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8.0   Latin Squares (8 x 8)

A Latin Square of order 8 is a 8 x 8 square filled with 8 different symbols, each occurring only once in each row and only once in each column.

Based on this definition 108.776.032.459.082.956.800 ea order 8 Latin Squares can be found (ref. OEIS A002860).

For the construction of order 8 Magic Squares normally only those Latin Squares are used for which the 8 different symbols occur also only once in each of the main diagonals (Latin Diagonal Squares).

8.1   Latin Diagonal Squares (8 x 8)

Based on the definition formulated above 300.286.741.708.800 Latin Diagonal Squares can be found (ref. OEIS A274806).

Consequently, pairs of suitable Latin Diagonal Squares (A, B) should be constructed separately rather than be selected from this massive amount.

8.2   Magic Squares, Natural Numbers

8.2.1 General

(Pan) Magic Square M of order 8 with the integers 1 ... 64 can be written as M = A + 8 * B + [1] where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6 and 7.

Consequently order 8 (Pan) Magic Squares can be based on pairs of Orthogonal (Latin Diagonal) Squares (A, B).

The construction of miscellaneous types Orthogonal (Latin Diagonal) Squares (A, B) has been discussed in detail in the sections listed below:

A few additional types of Orthogonal Squares (A, B) - and resulting Magic Squares - will be discussed in following sections.

8.2.2 Bordered Magic Squares

Order 8 Borders M can be constructed based on pairs of Orthogonal Semi-Latin Borders (A, B), as shown below for the symbols {ai, i = 1 ... 8} and {bj, j = 1 ... 8).

(A, B)
 a1, b1 a2, b8 a3, b8 a4, b1 a5, b1 a6, b8 a7, b8 a8, b1 a8, b2 - - - - - - a1, b7 a8, b3 - - - - - - a1, b6 a1, b4 - - - - - - a8, b5 a1, b5 - - - - - - a8, b4 a8, b6 - - - - - - a1, b3 a8, b7 - - - - - - a1, b2 a1, b8 a7, b1 a6, b1 a5, b8 a4, b8 a3, b1 a2, b1 a8, b8

All pairs of the resulting border (A, B) are distinct, as illustrated by following numerical example:

A
 0 1 2 3 4 5 6 7 7 - - - - - - 0 7 - - - - - - 0 0 - - - - - - 7 0 - - - - - - 7 7 - - - - - - 0 7 - - - - - - 0 0 6 5 4 3 2 1 7
B
 0 7 7 0 0 7 7 0 1 - - - - - - 6 2 - - - - - - 5 3 - - - - - - 4 4 - - - - - - 3 5 - - - - - - 2 6 - - - - - - 1 7 0 0 7 7 0 0 7
M = A + 8 * B + 1
 1 58 59 4 5 62 63 8 16 - - - - - - 49 24 - - - - - - 41 25 - - - - - - 40 33 - - - - - - 32 48 - - - - - - 17 56 - - - - - - 9 57 7 6 61 60 3 2 64

A pair of order 8 Orthogonal Semi-Latin Borders can be constructed for any pair of order 6 Orthogonal Semi-Latin Squares (A6, B6), of which a few types have been discussed in Section 6.2.

A numerical example of the construction of a Bordered Magic Square with a 'Square of the Sun' as Center Square is shown below:

A
 0 1 2 3 4 5 6 7 7 1 5 4 3 2 6 0 7 6 2 4 3 5 1 0 0 6 5 3 4 2 1 7 0 1 5 3 4 2 6 7 7 6 2 3 4 5 1 0 7 1 2 4 3 5 6 0 0 6 5 4 3 2 1 7
B
 0 7 7 0 0 7 7 0 1 1 6 6 1 6 1 6 2 5 2 5 5 2 2 5 3 4 4 3 3 3 4 4 4 3 3 4 4 4 3 3 5 2 5 2 2 5 5 2 6 6 1 1 6 1 6 1 7 0 0 7 7 0 0 7
M = A + 8 * B + 1
 1 58 59 4 5 62 63 8 16 10 54 53 12 51 15 49 24 47 19 45 44 22 18 41 25 39 38 28 29 27 34 40 33 26 30 36 37 35 31 32 48 23 43 20 21 46 42 17 56 50 11 13 52 14 55 9 57 7 6 61 60 3 2 64

Each pair of order 8 Orthogonal Semi-Latin Borders corresponds with 8 * (6!)2 = 4.147.200 pairs.

