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14.0   Special Magic Squares, Prime Numbers

14.7   Magic Squares (9 x 9)

14.7.1 Magic Squares (9 x 9), Overlapping Subsquares

Prime Number Magic Squares of order 9 with a Magic Sum s1 can be composed out of:

• Two each 4th order Pan Magic Corner Squares A and D (MC4 = 4 * s1 / 9) and
• Two each 5th order Pan Magic Corner Squares B and C (MC5 = 5 * s1 / 9) with the center element in common.

as illustrated below:

 a1 a2 a3 a4 b1 b2 b3 b4 b5 a5 a6 a7 a8 b6 b7 b8 b9 b10 a9 a10 a11 a12 b11 b12 b13 b14 b15 a13 a14 a15 a16 b16 b17 b18 b19 b20 c1 c2 c3 c4 c5/b21 b22 b23 b24 b25 c6 c7 c8 c9 c10 d1 d2 d3 d4 c11 c12 c13 c14 c15 d5 d6 d7 d8 c16 c17 c18 c19 c20 d9 d10 d11 d12 c21 c22 c23 c24 c25 d13 d14 d15 d16

In previous section a procedure has been applied to generate sets of 4th order Prime Number Pan Magic Squares with 16 different Prime Numbers for a Magic Sum MC4 (Priem4c).

A comparable routine (Priem5a) can be developed to generate sets of two 5th order Prime Number Ultra Magic Squares with 24 different Prime Numbers but the same center element for a Magic Sum MC5 (ref. Attachment 14.7.1).

A dedicated procedure (Priem4e), based on these two principles, can be used to transform the Ultra Magic Squares to Pan Magic Squares B and C, to generate the Pan Magic Squares A and D and to construct the 9 x 9 Magic Squares.

Attachment 14.7.2 shows for miscellaneous Magic Sums the first occurring 9th order Prime Number Magic Square with Overlapping Subsquares.

Based on the aspect of the Main Square shown above and one set of 4 unique Prime Number Pan Magic Squares 2 * 3842 * 2 * 11522 = 7,8 1011 Prime Number Magic Squares of the 9th order can be constructed.

14.7.2 Magic Squares (9 x 9), Overlapping Subsquares, Partly Symmetric

Composed Squares, as described in Section 14.7.1 above, can't be Center Symmetric because of the two 4th order Pan Magic Corner Squares.

However for certain Magic Sums the two 5th order Corner Squares can be constructed in such a way that the elements of both squares are complementary.

To achieve this it is sufficient to construct one 5th order Magic Square, further referred to as Anti Symmetric Magic Square, for which ci + cj ≠ 0.4 * MC5 for any i and j (i,j = 1 ... 25; i ≠ j).

Attachment 14.7.41 shows examples of 5th order Anti Symmetric Magic Squares for miscellaneous Magic Sums.

A dedicated procedure (Priem4f1) can be used to read the Anti Symmetric Magic Square B, to calculate the Complementary Magic Square C, to generate the Pan Magic Squares A and D and to construct the 9 x 9 Magic Squares.

Attachment 14.7.42 shows for miscellaneous Magic Sums the first occurring 9th order Partial Symmetric Prime Number Magic Square with Overlapping Sub Squares.

14.7.3 Magic Squares (9 x 9), Overlapping Subsquares, Associated

Finally, Prime Number Associated Magic Squares of order 9 can be composed out of:

• One set of overlapping Complementary Anti Symmetric (Pan) Magic Squares B and C (order 5);
• One set of Complementary Anti Symmetric (Semi) Magic Squares A and D (order 4).

A comparable procedure (Priem4f2) can be used:

• to read Complementary Anti Symmetric (Pan) Magic Squares B and C from Attachment 14.7.42;
• to generate Complementary Anti Symmetric Semi Magic Squares A and D and
• to complete the Associated Prime Number Magic Square.

Attachment 14.7.49 shows one Associated Prime Number Magic Square for each of the applicable Magic Sums.

14.7.4 Concentric Magic Squares (9 x 9)

A 9th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 7th order, as discussed in Section 14.5.1, with a border around it.

