Office Applications and Entertainment, Latin Squares  
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A Latin Square of order 5 is a 5 x 5 square filled with 5 different symbols, each occurring only once in each row and only once in each column.
5.1 Latin Diagonal Squares (5 x 5)
Attachment 3.7.1
shows the 960 ea order 5 Latin Diagonal Squares, which can be found based on the definition formulated above (ref. SudSqr5a).
5.2 Magic Squares, Natural Numbers
(Simple) Magic Square M of order 5 with the integers 1 ... 25 can be written as
M = A + 5 * B + [1]
where the squares A and B contain only the integers 0, 1, 2, 3 and 4.
Based on the 57600 Euler Squares found in Section 5.1 above, 57600 ea (Simple) Magic Squares can be constructed.
Attachment 3.7.2
shows the 240 ea order 5 Latin Diagonal Squares, with the Broken Diagonals summing to 10,
which can be filtered from the 960 Latin Diagonal squares found in Section 5.1 above.
All 28800 order 5 Pan Magic Squares
can be based on pairs of Orthogonal Latin Diagonal Squares (A, B)
out of this collection and are shown in Attachment 5.2.6.
Attachment 3.7.4
shows the 16 ea order 5 Symmetric Latin Diagonal Squares, with the Broken Diagonals summing to 10,
which can be filtered from the 960 Latin Diagonal squares found in Section 5.1 above.
All 128 order 5 Ultra Magic Squares
can be based on pairs of Orthogonal Symmetric Latin Diagonal Squares (A, B)
out of this collection and are shown in Attachment 5.4.1.
5.2.4 Associated Magic Squares
Symmetric Latin Diagonal Squares
Attachment 3.7.3
shows the 64 ea order 5 Symmetric Latin Diagonal Squares,
which can be filtered from the 960 Latin Diagonal Squares found in Section 5.1 above.
The 256 order 5 Associated Magic Squares, which
can be based on pairs of Orthogonal Symmetric Latin Diagonal Squares (A, B)
out of this collection, are shown in
Attachment 5.2.3.
Symmetric Latin Squares
Attachment 5.2.4
shows the 256 ea order 5 Symmetric Latin Squares,
which can be filtered from the 161280 Latin Squares found in Section 5.0 above.
Based on this collection 3072 order 5 Associated Magic Squares can be constructed.
Symmetric SemiLatin Squares
Alternatively order 5 Associated Magic Squares M can be constructed based on pairs of Orthogonal Symmetric SemiLatin Squares (A, B), as shown below for the symbols {a_{i}, i = 1 ... 5} and {b_{j}, j = 1 ... 5).
All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
Attachment 5.2.5 shows 960 ea order 5 Symmetric (Semi) Latin Squares,
which can be found with procedure SemiLat5a.
5.2.5 Crosswise Symmetric Magic Squares
Order 5 Crosswise Symmetric Magic Squares M can be constructed based on pairs of Orthogonal Crosswise Symmetric (Semi) Latin Squares (A, B), as summarized below:
For each type, all pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
Attachment 5.2.8 shows 256 ea order 5 Crosswise Symmetric Latin Squares,
which can be found with procedure SemiLat5d.
5.2.6 Concentric Magic Squares
Order 5 Concentric Magic Squares M can be constructed based on pairs of Orthogonal Concentric SemiLatin Squares (A, B), as shown below for the symbols {a_{i}, i = 1 ... 5} and {b_{j}, j = 1 ... 5).
All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
Attachment 5.2.6 shows 216 ea order 5 Concentric (Semi) Latin Squares,
which can be found with procedure SemiLat5b.
Order 5 Inlaid Magic Squares M can be constructed based on pairs of Orthogonal Inlaid SemiLatin Squares (A, B), as shown below for the symbols {a_{i}, i = 1 ... 5} and {b_{j}, j = 1 ... 5).
All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
Attachment 5.2.7 shows 284 ea order 5 Inlaid (Semi) Latin Squares,
which can be found with procedure SemiLat5c.
5.2.8 Evaluation of the Results
Following table compares the enumeration results for order 5 Magic Squares (ref. Section 5.8) with the results based on the construction methods described above:
The constructability by means of Orthogonal (Latin Diagonal) Squares can be considered as an additional property.
5.3 Magic Squares, Prime Numbers
When the elements {a_{i}, i = 1 ... 5} and {b_{j}, j = 1 ... 5) of a valid pair of Orthogonal Latin Diagonal Squares (A, B)  as applied in Section 5.2.1 above  comply with following condition:
the resulting square M = A + B will be an order 5 Prime Number Simple or Pan Magic Square.
Attachment 5.3, page 1 contains miscellaneous correlated unbalanced series
{a_{i}, i = 1 ... 5}
and
{b_{j}, j = 1 ... 5).
Based on Orthogonal Squares (A,B) as applied in Section 5.2.2 and correlated (unbalanced) series, the square M = A + B will be an order 5 Prime Number Pan Magic Square.
Attachment 5.3.2 contains the resulting Prime Number Pan Magic Squares for miscellaneous Magic Sums (Sm).
When the elements {a_{i}, i = 1 ... 5} and {b_{j}, j = 1 ... 5) of a valid pair of Orthogonal Latin Diagonal Squares (A, B)  as applied in Section 5.2.3 above  comply with following conditions:
The resulting square M = A + B will be an order 5 Prime Number Ultra Magic Square.
Attachment 5.3, page 2 contains miscellaneous correlated balanced series
{a_{i}, i = 1 ... 5}
and
{b_{j}, j = 1 ... 5).
5.3.4 Associated Magic Squares
Based on Orthogonal SemiLatin Squares (A,B) as applied in Section 5.2.4 and correlated balanced series, the square M = A + B will be an order 5 Prime Number Associated Magic Square.
Attachment 5.3.4 contains the resulting Prime Number Associated Magic Squares for miscellaneous Magic Sums (Sm).
5.3.5 Concentric Magic Squares
Based on Orthogonal SemiLatin Squares (A,B) as applied in Section 5.2.5 and correlated balanced series, the square M = A + B will be an order 5 Prime Number Concentric Magic Square.
Attachment 5.3.5 contains the resulting Prime Number Concentric Magic Squares for miscellaneous Magic Sums (Sm).
The obtained results regarding the order 5 Latin  and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:
Comparable methods as described above, can be used to construct order 6 SemiLatin  and related Magic Squares (with Symmetrical Main Diagonals), which will be described in following sections.

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