares with Order 3 Square Inlays:

a(47) = (10 * s1/7 - 2 * a(48) - a(39) - 2 * a(41) - a(46) - 2 * a(49))/2
a(44) =   4 * s1/7 - a(45) - a(47) - a(48)
a(43) =   3 * s1/7 - a(46) - a(49)
a(40) =  (    s1/7 - 2 * a(42) + a(46) + 2 * a(49))/2
a(37) =   3 * s1/7 - a(39) - a(41)
a(36) =   4 * s1/7 - a(38) - a(40) - a(42)
a(35) =   6 * s1/7 - a(42) - a(47) - a(48) - a(41) - a(49)
a(34) =   3 * s1/7 - a(40) - a(42) - a(48) + a(49)
a(33) =              a(38) - a(40) - a(45) - a(47) + a(41) + a(46) + a(49)
a(32) =       s1/7 - a(38) + a(40) + 2 * a(42) + a(45) + a(47) - 2 * a(46) - 2 * a(49)
a(31) = - 4 * s1/7 - a(38) + a(40) - a(45) + a(47) + 2 * a(48) + a(41) + a(46) + a(49)
a(30) = - 2 * s1/7 + a(40) + a(42) + a(45) + a(47) + a(48) - a(46) - a(49)
a(29) =   3 * s1/7 + a(38) - a(40) - a(42) - a(47) - a(48) - a(41) + a(46) + a(49)
a(28) =   4 * s1/7 - a(46) - 2 * a(49)
a(27) =   4 * s1/7 - a(39) - 2 * a(41)
a(26) =   8 * s1/7 - a(38) - a(40) - 2 * a(42) + a(45) - a(47) - a(39) - 2 * a(41)
a(25) =       s1/7

a routine can be written to generate subject Ultra Magic Squares of order 7 (ref. Priem7e5).

Attachment 14.6.16 shows for miscellaneous Magic Sums the first occurring Prime Number Ultra Magic Square.

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

14.5.12 Ultra Magic Squares, Order 3 Square and Diamond Inlay (a)

Based on the equations defining seventh order Ultra Magic Squares with Order 3 Square and Diamond Inlay:

a(44) =   4 * s1/7 - a(45) - a(47) - a(48)
a(43) =   3 * s1/7 - a(46) - a(49)
a(41) =   6 * s1/7 - a(45) - 2 * a(47) - a(48) - a(49)
a(40) =   4 * s1/7 - a(42) + a(45) - a(48) - a(39) - (a(33) + a(46))/2
a(37) = -     s1/7 - a(45) + a(48) + a(46) + a(49)
a(38) =            - a(42) + a(45) + a(47) + (a(33) - a(46))/2
a(36) = - 2 * s1/7 + a(42) + a(47) + a(48)
a(35) =            - a(42) + a(45) + a(47)
a(34) = -     s1/7 - a(45) + a(39) + a(49) + (a(33) + a(46))/2 
a(32) = - 2 * s1/7 + 2 * a(42) - a(45) + a(48) + a(39)
a(31) =   4 * s1/7 - a(33) - 2 * a(39)
a(30) =   4 * s1/7 - a(47) - a(49) - (a(33) + a(46))/2 
a(29) =   2 * s1/7 - a(42) + a(45) -  a(48)
a(28) =   4 * s1/7 - a(46) - 2 * a(49)
a(27) =   3 * s1/7 - a(33) - a(39)
a(26) =       s1/7 + a(45) + a(48) - a(33) - a(39)
a(25) =       s1/7

a routine can be written to generate subject Ultra Magic Squares of order 7 (ref. Priem7g1).

Attachment 14.6.17 shows for miscellaneous Magic Sums the first occurring Prime Number Ultra Magic Square.

It should be noted that the routine Priem7g1 searched only those solutions per Magic Sum which could be found within 30 seconds.

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

14.5.13 Ultra Magic Squares, Order 3 Square and Diamond Inlay (b)

Based on the equations defining seventh order Ultra Magic Squares with Order 3 Square and Diamond Inlay:

a(44) =   2 * s1/7 + a(45) - a(46) + a(47) - a(48) - a(39)
a(43) =   5 * s1/7 - 2 * a(45) - 2 * a(47) - a(49) + a(39)
a(40) = (19 * s1/7 - 2 * a(42) + 2 * a(45) - 2 * a(46) - 2*a(47) - 4*a(48) - 2*a(49) - 3*a(39) - 4*a(41))/2
a(38) = (     s1/7 - 2 * a(42) + a(39) + 2 * a(41))/2
a(37) =   3 * s1/7 - a(39) - a(41)
a(36) = - 6 * s1/7 + a(42) - a(45) + a(46) + a(47) + 2 * a(48) + a(49) + a(39) + a(41)
a(35) =   6 * s1/7 - a(42) - a(47) - a(48) - a(49) - a(41)
a(34) =   2 * s1/7 - a(38) - a(42) + a(45) - a(46) + a(47) - a(48) + a(41)
a(33) = -     s1/7 + 2 * a(41)
a(32) = - 6 * s1/7 + 2 * a(42) - 2 * a(45) + a(46) + 2 * a(48) + a(49) + 2 * a(39) + a(41)
a(31) =   5 * s1/7 - 2 * a(39) - 2 * a(41)
a(30) = -     s1/7 + a(38) + a(42) + a(48) - a(41)
a(29) =   2 * s1/7 - a(42) + a(45) - a(48)
a(28) = - 4 * s1/7 - 2 * a(45) + a(46) + 2 * a(48) + 2 * a(39) + 2 * a(41)
a(27) =   4 * s1/7 - a(39) - 2 * a(41)
a(26) = - 2 * s1/7 + a(46) + 2 * a(48) + a(49) - a(41)
a(25) =       s1/7

a routine can be written to generate subject Ultra Magic Squares of order 7 (ref. Priem7g2).

