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9.5 Self Orthogonal Latin Squares (9 x 9)
A Self Orthogonal Latin Square A is a Latin Square that is Orthogonal to its Transposed T(A).
The transposed square T(A) can be obtained by exchanging the rows and columns of A.
A construction example of a Simple Magic Square M = A + 9 * T(A) + [1] is shown below:
Unlike the order 4, 5, 7 and 8 Self Orthogonal Latin Diagonal Squares not all order 9 Self Orthogonal Latin Diagonal Squares are Double Self Orthogonal.
Each Self Orthogonal Latin Diagonal Square corresponds with 9! = 362.880 Self Orthogonal Latin Diagonal Squares,
which can be obtained by permutation of the integers {ai, i = 1 ... 9}.
In addition to the transformations and permutations described above, each Self Orthogonal Latin Diagonal Square A corresponds with 192 transformations, as described below.
The resulting number of transformations, excluding the 180o rotated aspects, is 16/2 * 24 = 192,
which are shown in Attachment 9.5.13.
9.5.2 Associated Magic Squares
A construction example of an Associated Magic Square M = A + 9 * T(A) + [1] is shown below:
A collection of 23232 Associated Idempotent Self Orthogonal Latin Diagonal Squares,
could be generated within about two hours (ref. SelfOrth9a).
A construction example of an Ultra Magic Square M = A + 9 * T(A) + [1] is shown below:
The total number of order 9 Self Orthogonal Ultra Magic Latin Diagonal Squares is 1296
and can be generated within 863 seconds (ref. SelfOrth9c).
9.5.4 Magic Squares, Compact, Pan Magic
The sub collection described in this section can be determined based on the defining properties rather then on permutations of the integers of Self Orthogonal Latin Diagonal Base Squares.
The total number of order 9 Self Orthogonal Compact Pan Magic Latin Diagonal Squares is 10368
and can be constructed based on the 3456 Ternary Compact Pan Magic Squares as deducted and discussed
in Section 9.5.3.
9.5.5 Magic Squares, Compact, Pan Magic A construction example of a Compact Pan Magic Square M = A + 9 * T(A) + [1] for which the elements of every third-row and third-column sum to s1/3 is shown below:
The total number of subject order 9 Self Orthogonal Magic Squares is 576 and can be
filtered from the collection of 10368 Compact Pan Magic Self Orthogonal Latin Diagonal Squares as deducted in Section 9.5.4 above.
9.5.6 Simple Magic Squares
The sub collection described in this section can be determined based on the defining properties rather then on permutations of the integers of Self Orthogonal Latin Diagonal Base Squares.
The total number of subject Simple Self Orthogonal Magic Squares
- excluding Associated and Compact Pan Magic Squares -
is 1160 (unique) and can be generated within 3,0 hrs
(ref. SelfOrth9e2).
9.6 Interesting Sub Collections (Associated)
The following Sub Collections of Associated Magic Squares can be found based on the Associated Idempotent Self Orthogonal Latin Diagonal Squares
found in Section 9.5.2 above.
A construction example of a Bordered Magic Square M = A + 9 * T(A) + [1] is shown below:
The total number of order 9 Self Orthogonal Bordered Latin Diagonal Squares is 5808
and can be generated within 860 seconds
(ref. SelfOrth9c).
9.6.2 Inlaid Magic Squares, Diamond (3 x 3)
A construction example of an Inlaid Magic Square M = A + 9 * T(A) + [1] with 3 x 3 Diamond Inlay is shown below:
The total number of order 9 Self Orthogonal Inlaid Latin Diagonal Squares with 3 x 3 Diamond Inlay is 56576
and can be generated within 860 seconds
(ref. SelfOrth9c).
9.6.3 Inlaid Magic Squares, Diamond (4 x 4)
A construction example of an Inlaid Magic Square M = A + 9 * T(A) + [1] with 4 x 4 Diamond Inlay is shown below:
The total number of order 9 Self Orthogonal Inlaid Latin Diagonal Squares with 4 x 4 Diamond Inlay is 96
and can be generated within 845 seconds
(ref. SelfOrth9c).
9.6.4 Inlaid Magic Squares, Diamond (5 x 5)
A construction example of an Inlaid Magic Square M = A + 9 * T(A) + [1] with 5 x 5 Diamond Inlay is shown below:
The total number of order 9 Self Orthogonal Inlaid Latin Diagonal Squares with 5 x 5 Diamond Inlay is 128
and can be generated within 615 seconds
(ref. SelfOrth9c).
