Office Applications and Entertainment, Latin Squares | ||
 |
Index | About the Author |
|
A Latin Square of order 8 is a 8 x 8 square filled with 8 different symbols, each occurring only once in each row and only once in each column.
8.1 Latin Diagonal Squares (8 x 8)
Based on the definition formulated above 300.286.741.708.800 Latin Diagonal Squares can be found (ref. OEIS A274806).
8.2 Magic Squares, Natural Numbers
(Pan) Magic Square M of order 8 with the integers 1 ... 64 can be written as
M = A + 8 * B + [1]
where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6 and 7.
A few additional types of Orthogonal Squares (A, B)
- and resulting Magic Squares -
will be discussed in following sections.
Order 8 Borders M can be constructed based on pairs of Orthogonal Semi-Latin Borders (A, B), as shown below for the symbols {ai, i = 1 ... 8} and {bj, j = 1 ... 8). (A, B) All pairs of the resulting border (A, B) are distinct, as illustrated by following numerical example:
A pair of order 8 Orthogonal Semi-Latin Borders can be constructed
for any pair of order 6 Orthogonal Semi-Latin Squares
(A6, B6),
of which a few types have been discussed in Section 6.2.
Each pair of order 8 Orthogonal Semi-Latin Borders corresponds with 8 * (6!)2 = 4.147.200 pairs.
8.2.3 Composed (Pan) Magic Squares
Pan Magic Sub Squares
Order 8 Composed (Pan) Magic Squares M can be constructed based on pairs of Orthogonal Composed
Semi-Latin Squares
(A, B)
for miscellaneous types of Sub Squares.
(A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
Attachment 8.2.2 contains 504 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals -
based on the properties mentioned above (ref. CompLat8a).
8.2.4 Composed Pan Magic Squares
Pan Magic Sub Squares
The Composed Pan Magic Squares as discussed in detail in Section 8.2.7 contain:
which limits the amount of possible Semi-Latin Squares considerable.
Order 8 Composed Pan Magic Squares M can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B), as shown below for the symbols {ai, i = 1 ... 8} and {bj, j = 1 ... 8). (A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
Attachment 8.2.3 contains 144 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals -
based on the properties summarized above (ref. CompLat8b).
8.2.5 Most Perfect Pan Magic Squares
Order 8 Most Perfect Pan Magic Squares, as discussed in detail in Section 8.5.5,
can be constructed based on pairs of Orthogonal Semi-Latin Squares
(A, B).
A numerical example is shown below: |
A
3 2 1 0 5 4 7 6 4 5 6 7 2 3 0 1 3 2 1 0 5 4 7 6 4 5 6 7 2 3 0 1 2 3 0 1 4 5 6 7 5 4 7 6 3 2 1 0 2 3 0 1 4 5 6 7 5 4 7 6 3 2 1 0 B = T(A)
3 4 3 4 2 5 2 5 2 5 2 5 3 4 3 4 1 6 1 6 0 7 0 7 0 7 0 7 1 6 1 6 5 2 5 2 4 3 4 3 4 3 4 3 5 2 5 2 7 0 7 0 6 1 6 1 6 1 6 1 7 0 7 0 M = A + 8 * B + 1
28 35 26 33 22 45 24 47 21 46 23 48 27 36 25 34 12 51 10 49 6 61 8 63 5 62 7 64 11 52 9 50 43 20 41 18 37 30 39 32 38 29 40 31 44 19 42 17 59 4 57 2 53 14 55 16 54 13 56 15 60 3 58 1
|
Attachment 8.25.1 shows 624 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals -
based on the properties mentioned above (ref. CompLat8c).
Order 8 Franklin Magic Squares, as discussed in detail in Section 8.4.1,
can be constructed based on pairs of Orthogonal Semi-Latin Squares
(A, B).