Based on the equations defining the border of a Concentric Magic Square (9 x 9) with Enclosed Magic Square (7 x 7):

```a(73) =     s1   - a(74) - a(75) - a(76) - a(77) - a(78) - a(79) - a(80) - a(81)
a(64) = 2 * s1/9 - a(72)
a(55) = 2 * s1/9 - a(63)
a(46) = 2 * s1/9 - a(54)
a(37) = 2 * s1/9 - a(45)
a(28) = 2 * s1/9 - a(36)
a(19) = 2 * s1/9 - a(27)
a(18) = 7 * s1/9 - a(27) - a(36) - a(45) - a(54) - a(63) - a(72) + a(73) - a(81)
a(10) = 2 * s1/9 - a(18)
a( 9) = 2 * s1/9 - a(73)
a( 8) = 2 * s1/9 - a(80)
a( 7) = 2 * s1/9 - a(79)
a( 6) = 2 * s1/9 - a(78)
a( 5) = 2 * s1/9 - a(77)
a( 4) = 2 * s1/9 - a(76)
a( 3) = 2 * s1/9 - a(75)
a( 2) = 2 * s1/9 - a(74)
a( 1) = 2 * s1/9 - a(81)
```

a routine can be written to generate Prime Number Concentric Magic Squares of order 9 (ref. Priem9a1).

Attachment 14.7.3 shows one Prime Number Concentric Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

Note:

For Prime Number Concentric Magic Squares of order 9 with Magic Sum s9, it is convenient to split the supplementary rows and columns into parts summing to s3 = 3 * s9 / 9:

 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(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81)

This results in following alternative border equations:

 a( 7) = s3 - a( 8) - a( 9) a(27) = s3 - a(18) - a( 9) a(73) = 2 * s3 / 3 - a( 9) a(79) = 2 * s3 / 3 - a( 7) a(80) = 2 * s3 / 3 - a( 8) a(10) = 2 * s3 / 3 - a(18) a(19) = 2 * s3 / 3 - a(27) a( 6) = s3 - s( 5) - a( 4) a(76) = 2 * s3 / 3 - a( 4) a(77) = 2 * s3 / 3 - a( 5) a(78) = 2 * s3 / 3 - a( 6) a( 3) = s3 - a( 2) - a( 1) a(64) = s3 - a(55) - a( 1) a(81) = 2 * s3 / 3 - a( 1) a(74) = 2 * s3 / 3 - a( 2) a(75) = 2 * s3 / 3 - a( 3) a(63) = 2 * s3 / 3 - a(55) a(72) = 2 * s3 / 3 - a(64) a(54) = s3 - s(45) - a(36) a(28) = 2 * s3 / 3 - a(36) a(37) = 2 * s3 / 3 - a(45) a(46) = 2 * s3 / 3 - a(54)

which enable the development of a much faster routine to generate Prime Number Concentric Magic Squares of order 7 (ref. Priem9a2).

Subject routine produced, based on 1689 previously generated Concentric Magic Squares of order 7, 1689 Prime Number Concentric Magic Square of order 9 within 100 seconds (one square per Magic Sum).

14.7.5 Bordered Magic Squares (9 x 9), Miscellaneous Inlays

Based on the collections of 7th order Inlaid and Ultra Magic Squares, as discussed in Section 14.5.2, 14.5.6 and 14.5.8, also following 9th order Bordered Magic Squares can be generated with routine Priem9a1 or Priem9a2:

• Concentric Center Square with Diamond Inlay      (ref. Attachment 14.7.04)

• Bordered Center Square with:

- Embedded Ultra Magic Square                    (ref. Attachment 14.7.02)
- Embedded Associated Square with Diamond Inlay  (ref. Attachment 14.7.03)
- Embedded Square with Square Inlay              (ref. Attachment 14.7.05)
- Embedded Square with Square and Diamond Inlay  (ref. Attachment 14.7.06)

• Associated Center Square with:

- Square Inlays Order 3 and 4                    (ref. Attachment 14.7.13)

• Ultra Magic Center Square                        (ref. Attachment 14.7.09)

• Ultra Magic Center Square with:

- Order 3 Concentric Square and Square Inlay (a) (ref. Attachment 14.7.14)
- Order 3 Concentric Square and Square Inlay (b) (ref. Attachment 14.7.15)
- Order 3 Square Inlays                          (ref. Attachment 14.7.16)
- Order 3 Square and Diamond Inlay (a)           (ref. Attachment 14.7.17)
- Order 3 Square and Diamond Inlay (b)           (ref. Attachment 14.7.18)
- Order 3 Concentric Square and Diamond Inlay    (ref. Attachment 14.7.19)

Each square shown corresponds with numerous squares for the same Magic Sum.