9.6.5 Magic Squares, Corner Squares (4 x 4)
A construction example of a Magic Square M = A + 9 * T(A) + [1] with 4 x 4 Corner Squares is shown below:
The total number of order 9 Self Orthogonal Latin Diagonal Squares with 4 x 4 Corner Squares is 3072
and can be generated within 810 seconds
(ref. SelfOrth9c).
9.6.6 Magic Squares, Corner Squares (5 x 5)
A construction example of a Magic Square M = A + 9 * T(A) + [1] with each other overlapping 5 x 5 Corner Squares is shown below:
The total number of order 9 Self Orthogonal Latin Diagonal Squares with each other overlapping 5 x 5 Corner Squares is 448
and can be generated within 810 seconds
(ref. SelfOrth9c).
9.6.7 Magic Squares, Compact Ultra Magic
A construction example of a Compact Ultra Magic Square M = A + 9 * T(A) + [1] is shown below:
The total number of order 9 Self Orthogonal Compact Ultra Magic Squares
is 576 and can be generated within 975 seconds
(ref. SelfOrth9c).
9.6.8 Magic Squares A construction example of a Magic Square M = A + 9 * T(A) + [1] for which the elements of every third-row and third-column sum to s1/3. is shown below:
The total number of subject order 9 Self Orthogonal Magic Squares
is 576 and can be generated within 865 seconds
(ref. SelfOrth9c).
9.7 Composed Latin Squares (36 x 36)
Order 9 Self Orthogonal Latin Diagonal Squares can be used to construct order 36 Self Orthogonal Composed Latin Diagonal Squares.
9.7.1 Composed Associated Squares
Order 9 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:
Sqrs9 The order 4 Self Orthogonal Associated Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A shown below. |
A, Associated
29 33 35 30 32 27 31 34 28 35 30 28 31 34 33 29 27 32 31 29 27 32 33 35 30 28 34 27 32 34 28 35 29 33 31 30 33 27 30 34 31 28 32 35 29 32 31 29 33 27 34 28 30 35 28 34 32 27 29 30 35 33 31 30 35 33 29 28 31 34 32 27 34 28 31 35 30 32 27 29 33
20 24 26 21 23 18 22 25 19 26 21 19 22 25 24 20 18 23 22 20 18 23 24 26 21 19 25 18 23 25 19 26 20 24 22 21 24 18 21 25 22 19 23 26 20 23 22 20 24 18 25 19 21 26 19 25 23 18 20 21 26 24 22 21 26 24 20 19 22 25 23 18 25 19 22 26 21 23 18 20 24
2 6 8 3 5 0 4 7 1 8 3 1 4 7 6 2 0 5 4 2 0 5 6 8 3 1 7 0 5 7 1 8 2 6 4 3 6 0 3 7 4 1 5 8 2 5 4 2 6 0 7 1 3 8 1 7 5 0 2 3 8 6 4 3 8 6 2 1 4 7 5 0 7 1 4 8 3 5 0 2 6
11 15 17 12 14 9 13 16 10 17 12 10 13 16 15 11 9 14 13 11 9 14 15 17 12 10 16 9 14 16 10 17 11 15 13 12 15 9 12 16 13 10 14 17 11 14 13 11 15 9 16 10 12 17 10 16 14 9 11 12 17 15 13 12 17 15 11 10 13 16 14 9 16 10 13 17 12 14 9 11 15
2 6 8 3 5 0 4 7 1 8 3 1 4 7 6 2 0 5 4 2 0 5 6 8 3 1 7 0 5 7 1 8 2 6 4 3 6 0 3 7 4 1 5 8 2 5 4 2 6 0 7 1 3 8 1 7 5 0 2 3 8 6 4 3 8 6 2 1 4 7 5 0 7 1 4 8 3 5 0 2 6
