6.0 Latin Squares (6 x 6)
A Latin Square of order 6 is a 6 x 6 square filled with 6 different symbols, each occurring only once in each row and only once in each column.
Based on this definition 812.851.200 ea order 6 Latin Squares can be found (ref. OEIS A002860).
6.1 Latin Diagonal Squares (6 x 6)
Latin Diagonal Squares
are Latin Squares for which the 6 different symbols occur also only once in each of the main diagonals.
Based on this definition 92.160 order 6 Latin Diagonal Squares can be found (ref. OEIS A274806).
6.2 Magic Squares, Natural Numbers
6.2.1 General
In spite of the vast amount of Latin (Diagonal) Squares, order 6 GrecoLatin Squares don't exist
(ref. Euler's 36 officers problem).
The nonexistence of order 6 GrecoLatin Squares was confirmed by Gaston Tarry through a proof by exhaustion (1901).
However it is possible to construct miscellaneous types of order 6 Magic Squares
based on pairs of
Orthogonal SemiLatin or
Orthogonal NonLatin
Squares (A, B), which will be illustrated in following sections.
6.2.2 Magic Squares, Symmetrical Diagonals
Order 6 Magic Squares with Symmetrical Diagonals M can be constructed based on pairs of Orthogonal Symmetric
SemiLatin Squares
(A, B),
as shown below for the symbols
{a_{i}, i = 1 ... 6}
and
{b_{j}, j = 1 ... 6).
A
a6 
a2 
a3 
a4 
a5 
a1 
a1 
a5 
a3 
a4 
a2 
a6 
a1 
a2 
a4 
a3 
a5 
a6 
a6 
a2 
a4 
a3 
a5 
a1 
a1 
a5 
a4 
a3 
a2 
a6 
a6 
a5 
a3 
a4 
a2 
a1 

B
b1 
b6 
b1 
b6 
b6 
b1 
b2 
b2 
b5 
b5 
b2 
b5 
b4 
b3 
b3 
b3 
b4 
b4 
b3 
b4 
b4 
b4 
b3 
b3 
b5 
b5 
b2 
b2 
b5 
b2 
b6 
b1 
b6 
b1 
b1 
b6 

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

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A
5 
1 
2 
3 
4 
0 
0 
4 
2 
3 
1 
5 
0 
1 
3 
2 
4 
5 
5 
1 
3 
2 
4 
0 
0 
4 
3 
2 
1 
5 
5 
4 
2 
3 
1 
0 

B
0 
5 
0 
5 
5 
0 
1 
1 
4 
4 
1 
4 
3 
2 
2 
2 
3 
3 
2 
3 
3 
3 
2 
2 
4 
4 
1 
1 
4 
1 
5 
0 
5 
0 
0 
5 

M = A + 6 * B +[1]
6 
32 
3 
34 
35 
1 
7 
11 
27 
28 
8 
30 
19 
14 
16 
15 
23 
24 
18 
20 
22 
21 
17 
13 
25 
29 
10 
9 
26 
12 
36 
5 
33 
4 
2 
31 

The amount of SemiLatin Squares with Symmetrical Diagonals is however so substantial, that it is more feasible to consider
Sub Collections based on additional (restricting) properties.
A few Sub Collections of pairs of Orthogonal SemiLatin Squares
(A, B)
will be discussed in following Sections.
6.2.3 Magic Squares of the Sun
The composite symmetry of the well known 'Magic Square of the Sun' consists of:

6 ea Diagonal Pairs

4 ea Border Line Center Pairs (blue)

4 ea Vertical Pairs

4 ea Horizontal Pairs
which limits the amount of Orthogonal Symmetric SemiLatin Squares (A, B) considerable.
Order 6 'Magic Squares of the Sun' M can be constructed based on pairs of Orthogonal
SemiLatin Squares
(A, B),
as shown below for the symbols
{a_{i}, i = 1 ... 6}
and
{b_{j}, j = 1 ... 6).
A
a1 
a5 
a4 
a3 
a2 
a6 
a6 
a2 
a4 
a3 
a5 
a1 
a6 
a5 
a3 
a4 
a2 
a1 
a1 
a5 
a3 
a4 
a2 
a6 
a6 
a2 
a3 
a4 
a5 
a1 
a1 
a2 
a4 
a3 
a5 
a6 

B
b1 
b6 
b6 
b1 
b6 
b1 
b5 
b2 
b5 
b5 
b2 
b2 
b4 
b4 
b3 
b3 
b3 
b4 
b3 
b3 
b4 
b4 
b4 
b3 
b2 
b5 
b2 
b2 
b5 
b5 
b6 
b1 
b1 
b6 
b1 
b6 

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

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A
0 
4 
3 
2 
1 
5 
5 
1 
3 
2 
4 
0 
5 
4 
2 
3 
1 
0 
0 
4 
2 
3 
1 
5 
5 
1 
2 
3 
4 
0 
0 
1 
3 
2 
4 
5 

B
0 
5 
5 
0 
5 
0 
4 
1 
4 
4 
1 
1 
3 
3 
2 
2 
2 
3 
2 
2 
3 
3 
3 
2 
1 
4 
1 
1 
4 
4 
5 
0 
0 
5 
0 
5 

M = A + 6 * B +[1]
1 
35 
34 
3 
32 
6 
30 
8 
28 
27 
11 
7 
24 
23 
15 
16 
14 
19 
13 
17 
21 
22 
20 
18 
12 
26 
9 
10 
29 
25 
31 
2 
4 
33 
5 
36 

Attachment 6.2.2 contains 384 ea order 6 Semi Latin Squares, based on the properties of the
'Square of the Sun'
(ref. SunLat6).
Based on this limited collection 36864 (= 2 * 384^{2} / 8) unique order 6 'Magic Squares of the Sun' can be constructed (ref. CnstrSqrs6a).
This collection is essential different from the 'Square of the Sun' collection as discussed by Francis Gaspalous in his study ‘Structure of Magic Squares including Transformations’.
6.2.4 Almost Associated Magic Squares
Order 6 Almost 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 ... 6}
and
{b_{j}, j = 1 ... 6).
A
a6 
a2 
a4 
a1 
a5 
a3 
a3 
a5 
a1 
a6 
a2 
a4 
a4 
a2 
a3 
a1 
a5 
a6 
a1 
a2 
a6 
a4 
a5 
a3 
a3 
a5 
a1 
a6 
a2 
a4 
a4 
a5 
a6 
a3 
a2 
a1 

B
b6 
b1 
b6 
b3 
b1 
b4 
b2 
b5 
b2 
b2 
b5 
b5 
b1 
b4 
b3 
b6 
b4 
b3 
b4 
b3 
b1 
b4 
b3 
b6 
b5 
b2 
b5 
b5 
b2 
b2 
b3 
b6 
b4 
b1 
b6 
b1 

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

All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A
5 
1 
3 
0 
4 
2 
2 
4 
0 
5 
1 
3 
3 
1 
2 
0 
4 
5 
0 
1 
5 
3 
4 
2 
2 
4 
0 
5 
1 
3 
3 
4 
5 
2 
1 
0 

B
5 
0 
5 
2 
0 
3 
1 
4 
1 
1 
4 
4 
0 
3 
2 
5 
3 
2 
3 
2 
0 
3 
2 
5 
4 
1 
4 
4 
1 
1 
2 
5 
3 
0 
5 
0 

M = A + 6 * B +[1]
36 
2 
34 
13 
5 
21 
9 
29 
7 
12 
26 
28 
4 
20 
15 
31 
23 
18 
19 
14 
6 
22 
17 
33 
27 
11 
25 
30 
8 
10 
16 
35 
24 
3 
32 
1 

Attachment 6.2.4 shows 1152 ea order 6 Almost Associated Semi Latin Squares
(ref. AssLat6).
Based on this collection 9216 (= 73728/8) unique order 6 Almost Associated Magic Squares can be constructed (ref. CnstrSqrs6a).
Notes
Each column of square A  and consequently each row of square B  contain
three (not necessarily different) paired integers (0,5), (1,4), (2,3).
This limits the number of Semi Latin squares considerable, however
ensures that Prime Number Magic Squares can be constructed based on the corresponding Euler Squares (A,B), as illustrated in Section 6.3.3 below.
6.2.5 Bordered Magic Squares
The 1152 order 4 Orthogonal Latin Diagonal Squares
(A4, B4),
as found in Section 4.2.1, have been used to construct a collection of 1152 Simple Magic Squares
based on the Balanced Series {0, 1, 2, 3}.
The Balanced Series {1, 2, 3, 4}, {0, 1, 4, 5} or {0, 2, 3, 5} can be used to construct Center Squares for order 6 Bordered Magic Squares.
Suitable Borders can be constructed for each of these three sets,
based on pairs of Non Latin but Orthogonal Borders
(A, B),
as illustrated by following numerical example:
A
0 
2 
1 
0 
5 
1 
5 
3 
4 
1 
2 
0 
5 
1 
2 
3 
4 
0 
5 
2 
1 
4 
3 
0 
2 
4 
3 
2 
1 
3 
4 
3 
4 
5 
0 
5 

B
0 
5 
5 
5 
1 
0 
4 
2 
1 
4 
3 
1 
3 
3 
4 
1 
2 
2 
2 
1 
2 
3 
4 
3 
0 
4 
3 
2 
1 
5 
5 
0 
0 
0 
4 
5 

M = A + 6 * B + 1
1 
33 
32 
31 
12 
2 
30 
16 
11 
26 
21 
7 
24 
20 
27 
10 
17 
13 
18 
9 
14 
23 
28 
19 
3 
29 
22 
15 
8 
34 
35 
4 
5 
6 
25 
36 

Attachment 6.2.3, page 1 contains the 89 ea Orthogonal Borders (Ai, Bi) for Center Squares {1, 2, 3, 4}
Attachment 6.2.3, page 2 contains the 238 ea Orthogonal Borders (Ai, Bi) for Center Squares {0, 1, 4, 5}
Attachment 6.2.3, page 3 contains the 145 ea Orthogonal Borders (Ai, Bi) for Center Squares {0, 2, 3, 5}
Each pair of order 6 Orthogonal Borders corresponds with 8 * (4!)^{2} = 4608 pairs.
6.2.6 Evaluation of the Results
Following table compares a few enumeration results for order 6 Magic Squares
with the results based on the construction methods described above:
Type

Enumerated

Source

Constructed

Type

Symm Diagonals

60.207.144.960

Francis Gaspalou

294.912

Square of the Sun





73.728

Almost Associated

Bordered

4.541.644.800)*



472.449.024

Att 6.2.3, page 1

1.263.403.008

Att 6.2.3, page 2

769.720.320

Att 6.2.3, page 3

)* Center Squares based on Consecutive Integers 11 ... 26
The constructability by means of Orthogonal (SemiLatin) Squares can be considered as an additional property.
6.3 Magic Squares, Prime Numbers
6.3.1 Magic Squares, Symmetrical Diagonals
When the elements
{a_{i}, i = 1 ... 6}
and
{b_{j}, j = 1 ... 6)
of a valid pair of Orthogonal SemiLatin Squares (A, B)
 as applied in
Section 6.2.2 above  complies with following condition:

m_{ij} = a_{i} + b_{j} = prime
for i = 1 ... 6 and j = 1 ... 6 (correlated)

a_{1} + a_{6} =
a_{2} + a_{5} =
a_{3} + a_{4} (balanced)
b_{1} + b_{6} =
b_{2} + b_{5} =
b_{3} + b_{4}
the resulting square M = A + B will be an order 6 Prime Number Magic Square with Symmetrical Diagonals.
Sa = 1578
479 
179 
215 
311 
347 
47 
47 
347 
215 
311 
179 
479 
47 
179 
311 
215 
347 
479 
479 
179 
311 
215 
347 
47 
47 
347 
311 
215 
179 
479 
479 
347 
215 
311 
179 
47 

