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A Latin Square of order 9 is a 9 x 9 square filled with 9 different symbols, each occurring only once in each row and only once in each column.
9.1 Latin Diagonal Squares (9 x 9)
Based on the definition formulated above 1.835.082.219.864.832.081.920 Latin Diagonal Squares can be found (ref. OEIS A274806).
9.2 Magic Squares, Natural Numbers
(Simple) Magic Square M of order 9 with the integers 1 ... 81 can be written as
M = A + 9 * B + [1]
where the squares A and B contain only the integers 0, 1, 2, 3, 4, 5, 6, 7 and 8.
A few additional types of Orthogonal Squares (A, B)
- and resulting Magic Squares -
will be discussed in following sections.
9.2.2 Concentric Magic Squares
Order 9 Concentric Magic Squares M can be constructed based on pairs of Orthogonal Concentric Semi-Latin Squares (A, B), as shown below for the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9). (A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A pair of order 9 Orthogonal Semi-Latin Borders can be constructed
for each pair of order 7 Orthogonal Concentric Semi-Latin Squares
(A7, B7),
as found in Section 7.2.3.
Diamond Inlay Order 3
Order 9 Bordered Magic Squares M can be constructed based on pairs of Orthogonal Bordered
Semi-Latin Squares
(A, B)
for miscellaneous types of Center Squares.
(A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A pair of order 9 Orthogonal Semi-Latin Borders can be constructed
for each pair of order 7 Orthogonal Inlaid Semi-Latin Squares
(A7, B7),
as found in Section 7.2.4.
Diamond Inlay Order 4
The example shown below is based on Center Squares with order 4 Diamond Inlays as discussed in Section 7.2.7 and the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9). (A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A pair of order 9 Orthogonal Semi-Latin Borders can be constructed
for each pair of order 7 Orthogonal Inlaid Semi-Latin Squares
(A7, B7),
as found in Section 7.2.7.
Overlapping Sub Squares Order 4
Another order 9 Bordered Magic Squares M, which can be constructed based on pairs of Orthogonal Bordered
Semi-Latin Squares
(A, B),
and the symbols
{ai, i = 1 ... 9}
and
{bj, j = 1 ... 9),
is shown below.
(A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A pair of order 9 Orthogonal Semi-Latin Borders can be constructed
for each pair of order 7 Orthogonal Semi-Latin Squares
(A7, B7)
with Overlapping Sub Squares,
as found in Section 7.2.6.
Associated Border
Order 9 Magic Squares M,
composed out of two order 3 Semi Magic, two order 4 Magic Center Squares and an Associated Border,
can be constructed based on pairs of Orthogonal Inlaid Semi-Latin Squares
(A, B).
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(A, B)
a1,b5 a4,b9 a8,b9 a7,b9 a1,b1 a7,b1 a8,b1 a4,b1 a5,b9 a9,b4 a2,b2 a3,b5 a5,b6 a6,b3 a6,b8 a3,b7 a2,b4 a9,b6 a9,b8 a5,b3 a6,b6 a2,b5 a3,b2 a3,b4 a2,b8 a6,b7 a9,b2 a9,b7 a6,b5 a5,b2 a3,b3 a2,b6 a2,b7 a6,b4 a3,b8 a9,b3 a9,b1 a3,b6 a2,b3 a6,b2 a5,b5 a4,b8 a8,b7 a7,b4 a1,b9 a1,b7 a4,b2 a7,b6 a8,b3 a8,b4 a7,b7 a5,b8 a4,b5 a1,b3 a1,b8 a7,b3 a8,b2 a4,b6 a7,b8 a8,b5 a4,b4 a5,b7 a1,b2 a1,b4 a8,b6 a4,b3 a7,b2 a4,b7 a5,b4 a7,b5 a8,b8 a1,b6 a5,b1 a6,b9 a2,b9 a3,b9 a9,b9 a3,b1 a2,b1 a6,b1 a9,b5
All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
A
0 3 7 6 0 6 7 3 4 8 1 2 4 5 5 2 1 8 8 4 5 1 2 2 1 5 8 8 5 4 2 1 1 5 2 8 8 2 1 5 4 3 7 6 0 0 3 6 7 7 6 4 3 0 0 6 7 3 6 7 3 4 0 0 7 3 6 3 4 6 7 0 4 5 1 2 8 2 1 5 8 Sa
12 8 16 20 B
4 8 8 8 0 0 0 0 8 3 1 4 5 2 7 6 3 5 7 2 5 4 1 3 7 6 1 6 4 1 2 5 6 3 7 2 0 5 2 1 4 7 6 3 8 6 1 5 2 3 6 7 4 2 7 2 1 5 7 4 3 6 1 3 5 2 1 6 3 4 7 5 0 8 8 8 8 0 0 0 4 Sb
12 16 8 20 M = A + 9 * B + 1
37 76 80 79 1 7 8 4 77 36 11 39 50 24 69 57 29 54 72 23 51 38 12 30 65 60 18 63 42 14 21 47 56 33 66 27 9 48 20 15 41 67 62 34 73 55 13 52 26 35 61 68 40 19 64 25 17 49 70 44 31 59 10 28 53 22 16 58 32 43 71 46 5 78 74 75 81 3 2 6 45 Sm
124 155 91 204
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The balanced series {0, 1, 2, 3, 4 ... 8} have been split into two unbalanced sub series:
Attachment 9.25.1 shows a few suitable sets (4 ea) of order 3/4 sub series for the integers 0 ... 8.
