Office Applications and Entertainment, Latin Squares  
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The main subject of this website is the construction of magic squares by means of elementary analytical methods.
2.0 Definition and Terminology
A Latin Square of order n is an n x n square filled with n different symbols, each occurring only once in each row and only once in each column, as illustrated below for a 3 x 3 Latin Square.
The concept of Latin Squares was introduced by Leonard Euler (1707  1783), who used Latin Characters rather than integers.
As Euler applied originally the symbols {A, B, C} and {α, β, γ} the resulting square C
has been traditionally referred to as GrecoLatin Square or Euler Square.
Attachment 3.1.1 shows the 12 ea order 3 Latin Squares, which can be found based on the definition formulated in Section 2.0 above (ref. LatSqr3).
Euler Squares can be found, by selecting pairs of Latin Squares (A, B)
while ensuring that the resulting square C contains 9 distinct pairs.
3.1 Magic Squares, Natural Numbers
(Semi) Magic Square M of order 3 with the integers 1 ... 9 can be written as
M = A + 3 * B + [1]
where the squares A and B contain only the integers 0, 1 and 2.
Attachment 3.1.3 shows the 72 ea (Semi) Magic Squares which can be constructed based on the
72 ea Euler Squares found in Section 3.0 above.
3.2 Magic Squares, Prime Numbers
When the elements {a_{i}, i = 1 ... 3} and {b_{j}, j = 1 ... 3) of a valid pair of Orthogonal Latin Squares (A, B)  as found in Section 3.1 above  comply with following conditions:
the resulting square M = A + B will be an order 3 Prime Number Magic Square:
Attachment 3.2.1 contains miscellaneous correlated balanced series
{a_{i}, i = 1 ... 3}
and
{b_{j}, j = 1 ... 3).
The obtained results regarding the order 3 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 4 Latin  and related (Pan) Magic Squares, which will be described in following sections.

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