11 15 17 12 14 9 13 16 10 17 12 10 13 16 15 11 9 14 13 11 9 14 15 17 12 10 16 9 14 16 10 17 11 15 13 12 15 9 12 16 13 10 14 17 11 14 13 11 15 9 16 10 12 17 10 16 14 9 11 12 17 15 13 12 17 15 11 10 13 16 14 9 16 10 13 17 12 14 9 11 15
29 33 35 30 32 27 31 34 28 35 30 28 31 34 33 29 27 32 31 29 27 32 33 35 30 28 34 27 32 34 28 35 29 33 31 30 33 27 30 34 31 28 32 35 29 32 31 29 33 27 34 28 30 35 28 34 32 27 29 30 35 33 31 30 35 33 29 28 31 34 32 27 34 28 31 35 30 32 27 29 33
20 24 26 21 23 18 22 25 19 26 21 19 22 25 24 20 18 23 22 20 18 23 24 26 21 19 25 18 23 25 19 26 20 24 22 21 24 18 21 25 22 19 23 26 20 23 22 20 24 18 25 19 21 26 19 25 23 18 20 21 26 24 22 21 26 24 20 19 22 25 23 18 25 19 22 26 21 23 18 20 24
11 15 17 12 14 9 13 16 10 17 12 10 13 16 15 11 9 14 13 11 9 14 15 17 12 10 16 9 14 16 10 17 11 15 13 12 15 9 12 16 13 10 14 17 11 14 13 11 15 9 16 10 12 17 10 16 14 9 11 12 17 15 13 12 17 15 11 10 13 16 14 9 16 10 13 17 12 14 9 11 15
2 6 8 3 5 0 4 7 1 8 3 1 4 7 6 2 0 5 4 2 0 5 6 8 3 1 7 0 5 7 1 8 2 6 4 3 6 0 3 7 4 1 5 8 2 5 4 2 6 0 7 1 3 8 1 7 5 0 2 3 8 6 4 3 8 6 2 1 4 7 5 0 7 1 4 8 3 5 0 2 6
20 24 26 21 23 18 22 25 19 26 21 19 22 25 24 20 18 23 22 20 18 23 24 26 21 19 25 18 23 25 19 26 20 24 22 21 24 18 21 25 22 19 23 26 20 23 22 20 24 18 25 19 21 26 19 25 23 18 20 21 26 24 22 21 26 24 20 19 22 25 23 18 25 19 22 26 21 23 18 20 24
29 33 35 30 32 27 31 34 28 35 30 28 31 34 33 29 27 32 31 29 27 32 33 35 30 28 34 27 32 34 28 35 29 33 31 30 33 27 30 34 31 28 32 35 29 32 31 29 33 27 34 28 30 35 28 34 32 27 29 30 35 33 31 30 35 33 29 28 31 34 32 27 34 28 31 35 30 32 27 29 33
20 24 26 21 23 18 22 25 19 26 21 19 22 25 24 20 18 23 22 20 18 23 24 26 21 19 25 18 23 25 19 26 20 24 22 21 24 18 21 25 22 19 23 26 20 23 22 20 24 18 25 19 21 26 19 25 23 18 20 21 26 24 22 21 26 24 20 19 22 25 23 18 25 19 22 26 21 23 18 20 24
29 33 35 30 32 27 31 34 28 35 30 28 31 34 33 29 27 32 31 29 27 32 33 35 30 28 34 27 32 34 28 35 29 33 31 30 33 27 30 34 31 28 32 35 29 32 31 29 33 27 34 28 30 35 28 34 32 27 29 30 35 33 31 30 35 33 29 28 31 34 32 27 34 28 31 35 30 32 27 29 33
11 15 17 12 14 9 13 16 10 17 12 10 13 16 15 11 9 14 13 11 9 14 15 17 12 10 16 9 14 16 10 17 11 15 13 12 15 9 12 16 13 10 14 17 11 14 13 11 15 9 16 10 12 17 10 16 14 9 11 12 17 15 13 12 17 15 11 10 13 16 14 9 16 10 13 17 12 14 9 11 15
2 6 8 3 5 0 4 7 1 8 3 1 4 7 6 2 0 5 4 2 0 5 6 8 3 1 7 0 5 7 1 8 2 6 4 3 6 0 3 7 4 1 5 8 2 5 4 2 6 0 7 1 3 8 1 7 5 0 2 3 8 6 4 3 8 6 2 1 4 7 5 0 7 1 4 8 3 5 0 2 6
Attachment 9.7.1
shows the resulting
order 36 Composed Associated Square based on the Self Orthogonal Composed Associated Latin Square shown above.