Pan Magic Sub Squares Order 4 and 5
Order 9 Composed Magic Squares, as discussed in detail in Section 9.7.2,
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:
Based on the distribution of the integers 0, 1 ... 8 over the four Sub Squares shown above,
589824 (= 162 * 482) Semi Latin Squares, with Latin Rows and - Diagonals can be constructed.
9.2.7 Associated Magic Squares
Sub Squares Order 4 and 5
Alternatively Order 9 Associated Composed Magic Squares can be constructed, based on pairs of Orthogonal Associated Composed Semi-Latin Squares (A, B) for miscellaneous types of Sub Squares. The example shown below is based on
and the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9). (A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
Attachment 9.2.6 contains 1008 ea order 9 Associated Semi-Latin Squares - with Latin Rows and Diagonals -
based on the properties mentioned above (ref. CompLat9b).
9.2.8 Associated Magic Squares
Diamond Inlays order 4 and 5
Order 9 Associated Magic Squares M, with order 4 and 5 Diamond Inlays, can be constructed based on pairs of Orthogonal Associated Semi-Latin Squares (A, B), as shown below for the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9). (A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
With procedure SemiLat9b
1090 pairs of Orthogonal Semi-Latin Associated Squares
(A, R(A))
with order 4 and 5 Diamond Inlays could be found,
of which 76 are shown in Attachment 9.2.9.
Sub Squares Order 3 and 6
Magic squares composed of order 3 (Semi) Magic Sub Squares and order 6 Associated or Pan Magic Sub Squares
can be obtained by transformation of Associated Compact Pan Magic Squares
(ref. Section 9.6.7).
Order 9 Associated Compact Pan Magic Squares, can be constructed based on pairs of Orthogonal Semi-Latin Squares (A, B) composed of order 3 Latin Sub Squares, based on the symbols {ai, i = 1 ... 9} and {bj, j = 1 ... 9). (A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
The corresponding transformations are illustrated below:
Attachment 9.2.7 contains 96 ea order 9 (Semi) Latin Associated Compact Pan Magic Squares - with Latin Rows and Diagonals -
based on the properties mentioned above (ref. CompLat9c).
Overlapping Sub Squares Order 3 and 7
Order 9 Composed Magic Squares, as discussed in detail in Section 9.6.6,
can be constructed based on pairs of Orthogonal Composed
Semi-Latin Squares
(A, B)
for miscellaneous types of order 7 Sub Squares.
(A, B) All pairs of the resulting square (A, B) are distinct, as illustrated by following numerical example:
The Frames (A', B'),
composed out of a 3 x 3 Semi-Magic Corner Square and two 2 x 6 Symmetric Magic Rectangles,
are Non-Latin but Orthogonal.
Attachment 9.2.8 contains 33 (unique) Non-Latin Orthogonal Frames,
based on the order 3 Semi Magic Sub Square as used in the example shown above (CompLat9d).
9.2.11 Evaluation of the Results
Following table compares a few enumeration results for order 9 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.
9.3 Magic Squares, Prime Numbers
When the elements {ai, i = 1 ... 9} and {bj, j = 1 ... 9) of a valid pair of Orthogonal Latin Diagonal Squares (A, B) complies with following condition:
the resulting square M = A + B will be an order 9 Prime Number Simple Magic Square.
Attachment 9.3 contains miscellaneous correlated magic series
{ai, i = 1 ... 9}
and
{bj, j = 1 ... 9).
Attachment 9.3.1 contains the resulting Prime Number Simple Magic Squares for miscellaneous Magic Sums (Sm).
9.3.21 Symmetric Magic Squares
Prime Number Symmetric Magic Squares require that the series {ai, i = 1 ... 9} and {bj, j = 1 ... 9) of a valid pair of Orthogonal Latin Diagonal Squares (A, B) comply with following conditions:
Such order 9 Correlated Balanced Magic Series, resulting in Prime Number Magic Squares, have not yet been found.
The obtained results regarding the order 9 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 10 (Semi) Latin - and related (Composed) Magic Squares, which will be described in following sections.
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