14.7.6 Bordered Magic Squares (9 x 9), Split Border

Alternatively a 9th order Bordered Magic Square with Magic Sum s9 can be constructed based on:

• an (Ultra) Magic Center Square of order 5 with Magic Sum s5 = 5 * s9 / 9;
• 28 pairs, each summing to 2 * s9 / 9, surrounding the Symmetric (Pan) Magic Center Square;
• a split of the supplementary rows and columns into three equal parts, each summing to s1 = s9 / 3.

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) 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(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81)

Based on the principles described in Section 14.7.1 above, a fast procedure (Priem9c) can be developed:

• to read the previously generated (Ultra) Magic Center Squares of order 5;
• to generate, based on the remainder of the available pairs, four Magic Squares of order 3;
• to transform these four Magic Squares into suitable Corner Squares, as shown above;
• to complete the Bordered Magic Squares of order 9 with the four remaining 2 x 3 Magic Rectangles.

Attachment 14.7.9 shows one Prime Number Bordered Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.7.7 Eccentric Magic Squares (9 x 9)

Also for Prime Number Eccentric Magic Squares of order 9 it is convenient to split the supplementary rows and columns into three equal parts each summing to s1 = s9 / 3:

 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(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81)

This enables, based on the same principles, the development of a fast procedure (ref. Priem9b):

• to read the previously generated Eccentric Magic Squares of order 7;
• to generate, based on the remainder of the available pairs, a suitable Corner Square of order 3;
• to complete the Main Diagonal and determine the related Border Pairs;
• to complete the Eccentric Magic Square of order 9 with the two remaining 2 x 3 Magic Rectangles.

Attachment 14.7.7 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.

14.7.8 Eccentric Magic Squares (9 x 9)
Overlapping Sub Squares

Prime Number Eccentric Magic Squares of order 9 with a Magic Sum s1 might contain:

• One 7th order (Pan) Magic Corner Square with Magic Sum s7 = 7 * s1 / 9 (bottom/right)
• One 3th order Semi Magic Corner Square with Magic Sum s3 = 3 * s1 / 9 (top/left)

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) 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(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81)

Based on this definition a dedicated procedure (Priem9g1) can be used:

• to read a square from a collection of previously generated 7th order Ultra Magic Squares (ref. Attachment 14.6.11);
• to transform the Ultra Magic Square to a suitable Pan Magic Corner Square;
• to calculate, based on the remainder of the available pairs, the 3th order Semi Magic Corner Square;
• to complete the Main Diagonal and determine the related Border Pairs;
• to complete the Eccentric Magic Square of order 9 with the two remaining 2 x 3 Magic Rectangles.

Attachment 14.7.31 shows for miscellaneous Magic Sums the first occurring order 9 Prime Number Eccentric Magic Square with Overlapping Sub Squares.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.7.9 Composed Magic Squares (9 x 9)
Associated Corner Squares, Composed Rectangles

The order 9 Magic Square shown below, with magic Sum s1, is composed out of:

• One 3th order Simple Magic Corner Square with Magic Sum s3 = 3 * s1 / 9 (top/left)
• One 6th order Associated Magic Corner Square with Magic Sum s6 = 6 * s1 / 9 (bottom/right)
• Two Composed Magic Rectangles order 3 x 6 with s3 = 3 * s1 / 9 and s6 = 6 * s1 / 9
 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(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81)

Based on this definition a dedicated procedure (ref. Priem9g2) can be used:

• to read the Border Square from a collection of previously generated 6th order Associated Magic Squares;
• to calculate, based on the remainder of the available pairs, the 3th order Magic Corner Square;
• to complete the Main Diagonal and determine related 3 x 3 Semi Magic Corner Squares (blue);
• to complete the Composed Magic Square of order 9 with the two remaining 3 x 3 Semi Magic Squares.

Attachment 14.7.32 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.7.10 Simple Magic Squares (9 x 9) Composed of (Semi) Magic Sub Squares (3 x 3)

Comparable with the method discussed in Section 14.4.10, Prime Number Magic Squares of order 9, with Magic Sum 3 * s1, can be composed out of one Magic Center Square and eight Semi Magic Border Squares of order 3 with Magic Sum s1.

The 8 border squares can be arranged in 8! ways around the center square, resulting in 8! * 8 * 124 * 244 = 2,22 1015 Magic Squares of the 9th order for subject Magic Sum.

Based on the previously applied procedures for Prime Number (Semi) Magic Squares with Magic Sum s1, a procedure can be developed to find sets of one Prime Number Magic - and eight (or more) Prime Number Semi Magic Squares with Magic Sum s1, each containing 9 different Prime Numbers (Priem3d1).

Attachment 14.7.43 shows for miscellaneous Magic Sums the first occurring Prime Number Simple Magic Square composed of one Magic Center - and eight Semi Magic Border Squares.

14.7.11 Simple Magic Squares (9 x 9) Composed of (Semi) Magic Sub Squares (3 x 3)

Comparable with the method discussed in Section 14.7.10 above, Prime Number Magic Squares of order 9, with Magic Sum 3 * s1, can be composed out of:

• One Magic Center Square,
• Four Semi Magic Corner Squares (7 Magic Lines) and
• Four Semi Magic Border Squares (6 Magic Lines)

of order 3 with Magic Sum s1.

The border and corner squares can each be arranged in 4! ways around the center square, resulting in (4!)2 * 8 * 124 * 724 = 2,57 1015 Magic Squares of the 9th order for subject Magic Sum.

Based on the previously applied procedures for Prime Number (Semi) Magic Squares with Magic Sum s1, a procedure can be developed to find sets of one Prime Number Magic and 2 * 4 Prime Number Semi Magic Squares with Magic Sum s1, each containing 9 different Prime Numbers (Priem3d2).

Attachment 14.7.45 shows for miscellaneous Magic Sums the first occurring Prime Number Simple Magic Square composed of one Magic Center, four Semi Magic Corner and four Semi Magic Border Squares.

14.7.12 Associated Magic Squares (9 x 9) Composed of (Semi) Magic Sub Squares (3 x 3)

Comparable with the method discussed in Section 14.7.11 above, Associated Prime Number Magic Squares of order 9, with Magic Sum 3 * s1, can be composed out of:

• One Magic Center Square,
• Two Complementary Pairs of Semi Magic Anti Symmetric Corner Squares (7 Magic Lines) and
• Two Complementary Pairs of Semi Magic Anti Symmetric Border Squares (6 Magic Lines)

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). Such squares occur in Complementary Pairs.

With some minor modifications to the procedure applied in Section 14.7.11 above (Priem3d2) a procedure can be obtained to generate subject Associated Composed Magic Squares (Priem3d3).

Attachment 14.7.46 shows for miscellaneous Magic Sums the first occurring Prime Number Associated Magic Square composed of one Magic Center, four Semi Magic Anti Symmetric Corner and four Semi Magic Anti Symmetric Border Squares.

14.7.13 Associated Magic Squares (9 x 9) with Associated Square Inlays Order 4 and 5

Associated Magic Squares of order 9 with Square Inlays of order 4 and 5 can be obtained by means of a transformation of order 9 Composed Magic Squares as illustrated below:

MC = 13023
 2887 1873 421 607 2857 2851 733 181 613 337 691 2143 2617 547 97 1033 2767 2791 277 751 2203 2557 103 127 1861 2797 2347 2287 2473 1021 7 2281 2713 2161 43 37 2833 223 211 2521 673 457 1723 2011 2371 1327 1201 1597 1663 1801 1741 2293 757 643 1471 1831 1063 1423 1987 2017 1447 877 907 1231 1297 1693 1567 2251 2137 601 1153 1093 373 2683 2671 61 523 883 1171 2437 2221
= > MC = 13023
 673 2833 457 223 1723 211 2011 2521 2371 2857 2887 2851 1873 733 421 181 607 613 1801 1327 1741 1201 2293 1597 757 1663 643 547 337 97 691 1033 2143 2767 2617 2791 1987 1471 2017 1831 1447 1063 877 1423 907 103 277 127 751 1861 2203 2797 2557 2347 2251 1231 2137 1297 601 1693 1153 1567 1093 2281 2287 2713 2473 2161 1021 43 7 37 523 373 883 2683 1171 2671 2437 61 2221

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

• One 4th order Associated Magic Corner Square with Magic Sum s4 = 4 * s1 / 9 (top/left)
• One 5th order Associated Magic Corner Square with Magic Sum s5 = 5 * s1 / 9 (bottom/right)
• Two Associated Magic Rectangles order 4 x 5 with s4 = 4 * s1 / 9 and s5 = 5 * s1 / 9

Based on this definition a routine can be developed to generate subject Composed Magic Squares (ref. Priem9f3).