9.7.2 Composed Pan Magic Squares (1)
Order 36 Self Orthogonal Composed Pan Magic and Complete Latin Diagonal Squares can be constructed based on Order 36 Self Orthogonal Composed Associated Latin Diagonal Squares as illustrated below (Euler): Sqrs9 The order 4 Self Orthogonal Associated Latin Square shown above is based on the first elements of the Sub Squares (before transformation) and has been used as a guideline for the construction of square A shown below. |
A, Pan Magic (Euler)
29 33 35 30 32 27 31 34 28 35 30 28 31 34 33 29 27 32 31 29 27 32 33 35 30 28 34 27 32 34 28 35 29 33 31 30 33 27 30 34 31 28 32 35 29 32 31 29 33 27 34 28 30 35 28 34 32 27 29 30 35 33 31 30 35 33 29 28 31 34 32 27 34 28 31 35 30 32 27 29 33
20 24 26 21 23 18 22 25 19 26 21 19 22 25 24 20 18 23 22 20 18 23 24 26 21 19 25 18 23 25 19 26 20 24 22 21 24 18 21 25 22 19 23 26 20 23 22 20 24 18 25 19 21 26 19 25 23 18 20 21 26 24 22 21 26 24 20 19 22 25 23 18 25 19 22 26 21 23 18 20 24
10 16 13 9 14 12 17 15 11 14 9 11 15 16 13 10 12 17 16 10 12 17 15 14 9 11 13 12 13 15 11 17 10 16 14 9 11 17 14 10 13 16 12 9 15 17 12 10 16 9 15 11 13 14 13 15 17 12 11 9 14 16 10 9 14 16 13 10 11 15 17 12 15 11 9 14 12 17 13 10 16
1 7 4 0 5 3 8 6 2 5 0 2 6 7 4 1 3 8 7 1 3 8 6 5 0 2 4 3 4 6 2 8 1 7 5 0 2 8 5 1 4 7 3 0 6 8 3 1 7 0 6 2 4 5 4 6 8 3 2 0 5 7 1 0 5 7 4 1 2 6 8 3 6 2 0 5 3 8 4 1 7
2 6 8 3 5 0 4 7 1 8 3 1 4 7 6 2 0 5 4 2 0 5 6 8 3 1 7 0 5 7 1 8 2 6 4 3 6 0 3 7 4 1 5 8 2 5 4 2 6 0 7 1 3 8 1 7 5 0 2 3 8 6 4 3 8 6 2 1 4 7 5 0 7 1 4 8 3 5 0 2 6
11 15 17 12 14 9 13 16 10 17 12 10 13 16 15 11 9 14 13 11 9 14 15 17 12 10 16 9 14 16 10 17 11 15 13 12 15 9 12 16 13 10 14 17 11 14 13 11 15 9 16 10 12 17 10 16 14 9 11 12 17 15 13 12 17 15 11 10 13 16 14 9 16 10 13 17 12 14 9 11 15
19 25 22 18 23 21 26 24 20 23 18 20 24 25 22 19 21 26 25 19 21 26 24 23 18 20 22 21 22 24 20 26 19 25 23 18 20 26 23 19 22 25 21 18 24 26 21 19 25 18 24 20 22 23 22 24 26 21 20 18 23 25 19 18 23 25 22 19 20 24 26 21 24 20 18 23 21 26 22 19 25
28 34 31 27 32 30 35 33 29 32 27 29 33 34 31 28 30 35 34 28 30 35 33 32 27 29 31 30 31 33 29 35 28 34 32 27 29 35 32 28 31 34 30 27 33 35 30 28 34 27 33 29 31 32 31 33 35 30 29 27 32 34 28 27 32 34 31 28 29 33 35 30 33 29 27 32 30 35 31 28 34
25 19 22 26 21 23 18 20 24 21 26 24 20 19 22 25 23 18 19 25 23 18 20 21 26 24 22 23 22 20 24 18 25 19 21 26 24 18 21 25 22 19 23 26 20 18 23 25 19 26 20 24 22 21 22 20 18 23 24 26 21 19 25 26 21 19 22 25 24 20 18 23 20 24 26 21 23 18 22 25 19