Consequently 294.912 * 4.147.200 = 1,223 1012 Bordered Magic Squares with 'Square of the Sun' Center Squares can be constructed based on the method described above.

8.2.3 Composed (Pan) Magic Squares

Pan Magic Sub Squares

Order 8 Composed (Pan) Magic Squares M can be constructed based on pairs of Orthogonal Composed Semi-Latin Squares (A, B) for miscellaneous types of Sub Squares.

The example shown below is based on the order 4 Latin Diagonal Pan Magic Squares, as discussed in Section 4.2.2, and the symbols {ai, i = 1 ... 8} and {bj, j = 1 ... 8).

(A, B)
 a8, b8 a4, b1 a5, b4 a1, b5 a3, b8 a2, b1 a7, b4 a6, b5 a1, b4 a5, b5 a4, b8 a8, b1 a6, b4 a7, b5 a2, b8 a3, b1 a4, b5 a8, b4 a1, b1 a5, b8 a2, b5 a3, b4 a6, b1 a7, b8 a5, b1 a1, b8 a8, b5 a4, b4 a7, b1 a6, b8 a3, b5 a2, b4 a8, b3 a4, b6 a5, b2 a1, b7 a3, b3 a2, b6 a7, b2 a6, b7 a1, b2 a5, b7 a4, b3 a8, b6 a6, b2 a7, b7 a2, b3 a3, b6 a4, b7 a8, b2 a1, b6 a5, b3 a2, b7 a3, b2 a6, b6 a7, b3 a5, b6 a1, b3 a8, b7 a4, b2 a7, b6 a6, b3 a3, b7 a2, b2

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 7 3 4 0 2 1 6 5 0 4 3 7 5 6 1 2 3 7 0 4 1 2 5 6 4 0 7 3 6 5 2 1 7 3 4 0 2 1 6 5 0 4 3 7 5 6 1 2 3 7 0 4 1 2 5 6 4 0 7 3 6 5 2 1
B
 7 0 3 4 7 0 3 4 3 4 7 0 3 4 7 0 4 3 0 7 4 3 0 7 0 7 4 3 0 7 4 3 2 5 1 6 2 5 1 6 1 6 2 5 1 6 2 5 6 1 5 2 6 1 5 2 5 2 6 1 5 2 6 1
M=A + 8 * B+1
 64 4 29 33 59 2 31 38 25 37 60 8 30 39 58 3 36 32 1 61 34 27 6 63 5 57 40 28 7 62 35 26 24 44 13 49 19 42 15 54 9 53 20 48 14 55 18 43 52 16 41 21 50 11 46 23 45 17 56 12 47 22 51 10

Attachment 8.2.2 contains 504 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat8a).

Based on this limited collection 21744 (= 2 * 10872) order 8 Composed Magic Squares can be generated (ref. CnstrSqrs8a).

Note

With the 'Latin Diagonal Check' disabled, routine CompLat8a returns 1680 Semi-Latin Squares.
This results in 87552 (= 2 * 43776) order 8 Composed Magic Squares.

8.2.4 Composed Pan Magic Squares

Pan Magic Sub Squares

The Composed Pan Magic Squares as discussed in detail in Section 8.2.7 contain:

• 4 ea Pan    Magic Corner Squares
• 4 ea Simple Magic Middle Squares
• 1 ea Simple Magic Center Square

which limits the amount of possible Semi-Latin Squares considerable.

Order 8 Composed Pan Magic Squares M can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B), as shown below for the symbols {ai, i = 1 ... 8} and {bj, j = 1 ... 8).

