14.0 Special Magic Squares, Prime Numbers
14.8 Magic Squares (10 x 10)
14.8.1 Magic Squares (10 x 10), Composed (1)
Prime Number Magic Squares of order 10 with a Magic Sum s10 can be composed out of:

Three each 4^{th} order Pan Magic Sub Squares A, B and C (s4 = 4 * s10 / 10),

One each 6^{th} order (Symmetric) Magic Corner Square D (s6 = 6 * s10 / 10) and

Eight supplementary pairs, each summing to 2 * s10 / 10
as illustrated below:
b1 
b2 
b3 
b4 
c1 
c2 
c3 
c4 
m9 
m10 
b5 
b6 
b7 
b8 
c5 
c6 
c7 
c8 
m19 
m20 
b9 
b10 
b11 
b12 
c9 
c10 
c11 
c12 
m29 
m30 
b13 
b14 
b15 
b16 
c13 
c14 
c15 
c16 
m39 
m40 
a1 
a2 
a3 
a4 
d1 
d2 
d3 
d4 
d5 
d6 
a5 
a6 
a7 
a8 
d7 
d8 
d9 
d10 
d11 
d12 
a9 
a10 
a11 
a12 
d13 
d14 
d15 
d16 
d17 
d18 
a13 
a14 
a15 
a16 
d19 
d20 
d21 
d22 
d23 
d24 
m81 
m82 
m83 
m84 
d25 
d26 
d27 
d28 
d29 
d30 
m91 
m92 
m93 
m94 
d31 
d32 
d33 
d34 
d35 
d36 
Based on the principles described in previous section a comparable procedure
(Priem10a1) can be developed:

to read previously generated (Symmetric) Magic Squares D,

to generate the Pan Magic Squares A, B and C and

to complete the 10 x 10 Composed Magic Squares M.
Following attachments show the first occurring 10^{th} order Prime Number Composed Magic Square for
miscellaneous 6^{th} order Corner Squares and Magic Sums:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.2 Magic Squares (10 x 10), Composed (2)
Alternatively Prime Number Magic Squares of order 10  composed out of Sub Squares as described in Section 14.8.1  can be arranged as shown below:
b1 
b2 
b3 
b4 
m9 
c1 
c2 
c3 
c4 
m10 
b5 
b6 
b7 
b8 
m19 
c5 
c6 
c7 
c8 
m20 
b9 
b10 
b11 
b12 
m29 
c9 
c10 
c11 
c12 
m30 
b13 
b14 
b15 
b16 
m39 
c13 
c14 
c15 
c16 
m40 
m81 
m82 
m83 
m84 
d1 
d2 
d3 
d4 
d5 
d6 
a1 
a2 
a3 
a4 
d7 
d8 
d9 
d10 
d11 
d12 
a5 
a6 
a7 
a8 
d13 
d14 
d15 
d16 
d17 
d18 
a9 
a10 
a11 
a12 
d19 
d20 
d21 
d22 
d23 
d24 
a13 
a14 
a15 
a16 
d25 
d26 
d27 
d28 
d29 
d30 
m91 
m92 
m93 
m94 
d31 
d32 
d33 
d34 
d35 
d36 
Based on the principles described in Section 14.8.1 above, a comparable procedure
(Priem10a2) can be developed.
Following attachments show the first occurring 10^{th} order Prime Number Composed Magic Square for
miscellaneous 6^{th} order Corner Squares and Magic Sums:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.3 Magic Squares (10 x 10), Composed (3)
Alternatively Prime Number Magic Squares of order 10 with a Magic Sum s10, can be composed out of:

Three each 4^{th} order Pan Magic Corner Squares A, B and C (s4 = 4 * s10 / 10),

One each 6^{th} order Eccentric Magic Corner Square D (s6 = 6 * s10 / 10) and

Eight supplementary pairs, each summing to 2 * s10 / 10
and arranged as illustrated below:
b1 
b2 
b3 
b4 
m5 
m6 
c1 
c2 
c3 
c4 
b5 
b6 
b7 
b8 
m15 
m16 
c5 
c6 
c7 
c8 
b9 
b10 
b11 
b12 
m25 
m26 
c9 
c10 
c11 
c12 
b13 
b14 
b15 
b16 
m35 
m36 
c13 
c14 
c15 
c16 
m41 
m42 
m43 
m44 
d1 
d2 
d3 
d4 
d5 
d6 
m51 
m52 
m53 
m54 
d7 
d8 
d9 
d10 
d11 
d12 
a1 
a2 
a3 
a4 
d13 
d14 
d15 
d16 
d17 
d18 
a5 
a6 
a7 
a8 
d19 
d20 
d21 
d22 
d23 
d24 
a9 
a10 
a11 
a12 
d25 
d26 
d27 
d28 
d29 
d30 
a13 
a14 
a15 
a16 
d31 
d32 
d33 
d34 
d35 
d36 
Based on the principles described above a comparable procedure (Priem10b) can be developed:

to read previously generated Eccentric Magic Squares D,

to generate the Pan Magic Squares A, B and C and

to complete the 10 x 10 Composed Magic Squares M.
Attachment 14.8.56 shows for miscellaneous Magic Sums the first occurring 10^{th} order Prime Number Composed Magic Square, based on 6^{th} order Eccentric Magic Squares (ref. Attachment 14.4.5).
Other Collections of Sub Squares D might be used, provided that the key condition d1 + d8 = d2 + d7
= s10 / 5 is fulfilled.
Following attachments show such collections:
The elements of both collections can be obtained by moving the row and columns such that the original 2 x 2 Center Square becomes the TopLeft Corner Square. The resulting Magic Squares are still (Partly) Compact and Pan Diagonal.
Following attachments show the first occurring 10^{th} order Prime Number Composed Magic Square for
subject 6^{th} order Corner Squares and miscellaneous Magic Sums:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.4 Magic Squares (10 x 10), Composed (4)
Alternatively Prime Number Magic Squares of order 10 with a Magic Sum s10, can be composed out of:

Four each 4^{th} order Pan Magic Corner Squares (s4 = 4 * s10 / 10),

Eighteen supplementary pairs, each summing to 2 * s10 / 10
and arranged as illustrated below:
a1 
a2 
a3 
a4 
a5 
a6 
a7 
a8 
a9 
a10 
a11 
a12 
a13 
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The Center Cross might be constructed with the method of Al Antaki (10^{th} century):

Construct a 6 x 6 Center Cross (s6 = 6 * s10 / 10)

Complete the 10 x 10 Center Cross with the remaining pairs
A procedure (Priem10g) can be developed to:

Generate the 4 x 4 Pan Magic Squares

Generate the 10 x 10 Center Crosses
Attachment 14.8.41 shows the first occurring 10^{th} order Prime Number Composed Magic Square for a few Magic Sums.
Each Center Cross corresponds with (8!) * (8!) = 1,6 10^{9} Center Crosses,
which can be obtained by permutation of the horizontal and vertical pairs.
With the Center Cross fixed, each square corresponds with 4! * 384^{4} = 0,5 10^{12}
squares.
14.8.5 Magic Squares (10 x 10)
Corner Squares Order 4 and 6, Associated Rectangles
Alternatively Prime Number Magic Squares of order 10 with a Magic Sum s10, can be composed out of:

One 4^{th} order Pan Magic Corner Square A (s4 = 4 * s10 / 10),

One 6^{th} order Magic Corner Square D (s6 = 6 * s10 / 10) and

Two order 4 x 6 Associated Magic Rectangles B and C (s4 = 4 * s10 / 10 and s6 = 6 * s10 / 10)
as illustrated below:
a1 
a2 
a3 
a4 
b1 
b2 
b3 
b4 
b5 
b6 
a5 
a6 
a7 
a8 
b7 
b8 
b9 
b10 
b11 
b12 
a9 
a10 
a11 
a12 
b13 
b14 
b15 
b16 
b17 
b18 
a13 
a14 
a15 
a16 
b19 
b20 
b21 
b22 
b23 
b24 
c1 
c2 
c3 
c4 
d1 
d2 
d3 
d4 
d5 
d6 
c5 
c6 
c7 
c8 
d7 
d8 
d9 
d10 
d11 
d12 
c9 
c10 
c11 
c12 
d13 
d14 
d15 
d16 
d17 
d18 
c13 
c14 
c15 
c16 
d19 
d20 
d21 
d22 
d23 
d24 
c17 
c18 
c19 
c20 
d25 
d26 
d27 
d28 
d29 
d30 
c21 
c22 
c23 
c24 
d31 
d32 
d33 
d34 
d35 
d36 
Based on this definition a comparable procedure (ref. Priem10c2) can be developed:

to read previously generated (Symmetric) Magic Squares D;

to generate the Associated Magic Rectangles B and C;

to complete the 10 x 10 Composed Magic Squares M with the Pan Magic Square A.
Following attachments show the first occurring 10^{th} order Prime Number Composed Magic Square for
miscellaneous 6^{th} order Corner Squares and Magic Sums:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.6 Associated Magic Squares (10 x 10)
Composed of Simple Magic Squares (5 x 5)
Associated Magic Squares, composed of four each Simple Magic Squares, contain two sets of Complementary Anti Symmetric Magic Squares,
as discussed in Section 14.3.10.

Attachment 14.8.65 shows examples of such suitable 5^{th} order Anti Symmetric Magic Squares;

Attachment 14.8.66 shows for miscellaneous Magic Sums the related 10^{th} order Associated Magic Squares;

Attachment 14.8.67 shows the corresponding Pan Magic and Complete Magic Squares (Eulers Transformation).
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.7 Concentric Magic Squares (10 x 10)
A 10^{th} order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 8^{th} order with a border around it.
For Prime Number Concentric Magic Squares of order 10 with Magic Sum s10, it is convenient to split the supplementary rows and columns into parts summing to s5 = s10 / 2:
a1

a2

a3

a4

a5

a6

a7

a8

a9

a10

a11

a12

a13

a14

a15

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a100

This results in following border equations:
a(10) = s5  a( 9)  a( 8)  a( 7)  a(6)
a(50) = s5  a(40)  a(30)  a(20) a(10)
a(91) = s10/2  a(10)
a(96) = s10/2  a( 6)
a(97) = s10/2  a( 7)
a(98) = s10/2  a( 8)
a(99) = s10/2  a( 9)
a(11) = s10/2  a(20)
a(21) = s10/2  a(30)
a(31) = s10/2  a(40)
a(41) = s10/2  a(50)

a( 1) = s5  a( 2)  a( 3)  a( 4)  a(5)
a( 81) = s5  a(71)  a(61)  a(51)  a(1)
a(100) = s10/2  a( 1)
a( 92) = s10/2  a( 2)
a( 93) = s10/2  a( 3)
a( 94) = s10/2  a( 4)
a( 95) = s10/2  a( 5)
a( 60) = s10/2  a(51)
a( 70) = s10/2  a(61)
a( 80) = s10/2  a(71)
a( 90) = s10/2  a(81)

which enable the development of a fast procedure to generate Prime Number Concentric Magic Squares of order 10 (ref. Priem10c).
Miscellaneous Prime Number Concentric Magic Squares of order 10, based on the 8^{th} order Concentric Magic Squares as discussed in Section 14.6.4, are shown in following attachments:
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.8 Bordered Magic Squares (10 x 10), Miscellaneous Inlays
Based on the collections of 8^{th} order Magic Squares, as deducted in
Section 14.6.1 thru 14.6.3,
Section 14.6.5,
and
Section 14.6.7 thru 14.6.10,
also following Bordered Magic Squares can be generated with routine Priem10c:
It should be noted that the Attachments listed above contain only those solutions which could be found within 100 seconds.
Each square shown corresponds with numerous squares for the same Magic Sum.
14.8.9 Bordered Magic Squares (10 x 10), Split Border
Alternatively a 10^{th} order Bordered Magic Square with Magic Sum s10 can be constructed based on:

a Magic Center Square of order 6 with Magic Sum s6 = 6 * s10 / 10;