Attachment 14.7.11 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square.

Attachment 14.7.12 shows the corresponding Associated Magic Squares with order 4 and 5 Square Inlays.

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

14.7.14 Associated Magic Squares (9 x 9) with Associated Diamond Inlays Order 4 and 5

Associated Magic Squares of order 9 with Associated Diamond Inlays of order 4 and 5

 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(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81)

can be constructed as follows:

• Read previously generated Order 5 Associated - or Ultra Magic Diamonds with Magic Sum s5 = 5 * s9 / 9;
• Generate Order 4 Associated Magic Diamonds with Magic Sum s4 = 4 * s9 / 9;
• Complete the Order 9 Associated Magic Squares with the remaining Border Pairs.

It is convenient to split the two bottom rows and right columns into parts summing to s3 = s9 / 3 and s6 = 2 * s9 / 3, which results in following border equations:

 a9(79) = s3 - a9(80) - a9(81) a9(70) = s3 - a9(71) - a9(72) a9(63) = s3 - a9(72) - a9(81) a9(62) = s3 - a9(71) - a9(80) a9( 3) = 2 * s3/3 - a9(79) a9(12) = 2 * s3/3 - a9(70) a9(19) = 2 * s3/3 - a9(63) a9(20) = 2 * s3/3 - a9(62) a9( 1) = 2 * s3/3 - a9(81) a9( 2) = 2 * s3/3 - a9(80) a9(10) = 2 * s3/3 - a9(72) a9(11) = 2 * s3/3 - a9(71)
```a9( 4) = 2 * s9/9 - a9(78)
a9(76) =     s9   - a9( 4) - a9(13) - a9(22) - a9(31) - a9(40) - a9(49) - a9(58) - a9(67)
a9(75) =     s9   - a9( 3) - a9(12) - a9(21) - a9(30) - a9(39) - a9(48) - a9(57) - a9(66)
a9(55) =     s9   - a9(56) - a9(57) - a9(58) - a9(59) - a9(60) - a9(61) - a9(62) - a9(63)
a9(64) =     s9   - a9(65) - a9(66) - a9(67) - a9(68) - a9(69) - a9(70) - a9(71) - a9(72)
a9(74) =     s9   - a9( 2) - a9(11) - a9(20) - a9(29) - a9(38) - a9(47) - a9(56) - a9(65)
a9(73) =     s9   - a9(74) - a9(75) - a9(76) - a9(77) - a9(78) - a9(79) - a9(80) - a9(81)
a9(54) =(7 * s9 - 4 * a9(55) - 5 * a9(56) - 8 * a9(64) -  9 * a9(65) - 10 * a9(66) - 10 * a9(70)  +
- a9(74) - 2 * a9(75) - 3 * a9(76) + 12 * a9(77) -  5 * a9(78) - 10 * a9(79)) / 8
a9(46) =     s9     - a9(47) - a9(48) - a9(49) - a9(50) - a9(51) - a9(52) - a9(53) - a9(54)
```
 a9( 6) = 2 * s9/9 - a9(76) a9( 7) = 2 * s9/9 - a9(75) a9( 8) = 2 * s9/9 - a9(74) a9( 9) = 2 * s9/9 - a9(73) a9(16) = 2 * s9/9 - a9(66) a9(17) = 2 * s9/9 - a9(65) a9(18) = 2 * s9/9 - a9(64) a9(26) = 2 * s9/9 - a9(56) a9(27) = 2 * s9/9 - a9(55) a9(28) = 2 * s9/9 - a9(54) a9(36) = 2 * s9/9 - a9(46)

with a9(81), a9(80), a9(72), a9(71), a9(78), a9(66), a9(56) and a9(65) the independent variables.

Subject equations can be incorporated in a fast routine to generate the defined Associated Magic Squares (ref. Priem9f4).