(A, B)
 a4, b4 a1, b6 a8, b1 a5, b7 a3, b4 a2, b6 a7, b1 a6, b7 a6, b1 a7, b7 a2, b4 a3, b6 a5, b1 a8, b7 a1, b4 a4, b6 a1, b8 a4, b2 a5, b5 a8, b3 a2, b8 a3, b2 a6, b5 a7, b3 a7, b5 a6, b3 a3, b8 a2, b2 a8, b5 a5, b3 a4, b8 a1, b2 a4, b3 a1, b5 a8, b2 a5, b8 a3, b3 a2, b5 a7, b2 a6, b8 a6, b2 a7, b8 a2, b3 a3, b5 a5, b2 a8, b8 a1, b3 a4, b5 a1, b7 a4, b1 a5, b6 a8, b4 a2, b7 a3, b1 a6, b6 a7, b4 a7, b6 a6, b4 a3, b7 a2, b1 a8, b6 a5, b4 a4, b7 a1, b1

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:

A
 3 0 7 4 2 1 6 5 5 6 1 2 4 7 0 3 0 3 4 7 1 2 5 6 6 5 2 1 7 4 3 0 3 0 7 4 2 1 6 5 5 6 1 2 4 7 0 3 0 3 4 7 1 2 5 6 6 5 2 1 7 4 3 0
B
 3 5 0 6 3 5 0 6 0 6 3 5 0 6 3 5 7 1 4 2 7 1 4 2 4 2 7 1 4 2 7 1 2 4 1 7 2 4 1 7 1 7 2 4 1 7 2 4 6 0 5 3 6 0 5 3 5 3 6 0 5 3 6 0
M=A + 8 * B + 1
 28 41 8 53 27 42 7 54 6 55 26 43 5 56 25 44 57 12 37 24 58 11 38 23 39 22 59 10 40 21 60 9 20 33 16 61 19 34 15 62 14 63 18 35 13 64 17 36 49 4 45 32 50 3 46 31 47 30 51 2 48 29 52 1

Attachment 8.2.3 contains 144 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals - based on the properties summarized above (ref. CompLat8b).

Based on this limited collection 13728 (= 2 * 6844) order 8 Composed Pan Magic Squares can be constructed (ref. CnstrSqrs8a).

Note

With the 'Latin Diagonal Check' disabled, routine CompLat8b returns 384 Semi-Latin Squares.
This results in 22944 (= 2 * 11472) order 8 Composed Pan Magic Squares.

It is not really necessary to limit the integers of the Latin Sub Squares to two out of four pairs.
(As illustrated by the example above)

8.2.5 Most Perfect Pan Magic Squares

Order 8 Most Perfect Pan Magic Squares, as discussed in detail in Section 8.5.5, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B).

Typical for this order 8 Most Perfect Pan Magic Squares is that:

• The integers of each 2 × 2 sub square sum to s8/2 = 130 (Compact)
• All pairs of integers distant n/2 along a (main) diagonal sum to s8/4 = 65 (Complete)

A numerical example is shown below:

A
 3 2 1 0 5 4 7 6 4 5 6 7 2 3 0 1 3 2 1 0 5 4 7 6 4 5 6 7 2 3 0 1 2 3 0 1 4 5 6 7 5 4 7 6 3 2 1 0 2 3 0 1 4 5 6 7 5 4 7 6 3 2 1 0
B = T(A)
 3 4 3 4 2 5 2 5 2 5 2 5 3 4 3 4 1 6 1 6 0 7 0 7 0 7 0 7 1 6 1 6 5 2 5 2 4 3 4 3 4 3 4 3 5 2 5 2 7 0 7 0 6 1 6 1 6 1 6 1 7 0 7 0
M = A + 8 * B + 1
 28 35 26 33 22 45 24 47 21 46 23 48 27 36 25 34 12 51 10 49 6 61 8 63 5 62 7 64 11 52 9 50 43 20 41 18 37 30 39 32 38 29 40 31 44 19 42 17 59 4 57 2 53 14 55 16 54 13 56 15 60 3 58 1
 Attachment 8.25.1 shows 624 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat8c). Attachment 8.25.2 shows 624 ea order 8 Most Perfect Pan Magic Squares based on M = A + 8 * T(A) + 1, with T(A) the transposed square of A (rows and columns exchanged). Based on the total collection {Ai, T(Aj)}, 322.048 (= 2 * 161.024) order 8 Most Perfect Pan Magic Squares can be generated (ref. CnstrSqrs8a). 8.2.6 Franklin Magic Squares Order 8 Franklin Magic Squares, as discussed in detail in Section 8.4.1, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B). Typical for this order 8 Franklin Magic Squares is that: The numbers of every row and column sum to s8 = 260 In every half-row and half-column the numbers sum to S8/2 = 130 The numbers of the main bent diagonals and all the bent diagonals parallel to it sum to s8 = 260 The integers of each 2 × 2 sub square sum to s8/2 = 130 (Compact) A numerical example is shown below:
A
 3 2 5 4 1 0 7 6 4 5 2 3 6 7 0 1 2 3 4 5 0 1 6 7 5 4 3 2 7 6 1 0 3 2 5 4 1 0 7 6 4 5 2 3 6 7 0 1 2 3 4 5 0 1 6 7 5 4 3 2 7 6 1 0
B = T(A)
 3 4 2 5 3 4 2 5 2 5 3 4 2 5 3 4 5 2 4 3 5 2 4 3 4 3 5 2 4 3 5 2 1 6 0 7 1 6 0 7 0 7 1 6 0 7 1 6 7 0 6 1 7 0 6 1 6 1 7 0 6 1 7 0
M = A + 8 * B + 1
 28 35 22 45 26 33 24 47 21 46 27 36 23 48 25 34 43 20 37 30 41 18 39 32 38 29 44 19 40 31 42 17 12 51 6 61 10 49 8 63 5 62 11 52 7 64 9 50 59 4 53 14 57 2 55 16 54 13 60 3 56 15 58 1
 Attachment 8.26.1 shows 960 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat8d). Attachment 8.26.2 shows 960 ea order 8 Franklin Magic Squares based on M = A + 8 * T(A) + 1, with T(A) the transposed square of A (rows and columns exchanged). Based on the total collection {Ai, T(Aj)}, 442.368 (= 2 * 221.184) order 8 Franklin Magic Squares can be generated (ref. CnstrSqrs8a). 8.2.7 Compact Associated Pan Magic Squares Compact Associated Pan Magic Squares, as discussed in detail in Section 8.6.5, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B). A numerical example is shown below:
A1
 7 4 3 0 2 1 6 5 2 1 6 5 7 4 3 0 5 6 1 2 0 3 4 7 0 3 4 7 5 6 1 2 5 6 1 2 0 3 4 7 0 3 4 7 5 6 1 2 7 4 3 0 2 1 6 5 2 1 6 5 7 4 3 0
B = T(A39)
 7 0 3 4 7 0 3 4 0 7 4 3 0 7 4 3 6 1 2 5 6 1 2 5 1 6 5 2 1 6 5 2 5 2 1 6 5 2 1 6 2 5 6 1 2 5 6 1 4 3 0 7 4 3 0 7 3 4 7 0 3 4 7 0
M = A + 8 * B + 1
 64 5 28 33 59 2 31 38 3 58 39 30 8 61 36 25 54 15 18 43 49 12 21 48 9 52 45 24 14 55 42 19 46 23 10 51 41 20 13 56 17 44 53 16 22 47 50 11 40 29 4 57 35 26 7 62 27 34 63 6 32 37 60 1

Attachment 8.27.1 shows 432 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals - based on the properties mentioned above (ref. CompLat8e).

Based on the total collection {Ai, T(Aj)}, 41472 (= 2 * 432 * 48) order 8 Compact Associated Magic Squares can be generated (ref. CnstrSqrs8a).

Attachment 8.27.2 shows the first 432 (= 9 * 48) ea order 8 Compact Associated Pan Magic Squares of this collection.

8.2.8 Evaluation of the Results

Following table compares a few enumeration results for order 8 Magic Squares with the results based on the construction methods described above:

 Type Enumerated Source Constructed Base Borders 1,18154 1011)** 4.147.200 Semi-Latin Composed Squares 1,55 1016)* 87.552 Semi-Latin Composed, Pan Magic 46.080)* 22.944 Semi-Latin Most Perfect 2.949.120 322.048 Semi-Latin Franklin Squares 8.847.360 442.368 Semi-Latin Compact Associated Pan Magic 46.080 41.472 Semi-Latin

)** Based on integers 1 ... 14 and 51 ... 64
)*  Pan Magic Sub Squares

The constructability by means of Orthogonal (Latin Diagonal) Squares can be considered as an additional property.