Attachment 14.6.18 shows for miscellaneous Magic Sums the first occurring Prime Number Ultra Magic Square.

It should be noted that the routine Priem7g2 searched only those solutions per Magic Sum which could be found within 30 seconds.

Each square shown corresponds with eight (Check) squares for the applicable Magic Sum and variable values {a_i}.

14.5.14 Ultra Magic Squares, Order 3 Concentric Square and Diamond Inlay

Based on the equations defining seventh order Ultra Magic Squares with Order 3 Concentric Square and Diamond Inlay:

a(43) =       s1   - a(44) - a(45) - a(46) - a(47) - a(48) - a(49)
a(42) = (-3 * s1/7 + 2 * a(45) + a(33) + 2 * a(39))/2 
a(41) =   6 * s1/7 + a(44) - a(45) - a(47) - a(48) - a(49) - a(33) - a(39)
a(40) = (     s1   - a(44) + a(45) - a(46) - a(47) - a(48) - 2 * a(39))/2
a(38) =            a(40) - a(45) + a(47)
a(37) = - 5 * s1/7 + a(46) + 2 * a(48) + a(49) + a(33) + a(39)
a(36) = (     s1/7 + 2 * a(47) - a(33)) / 2
a(35) =   3 * s1/7 + a(42) - a(44) - a(45) - a(39)
a(34) = - 2 * s1/7 - a(40) + a(42) + a(48) + a(49) + a(33)
a(32) = -     s1/7 + 2 * a(39)
a(31) =   4 * s1/7 - a(33) - 2 * a(39)
a(30) =   8 * s1/7 - a(40) - a(42) + a(45) - a(46) - a(47) - a(48) - a(49) - a(33) - a(39)
a(29) = (     s1   - 2 * a(48) - a(33) - 2 * a(39))/2
a(28) =  12 * s1/7 - 2 * a(45) - a(46) - 2 * a(48) - 2 * a(49) - 2 * a(33) - 2 * a(39)
a(27) =   3 * s1/7 - a(33) - a(39)
a(26) =   5 * s1/7 - 2 * a(33) - 2 * a(39)
a(25) =       s1/7

a routine can be written to generate subject Ultra Magic Squares of order 7 (ref. Priem7h).

Attachment 14.6.19 shows for miscellaneous Magic Sums the first occurring Prime Number Ultra Magic Square.

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

14.5.15 Ultra Magic Squares, Three Cell Patterns

Based on the equations defining seventh order Ultra Magic Squares, for which all Three Cell patterns sum to 3*s1/7:

a(33) =  3*s1 / 7 - a(34) - a(40)
a(30) =  3*s1 / 7 - a(31) - a(38)
a(46) = -2*s1 / 7 + a(31) + 2 * a(38)
a(32) =  5*s1 / 7 - a(39) - a(31) - 2 * a(38)
a(28) =    s1 / 7 + a(34) - a(40)
a(26) =  2*s1 / 7 - a(27) - a(34) + a(40)
a(47) =  3*s1 / 7 - a(48) - a(49)
a(45) =  7*s1 / 7 - a(48) - a(49) - a(27) - a(31) - a(38) - 2 * a(34) + a(40)
a(44) =  3*s1 / 7 - a(48) + a(39) - a(38) - a(40)
a(43) = -4*s1 / 7 + 2 * a(48) + a(49) + a(27) - a(39) + 2 * a(34)
a(42) =  8*s1 / 7 - a(48) - a(49) - a(31) - 2 * a(38) - 2 * a(34)
a(41) =             a(49) + a(34) - a(40)
a(37) = -7*s1 / 7 + 2 * a(48) + a(49) + a(27) - a(39) + a(31) + 2 * a(38) + 2 * a(34)
a(36) =  6*s1 / 7 - a(48) - a(49) - a(27) - a(38) - a(34)
a(35) = -5*s1 / 7 + a(48) + a(31) + 2 * a(38) + a(34) + a(40)
a(29) =    s1 / 7 - a(48) + a(39) + a(38) - a(34)
a(25) =    s1 / 7

a routine can be written to generate subject Ultra Magic Squares of order 7 (ref. Priem7d).

Attachment 14.6.20 shows for a few Magic Sums the first occurring Prime Number Ultra Magic Square.

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

14.5.16 Summary

The obtained results regarding the miscellaneous types of order 7 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 8, which will be described in following sections.