Attachment 14.7.20 shows for miscellaneous Magic Sums the first occurring Associated Magic Squares with order 4 and 5 Diamond Inlays.

14.7.15 Inlaid Magic Squares (9 x 9), Square Inlays Order 3 and 4 (Overlapping)

The 9th order (Simple) Inlaid Magic Square 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(50) a(51) a(52) a(53) a(54) a(55) a(56) a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64) a(65) a(66) a(67) a(68) a(69) a(70) a(71) a(72) a(73) a(74) a(75) a(76) a(77) a(78) a(79) a(80) a(81)

contains following inlays:

• two each 4th order Simple Magic Squares - Magic Sums s(1) and s(4) - with the center element in common,
• two each 3th order Simple Magic Squares with Magic Sums s(2) and s(3) .

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

```s(1) = 8 * s9 / 9 - s(4)
s(2) = 6 * s9 / 9 - s(3)
```

With s9 the Magic Sum of the 9th order Inlaid Magic Square.

Based on the equations describing the Associated Border:

```a(76) =    - s9 / 9 + a(78) - s(3) + s(4)
a(75) =    - s9 / 9 + a(79) - s(3) + s(4)
a(73) = 12 * s9 / 9 - a(77) - 2 * a(78) - 2 * a(79) - 2 * a(80) - a(81) + 3 * s(3) - 3 * s(4)
a(64) =      s9     - a(72) - s(3) - s(4)
a(55) =      s9     - a(63) - s(3) - s(4)
a(46) =      s9     - a(54) - s(3) - s(4)
a(45) = 40 * s9 / 9 - 2*a(54) - 2*a(63) - 2*a(72) - a(77) - 2*a(78) - 2*a(79) - 2*a(80) - 2*a(81) - 6*s(4)
```
 a(37) = 2 * s9 / 9 - a(45) a(36) = 2 * s9 / 9 - a(46) a(28) = 2 * s9 / 9 - a(54) a(27) = 2 * s9 / 9 - a(55) a(19) = 2 * s9 / 9 - a(63) a(18) = 2 * s9 / 9 - a(64) a(10) = 2 * s9 / 9 - a(72) a( 9) = 2 * s9 / 9 - a(73) a( 8) = 2 * s9 / 9 - a(74) a( 7) = 2 * s9 / 9 - a(75) a(6) = 2 * s9 / 9 - a(76) a(5) = 2 * s9 / 9 - a(77) a(4) = 2 * s9 / 9 - a(78) a(3) = 2 * s9 / 9 - a(79) a(2) = 2 * s9 / 9 - a(80) a(1) = 2 * s9 / 9 - a(81)

a procedure can be developed

• to collect previously generated Simple Magic Squares of the 3th order (ref. Section 14.1.1),
• to generate Simple Magic Squares of the 4th order (ref. Section 14.2.1) and
• to complete subject 9th order Prime Number Inlaid Magic Squares (ref. Priem9k).

Attachment 14.7.21 shows for miscellaneous Magic Sums the first occurring Prime Number Inlaid Magic Square of order 9.

More unique solutions per Magic Sum s9 are possible, by variation of the magic sums s(1) ... s(4).

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.

14.7.16 Summary

The obtained results regarding the miscellaneous types of order 9 Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table:

 Type Characteristics Subroutine Results Composed (1) Overlapping Sub Squares Overlapping Sub Squares, Partly Symmetric Overlapping Sub Squares, Associated Associated Corner Squares Order 3 and 6 Associated Corner Squares Order 4 and 5 Concentric - Bordered Miscellaneous Types Split Border Lines, Center Square order 5 Eccentric Split Border Lines Overlapping Sub Squares Associated Corner Squares Composed (2) Eight Semi Magic Border Squares One Magic Center Square Four Semi Magic Border Squares Four Semi Magic Corner Squares One Magic Center Square Four Semi Magic Anti Symmetric Border Squares Four Semi Magic Anti Symmetric Corner Squares One Magic Center Square Associated Square  Inlays Order 4 and 5 - Diamond Inlays Order 4 and 5 Inlaid Square Inlays Order 3 and 4 (Overlapping)
 Comparable routines as listed above, can be used to generate Prime Number Magic Squares of order 10, which will be described in following sections.