34 28 31 35 30 32 27 29 33 30 35 33 29 28 31 34 32 27 28 34 32 27 29 30 35 33 31 32 31 29 33 27 34 28 30 35 33 27 30 34 31 28 32 35 29 27 32 34 28 35 29 33 31 30 31 29 27 32 33 35 30 28 34 35 30 28 31 34 33 29 27 32 29 33 35 30 32 27 31 34 28
6 2 0 5 3 8 4 1 7 0 5 7 4 1 2 6 8 3 4 6 8 3 2 0 5 7 1 8 3 1 7 0 6 2 4 5 2 8 5 1 4 7 3 0 6 3 4 6 2 8 1 7 5 0 7 1 3 8 6 5 0 2 4 5 0 2 6 7 4 1 3 8 1 7 4 0 5 3 8 6 2
15 11 9 14 12 17 13 10 16 9 14 16 13 10 11 15 17 12 13 15 17 12 11 9 14 16 10 17 12 10 16 9 15 11 13 14 11 17 14 10 13 16 12 9 15 12 13 15 11 17 10 16 14 9 16 10 12 17 15 14 9 11 13 14 9 11 15 16 13 10 12 17 10 16 13 9 14 12 17 15 11
16 10 13 17 12 14 9 11 15 12 17 15 11 10 13 16 14 9 10 16 14 9 11 12 17 15 13 14 13 11 15 9 16 10 12 17 15 9 12 16 13 10 14 17 11 9 14 16 10 17 11 15 13 12 13 11 9 14 15 17 12 10 16 17 12 10 13 16 15 11 9 14 11 15 17 12 14 9 13 16 10
7 1 4 8 3 5 0 2 6 3 8 6 2 1 4 7 5 0 1 7 5 0 2 3 8 6 4 5 4 2 6 0 7 1 3 8 6 0 3 7 4 1 5 8 2 0 5 7 1 8 2 6 4 3 4 2 0 5 6 8 3 1 7 8 3 1 4 7 6 2 0 5 2 6 8 3 5 0 4 7 1
33 29 27 32 30 35 31 28 34 27 32 34 31 28 29 33 35 30 31 33 35 30 29 27 32 34 28 35 30 28 34 27 33 29 31 32 29 35 32 28 31 34 30 27 33 30 31 33 29 35 28 34 32 27 34 28 30 35 33 32 27 29 31 32 27 29 33 34 31 28 30 35 28 34 31 27 32 30 35 33 29
24 20 18 23 21 26 22 19 25 18 23 25 22 19 20 24 26 21 22 24 26 21 20 18 23 25 19 26 21 19 25 18 24 20 22 23 20 26 23 19 22 25 21 18 24 21 22 24 20 26 19 25 23 18 25 19 21 26 24 23 18 20 22 23 18 20 24 25 22 19 21 26 19 25 22 18 23 21 26 24 20
Attachment 9.7.2
shows the resulting
order 36 Composed Pan Magic and Complete Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.
9.7.3 Composed Pan Magic Squares (2)
Order 9 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:
Sqrs9 The order 4 Self Orthogonal Pan Magic Latin Square shown above is based on first elements of the Sub Squares and has been used as a guideline for the construction of square A shown below. |
A, Pan Magic
10 17 12 13 11 15 16 14 9 14 9 16 17 12 10 11 15 13 15 13 11 9 16 14 12 10 17 11 15 13 14 9 16 17 12 10 12 10 17 15 13 11 9 16 14 16 14 9 10 17 12 13 11 15 9 16 14 12 10 17 15 13 11 13 11 15 16 14 9 10 17 12 17 12 10 11 15 13 14 9 16
1 8 3 4 2 6 7 5 0 5 0 7 8 3 1 2 6 4 6 4 2 0 7 5 3 1 8 2 6 4 5 0 7 8 3 1 3 1 8 6 4 2 0 7 5 7 5 0 1 8 3 4 2 6 0 7 5 3 1 8 6 4 2 4 2 6 7 5 0 1 8 3 8 3 1 2 6 4 5 0 7
28 35 30 31 29 33 34 32 27 32 27 34 35 30 28 29 33 31 33 31 29 27 34 32 30 28 35 29 33 31 32 27 34 35 30 28 30 28 35 33 31 29 27 34 32 34 32 27 28 35 30 31 29 33 27 34 32 30 28 35 33 31 29 31 29 33 34 32 27 28 35 30 35 30 28 29 33 31 32 27 34
19 26 21 22 20 24 25 23 18 23 18 25 26 21 19 20 24 22 24 22 20 18 25 23 21 19 26 20 24 22 23 18 25 26 21 19 21 19 26 24 22 20 18 25 23 25 23 18 19 26 21 22 20 24 18 25 23 21 19 26 24 22 20 22 20 24 25 23 18 19 26 21 26 21 19 20 24 22 23 18 25