32 pairs, each summing to 2 * s10 / 10, surrounding the Symmetric (Pan) Magic Center Square;

a split of the supplementary rows and columns into three parts:
two summing to s3 = 3 * s10 / 10 and one to s4 = 4 * s10 / 10
as illustrated below:
a1

a2

a3

a4

a5

a6

a7

a8

a9

a10

a11

a12

a13

a14

a15

a16

a17

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As the first border  as specified above  occurs for s10 = 10850, it is convenient to construct the border first and the 6^{th} order Magic Center Square later (based on the remainder of the available pairs).
Based on the principles described in previous sections, a fast procedure (Priem10e1) can be developed:

to generate, four Magic Squares of order 3;

to transform these four Magic Squares into suitable Corner Squares, as shown above;

to complete the Border of order 10 with the four remaining 2 x 4 Magic Rectangles.
The Magic Center Square can be added with a separate routine e.g. Priem10e2 for Concentric, Partly Compact, Magic Center Squares as discussed in Section 14.4.4.
Attachment 14.8.81 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.
14.8.10 Bordered Magic Squares (10 x 10), Composed Border
Another type of order 10 Bordered Magic Squares with Magic Sum s10 can be constructed based on:

a Border composed out of:
 4 Semi Magic Squares of order 3 with Magic Sum s3 = 3 * s10 / 10;
 4 Associated Magic Rectangles order 3 x 4 with s3 = 3 * s10 / 10 and s4 = 4 * s10 / 10;

a Magic Center Square of order 4 with Magic Sum s4 = 4 * s10 / 10
as illustrated below:
a1

a2

a3

a4

a5

a6

a7

a8

a9

a10

a11

a12

a13

a14

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It is convenient to construct the Composed Border first and the 4^{th} order Magic Center Square later (based on the remainder of the available pairs).
Based on the principles described in previous sections, a fast procedure (Priem10f1) can be developed:

to retrieve, the four Semi Magic Squares of order 3 from previously generated Composed Magic Squares of order 6 (ref. Section 14.4.11);

to complete the Composed Border of order 10 with the four 3 x 4 Magic Rectangles.
The Magic Center Square can be added with a separate routine e.g. Priem10f2 for Associated Magic Center Squares as discussed in Section 14.2.3.
Attachment 14.8.83 shows one Prime Number Composed Bordered Magic Square for some of the occurring Magic Sums.
Each square shown corresponds with numerous squares for the same Magic Sum.
Note:
As a consequence of the applied properties:

The opposite Semi Magic Corner Squares are Anti Symmetric and Complementary;

The Magic Center Square is Center Symmetric (Associated)
the 11^{th} order Composed Magic Square will be associated, if the opposite Magic Rectangles (3 x 4) are Anti Symmetric and Complementary as well.
Attachment 14.8.84 shows one Associated Composed Magic Square for some of the occurring Magic Sums (ref. Priem10f3).
14.8.11 Eccentric Magic Squares (10 x 10)
Also for Prime Number Eccentric Magic Squares of order 10 it is convenient to split the supplementary rows and columns into:
two parts summing to s3 = 3 * s10 / 10 and one part summing to s4 = 4 * s10 / 10.
a1

a2

a3

a4

a5

a6

a7

a8

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a11

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This enables, based on the same principles, the development of a set of fast procedures (ref. Priem10d):

to read the previously generated Eccentric Magic Squares of order 8;

to complete the Main Diagonal and determine the related Border Pairs;

to generate, based on the remainder of the available pairs, a suitable Corner Square of order 3;

to complete the Eccentric Magic Square of order 10 with the two remaining 2 x 4 Magic Rectangles.
Attachment 14.8.82 shows,
based on the 8^{th} order Eccentric Magic Squares as discussed in Section 14.6.6,
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.
14.8.12 Summary
The obtained results regarding the miscellaneous types of order 10 Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table:
