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 Greco-Latin Squares don't exist
(ref. Euler's 36 officers problem).
The non-existence of order 6 Greco-Latin 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 Semi-Latin or
Orthogonal Non-Latin
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
Semi-Latin Squares
(A, B),
as shown below for the symbols
{ai, i = 1 ... 6}
and
{bj, 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 Semi-Latin 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 Semi-Latin 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 Semi-Latin Squares (A, B) considerable.
Order 6 'Magic Squares of the Sun' M can be constructed based on pairs of Orthogonal
Semi-Latin Squares
(A, B),
as shown below for the symbols
{ai, i = 1 ... 6}
and
{bj, 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 * 3842 / 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
Semi-Latin Squares
(A, B),
as shown below for the symbols
{ai, i = 1 ... 6}
and
{bj, 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 (Semi-Latin) Squares can be considered as an additional property.
6.3 Magic Squares, Prime Numbers
6.3.1 Magic Squares, Symmetrical Diagonals
When the elements
{ai, i = 1 ... 6}
and
{bj, j = 1 ... 6)
of a valid pair of Orthogonal Semi-Latin Squares (A, B)
- as applied in
Section 6.2.2 above - complies with following condition:
-
mij = ai + bj = prime
for i = 1 ... 6 and j = 1 ... 6 (correlated)
-
a1 + a6 =
a2 + a5 =
a3 + a4 (balanced)
b1 + b6 =
b2 + b5 =
b3 + b4
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
{ai, i = 1 ... 6}
and
{bj, 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 Semi-Latin 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 Semi-Latin 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 Semi-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 7 Latin Diagonal - and related (Pan) Magic Squares,
which will be described in following sections.
|