Sb = 3102
62 
972 
62 
972 
972 
62 
152 
152 
882 
882 
152 
882 
572 
462 
462 
462 
572 
572 
462 
572 
572 
572 
462 
462 
882 
882 
152 
152 
882 
152 
972 
62 
972 
62 
62 
972 

Sm = 4680
541 
1151 
277 
1283 
1319 
109 
199 
499 
1097 
1193 
331 
1361 
619 
641 
773 
677 
919 
1051 
941 
751 
883 
787 
809 
509 
929 
1229 
463 
367 
1061 
631 
1451 
409 
1187 
373 
241 
1019 

Attachment 6.3 contains miscellaneous correlated balanced series
{a_{i}, i = 1 ... 6}
and
{b_{j}, j = 1 ... 6).
Attachment 6.3.1 contains the resulting Prime Number Magic Squares for miscellaneous Magic Sums (Sm).
Each square shown corresponds with numerous Prime Number Magic Squares with Symmetrical Diagonals.
6.3.2 Magic Squares, Square of the Sun
Based on Orthogonal SemiLatin Squares (A,B) as applied in
Section 6.2.3 and correlated balanced series,
the square M = A + B will be an order 6 Prime Number Magic Square (Square of the Sun).
Sa = 1578
47 
347 
311 
215 
179 
479 
479 
179 
311 
215 
347 
47 
479 
347 
215 
311 
179 
47 
47 
347 
215 
311 
179 
479 
479 
179 
215 
311 
347 
47 
47 
179 
311 
215 
347 
479 

Sb = 3102
62 
972 
972 
62 
972 
62 
882 
152 
882 
882 
152 
152 
572 
572 
462 
462 
462 
572 
462 
462 
572 
572 
572 
462 
152 
882 
152 
152 
882 
882 
972 
62 
62 
972 
62 
972 

Sm = 4680
109 
1319 
1283 
277 
1151 
541 
1361 
331 
1193 
1097 
499 
199 
1051 
919 
677 
773 
641 
619 
509 
809 
787 
883 
751 
941 
631 
1061 
367 
463 
1229 
929 
1019 
241 
373 
1187 
409 
1451 

Attachment 6.3.2 contains the resulting Prime Number Magic Squares for miscellaneous Magic Sums (Sm).
Each square shown corresponds with numerous Prime Number 'Magic Squares of the Sun'.
6.3.3 Magic Squares, Almost Associated
Based on Orthogonal SemiLatin Squares (A,B) as applied in
Section 6.2.4 and correlated balanced series,
the square M = A + B will be an order 6 Prime Number Almost Associated Magic Square.
Sa = 1578
479 
179 
311 
47 
347 
215 
215 
347 
47 
479 
179 
311 
311 
179 
215 
47 
347 
479 
47 
179 
479 
311 
347 
215 
215 
347 
47 
479 
179 
311 
311 
347 
479 
215 
179 
47 

Sb = 3102
972 
62 
972 
462 
62 
572 
152 
882 
152 
152 
882 
882 
62 
572 
462 
972 
572 
462 
572 
462 
62 
572 
462 
972 
882 
152 
882 
882 
152 
152 
462 
972 
572 
62 
972 
62 

Sm = 4680
1451 
241 
1283 
509 
409 
787 
367 
1229 
199 
631 
1061 
1193 
373 
751 
677 
1019 
919 
941 
619 
641 
541 
883 
809 
1187 
1097 
499 
929 
1361 
331 
463 
773 
1319 
1051 
277 
1151 
109 

Attachment 6.3.3 contains the resulting Prime Number Magic Squares for miscellaneous Magic Sums (Sm).
Each square shown corresponds with numerous Prime Number Almost Associated Magic Squares.
6.4 Summary
The obtained results regarding the order 6 SemiLatin  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 7 Latin Diagonal  and related (Pan) Magic Squares,
which will be described in following sections.