8.3   Magic Squares, Prime Numbers

8.3.1 Simple Magic Squares

When the elements {ai, i = 1 ... 8} and {bj, j = 1 ... 8) of a valid pair of Orthogonal Latin Diagonal Squares (A, B) comply with following condition:

• mij = ai + bj = prime for i = 1 ... 8 and j = 1 ... 8 (correlated)

the resulting square M = A + B will be an order 8 Prime Number Simple Magic Square, as illustrated below.

Sa = 1366
 1 3 9 79 93 289 309 583 93 289 309 583 1 3 9 79 79 9 3 1 583 309 289 93 583 309 289 93 79 9 3 1 309 583 93 289 9 79 1 3 9 79 1 3 309 583 93 289 289 93 583 309 3 1 79 9 3 1 79 9 289 93 583 309
Sb = 3956
 4 58 70 100 430 568 598 2128 70 100 4 58 598 2128 430 568 430 568 598 2128 4 58 70 100 598 2128 430 568 70 100 4 58 568 430 2128 598 58 4 100 70 2128 598 568 430 100 70 58 4 58 4 100 70 568 430 2128 598 100 70 58 4 2128 598 568 430
Sm = 5322
 5 61 79 179 523 857 907 2711 163 389 313 641 599 2131 439 647 509 577 601 2129 587 367 359 193 1181 2437 719 661 149 109 7 59 877 1013 2221 887 67 83 101 73 2137 677 569 433 409 653 151 293 347 97 683 379 571 431 2207 607 103 71 137 13 2417 691 1151 739

Attachment 8.3, page 1 contains miscellaneous correlated unbalanced series {ai, i = 1 ... 8} and {bj, j = 1 ... 8).

Attachment 8.3.2 contains the resulting Prime Number Simple Magic Squares for miscellaneous Magic Sums (Sm).

Each square shown corresponds with numerous Prime Number Simple Magic Squares.

8.3.2 Pan Magic Squares, Most Perefect

When the elements {ai, i = 1 ... 8} and {bj, j = 1 ... 8) of a valid pair of Orthogonal Squares (A, B) comply with following conditions:

• mij = ai + bj = prime for i = 1 ... 8 and j = 1 ... 8 (correlated)
• a1 + a8 = a2 + a7 = a3 + a6 = a4 + a5                 (balanced)
b1 + b8 = b2 + b7 = b3 + b6 = b4 + b5

The resulting square M = A + B will be an order 8 Prime Number Pan Magic Square (Most Perfect).

Sa = 19248
 19 83 1019 1583 4793 4729 3793 3229 4793 4729 3793 3229 19 83 1019 1583 19 83 1019 1583 4793 4729 3793 3229 4793 4729 3793 3229 19 83 1019 1583 19 83 1019 1583 4793 4729 3793 3229 4793 4729 3793 3229 19 83 1019 1583 19 83 1019 1583 4793 4729 3793 3229 4793 4729 3793 3229 19 83 1019 1583
Sb = 4776
 0 1194 0 1194 0 1194 0 1194 84 1110 84 1110 84 1110 84 1110 480 714 480 714 480 714 480 714 504 690 504 690 504 690 504 690 1194 0 1194 0 1194 0 1194 0 1110 84 1110 84 1110 84 1110 84 714 480 714 480 714 480 714 480 690 504 690 504 690 504 690 504
Sm = 24024
 19 1277 1019 2777 4793 5923 3793 4423 4877 5839 3877 4339 103 1193 1103 2693 499 797 1499 2297 5273 5443 4273 3943 5297 5419 4297 3919 523 773 1523 2273 1213 83 2213 1583 5987 4729 4987 3229 5903 4813 4903 3313 1129 167 2129 1667 733 563 1733 2063 5507 5209 4507 3709 5483 5233 4483 3733 709 587 1709 2087

Attachment 8.3, page 2 contains miscellaneous correlated balanced series {ai, i = 1 ... 8} and {bj, j = 1 ... 8).

Attachment 8.3.3 contains the resulting Prime Number Most Perfect Pan Magic Squares for miscellaneous Magic Sums (Sm).

Each square shown corresponds with numerous Prime Number Most Perfect Pan Magic Squares.

