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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:

 Attachment Subject Subroutine Semi-Latin Squares (Square of the Sun) Semi-Latin Squares (Almost Associated) Orthogonal Borders - - - Correlated Balanced Series - Prime Number Magic Squares, Symm Diagonals Prime Number Magic Squares, Square of the Sun Prime Number Magic Squares, Almost Associated - - -

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.