A numerical example is shown below: |
A
3 2 5 4 1 0 7 6 4 5 2 3 6 7 0 1 2 3 4 5 0 1 6 7 5 4 3 2 7 6 1 0 3 2 5 4 1 0 7 6 4 5 2 3 6 7 0 1 2 3 4 5 0 1 6 7 5 4 3 2 7 6 1 0 B = T(A)
3 4 2 5 3 4 2 5 2 5 3 4 2 5 3 4 5 2 4 3 5 2 4 3 4 3 5 2 4 3 5 2 1 6 0 7 1 6 0 7 0 7 1 6 0 7 1 6 7 0 6 1 7 0 6 1 6 1 7 0 6 1 7 0 M = A + 8 * B + 1
28 35 22 45 26 33 24 47 21 46 27 36 23 48 25 34 43 20 37 30 41 18 39 32 38 29 44 19 40 31 42 17 12 51 6 61 10 49 8 63 5 62 11 52 7 64 9 50 59 4 53 14 57 2 55 16 54 13 60 3 56 15 58 1
|
Attachment 8.26.1 shows 960 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals -
based on the properties mentioned above (ref. CompLat8d).
8.2.7 Compact Associated Pan Magic Squares
Compact Associated Pan Magic Squares, as discussed in detail in Section 8.6.5,
can be constructed based on pairs of Orthogonal Semi-Latin Squares
(A, B).
|
A1
7 4 3 0 2 1 6 5 2 1 6 5 7 4 3 0 5 6 1 2 0 3 4 7 0 3 4 7 5 6 1 2 5 6 1 2 0 3 4 7 0 3 4 7 5 6 1 2 7 4 3 0 2 1 6 5 2 1 6 5 7 4 3 0 B = T(A39)
7 0 3 4 7 0 3 4 0 7 4 3 0 7 4 3 6 1 2 5 6 1 2 5 1 6 5 2 1 6 5 2 5 2 1 6 5 2 1 6 2 5 6 1 2 5 6 1 4 3 0 7 4 3 0 7 3 4 7 0 3 4 7 0 M = A + 8 * B + 1
64 5 28 33 59 2 31 38 3 58 39 30 8 61 36 25 54 15 18 43 49 12 21 48 9 52 45 24 14 55 42 19 46 23 10 51 41 20 13 56 17 44 53 16 22 47 50 11 40 29 4 57 35 26 7 62 27 34 63 6 32 37 60 1
|
Attachment 8.27.1 shows 432 ea order 8 Semi-Latin Squares - with Latin Rows and Diagonals -
based on the properties mentioned above (ref. CompLat8e).
8.2.8 Evaluation of the Results
Following table compares a few enumeration results for order 8 Magic Squares 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.
8.3 Magic Squares, Prime Numbers
When the elements {ai, i = 1 ... 8} and {bj, j = 1 ... 8) of a valid pair of Orthogonal Latin Diagonal Squares (A, B) comply with following condition:
the resulting square M = A + B will be an order 8 Prime Number Simple Magic Square, as illustrated below.
Attachment 8.3, page 1 contains miscellaneous correlated unbalanced series
{ai, i = 1 ... 8}
and
{bj, j = 1 ... 8).
Attachment 8.3.2 contains the resulting Prime Number Simple Magic Squares for miscellaneous Magic Sums (Sm).
8.3.2 Pan Magic Squares, Most Perefect
When the elements {ai, i = 1 ... 8} and {bj, j = 1 ... 8) of a valid pair of Orthogonal Squares (A, B) comply with following conditions:
The resulting square M = A + B will be an order 8 Prime Number Pan Magic Square (Most Perfect).
Attachment 8.3, page 2 contains miscellaneous correlated balanced series
{ai, i = 1 ... 8}
and
{bj, j = 1 ... 8).
Attachment 8.3.3 contains the resulting Prime Number Most Perfect Pan Magic Squares for miscellaneous Magic Sums (Sm).
Center Square, Square of the Sun
Based on Orthogonal Semi-Latin Squares (A,B) as applied in Section 8.2.2 and correlated balanced series, the square M = A + B will be an order 8 Prime Number Concentric Magic Square.
Attachment 8.3.4 contains the resulting Prime Number Concentric Magic Squares for miscellaneous Magic Sums (Sm).
8.3.4 Composed Pan Magic Squares
Pan Magic Sub Squares
Based on Orthogonal Semi-Latin Squares (A,B) as applied in Section 8.2.3 and correlated balanced series, the square M = A + B will be an order 8 Prime Number Composed Pan Magic Square.
Attachment 8.3.5 contains the resulting Prime Number Composed Pan Magic Squares for miscellaneous Magic Sums (Sm).
The obtained results regarding the order 8 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 9 (Semi) Latin - and related (Pan) Magic Squares, which will be described in following sections.
|
|
|
Index | About the Author |