8.3.3 Bordered Magic Squares

Center Square, Square of the Sun

Based on Orthogonal Semi-Latin Squares (A,B) as applied in Section 8.2.2 and correlated balanced series, the square M = A + B will be an order 8 Prime Number Concentric Magic Square.

Sa = 19248
 19 83 1019 1583 3229 3793 4729 4793 4793 83 3793 3229 1583 1019 4729 19 4793 4729 1019 3229 1583 3793 83 19 19 4729 3793 1583 3229 1019 83 4793 19 83 3793 1583 3229 1019 4729 4793 4793 4729 1019 1583 3229 3793 83 19 4793 83 1019 3229 1583 3793 4729 19 19 4729 3793 3229 1583 1019 83 4793
Sb = 4776
 0 1194 1194 0 0 1194 1194 0 84 84 1110 1110 84 1110 84 1110 480 714 480 714 714 480 480 714 504 690 690 504 504 504 690 690 690 504 504 690 690 690 504 504 714 480 714 480 480 714 714 480 1110 1110 84 84 1110 84 1110 84 1194 0 0 1194 1194 0 0 1194
Sm = 24024
 19 1277 2213 1583 3229 4987 5923 4793 4877 167 4903 4339 1667 2129 4813 1129 5273 5443 1499 3943 2297 4273 563 733 523 5419 4483 2087 3733 1523 773 5483 709 587 4297 2273 3919 1709 5233 5297 5507 5209 1733 2063 3709 4507 797 499 5903 1193 1103 3313 2693 3877 5839 103 1213 4729 3793 4423 2777 1019 83 5987

Attachment 8.3.4 contains the resulting Prime Number Concentric Magic Squares for miscellaneous Magic Sums (Sm).

Each square shown corresponds with numerous Prime Number Concentric Magic Squares.

8.3.4 Composed Pan Magic Squares

Pan Magic Sub Squares

Based on Orthogonal Semi-Latin Squares (A,B) as applied in Section 8.2.3 and correlated balanced series, the square M = A + B will be an order 8 Prime Number Composed Pan Magic Square.

Sa = 19248
 4793 1583 3229 19 1019 83 4729 3793 19 3229 1583 4793 3793 4729 83 1019 1583 4793 19 3229 83 1019 3793 4729 3229 19 4793 1583 4729 3793 1019 83 4793 1583 3229 19 1019 83 4729 3793 19 3229 1583 4793 3793 4729 83 1019 1583 4793 19 3229 83 1019 3793 4729 3229 19 4793 1583 4729 3793 1019 83
Sb = 4776
 1194 0 504 690 1194 0 504 690 504 690 1194 0 504 690 1194 0 690 504 0 1194 690 504 0 1194 0 1194 690 504 0 1194 690 504 480 714 84 1110 480 714 84 1110 84 1110 480 714 84 1110 480 714 1110 84 714 480 1110 84 714 480 714 480 1110 84 714 480 1110 84
Sm = 24024
 5987 1583 3733 709 2213 83 5233 4483 523 3919 2777 4793 4297 5419 1277 1019 2273 5297 19 4423 773 1523 3793 5923 3229 1213 5483 2087 4729 4987 1709 587 5273 2297 3313 1129 1499 797 4813 4903 103 4339 2063 5507 3877 5839 563 1733 2693 4877 733 3709 1193 1103 4507 5209 3943 499 5903 1667 5443 4273 2129 167

Attachment 8.3.5 contains the resulting Prime Number Composed Pan Magic Squares for miscellaneous Magic Sums (Sm).

Each square shown corresponds with numerous Prime Number Composed Pan Magic Squares.

8.4   Summary

The obtained results regarding the order 8 Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

 Attachment Subject Subroutine Composed (Pan) Magic Semi-Latin Squares Composed Pan Magic Semi-Latin Squares Most Perfect Pan Magic Semi-Latin Squares Semi-Latin Franklin Magic Squares Semi-Latin Compact Asociated PM Squares - - - Correlated Magic Series - Prime Number Simple Magic Squares Prime Number Pan Magic Squares (Most Perfect) Prime Number Bordered Magic Squares Prime Number Composed Pan Magic Squares

Comparable methods as described above, can be used to construct order 9 (Semi) Latin - and related (Pan) Magic Squares, which will be described in following sections.