19 26 21 22 20 24 25 23 18 23 18 25 26 21 19 20 24 22 24 22 20 18 25 23 21 19 26 20 24 22 23 18 25 26 21 19 21 19 26 24 22 20 18 25 23 25 23 18 19 26 21 22 20 24 18 25 23 21 19 26 24 22 20 22 20 24 25 23 18 19 26 21 26 21 19 20 24 22 23 18 25
28 35 30 31 29 33 34 32 27 32 27 34 35 30 28 29 33 31 33 31 29 27 34 32 30 28 35 29 33 31 32 27 34 35 30 28 30 28 35 33 31 29 27 34 32 34 32 27 28 35 30 31 29 33 27 34 32 30 28 35 33 31 29 31 29 33 34 32 27 28 35 30 35 30 28 29 33 31 32 27 34
1 8 3 4 2 6 7 5 0 5 0 7 8 3 1 2 6 4 6 4 2 0 7 5 3 1 8 2 6 4 5 0 7 8 3 1 3 1 8 6 4 2 0 7 5 7 5 0 1 8 3 4 2 6 0 7 5 3 1 8 6 4 2 4 2 6 7 5 0 1 8 3 8 3 1 2 6 4 5 0 7
10 17 12 13 11 15 16 14 9 14 9 16 17 12 10 11 15 13 15 13 11 9 16 14 12 10 17 11 15 13 14 9 16 17 12 10 12 10 17 15 13 11 9 16 14 16 14 9 10 17 12 13 11 15 9 16 14 12 10 17 15 13 11 13 11 15 16 14 9 10 17 12 17 12 10 11 15 13 14 9 16
1 8 3 4 2 6 7 5 0 5 0 7 8 3 1 2 6 4 6 4 2 0 7 5 3 1 8 2 6 4 5 0 7 8 3 1 3 1 8 6 4 2 0 7 5 7 5 0 1 8 3 4 2 6 0 7 5 3 1 8 6 4 2 4 2 6 7 5 0 1 8 3 8 3 1 2 6 4 5 0 7
10 17 12 13 11 15 16 14 9 14 9 16 17 12 10 11 15 13 15 13 11 9 16 14 12 10 17 11 15 13 14 9 16 17 12 10 12 10 17 15 13 11 9 16 14 16 14 9 10 17 12 13 11 15 9 16 14 12 10 17 15 13 11 13 11 15 16 14 9 10 17 12 17 12 10 11 15 13 14 9 16
19 26 21 22 20 24 25 23 18 23 18 25 26 21 19 20 24 22 24 22 20 18 25 23 21 19 26 20 24 22 23 18 25 26 21 19 21 19 26 24 22 20 18 25 23 25 23 18 19 26 21 22 20 24 18 25 23 21 19 26 24 22 20 22 20 24 25 23 18 19 26 21 26 21 19 20 24 22 23 18 25
28 35 30 31 29 33 34 32 27 32 27 34 35 30 28 29 33 31 33 31 29 27 34 32 30 28 35 29 33 31 32 27 34 35 30 28 30 28 35 33 31 29 27 34 32 34 32 27 28 35 30 31 29 33 27 34 32 30 28 35 33 31 29 31 29 33 34 32 27 28 35 30 35 30 28 29 33 31 32 27 34
28 35 30 31 29 33 34 32 27 32 27 34 35 30 28 29 33 31 33 31 29 27 34 32 30 28 35 29 33 31 32 27 34 35 30 28 30 28 35 33 31 29 27 34 32 34 32 27 28 35 30 31 29 33 27 34 32 30 28 35 33 31 29 31 29 33 34 32 27 28 35 30 35 30 28 29 33 31 32 27 34
19 26 21 22 20 24 25 23 18 23 18 25 26 21 19 20 24 22 24 22 20 18 25 23 21 19 26 20 24 22 23 18 25 26 21 19 21 19 26 24 22 20 18 25 23 25 23 18 19 26 21 22 20 24 18 25 23 21 19 26 24 22 20 22 20 24 25 23 18 19 26 21 26 21 19 20 24 22 23 18 25
10 17 12 13 11 15 16 14 9 14 9 16 17 12 10 11 15 13 15 13 11 9 16 14 12 10 17 11 15 13 14 9 16 17 12 10 12 10 17 15 13 11 9 16 14 16 14 9 10 17 12 13 11 15 9 16 14 12 10 17 15 13 11 13 11 15 16 14 9 10 17 12 17 12 10 11 15 13 14 9 16
1 8 3 4 2 6 7 5 0 5 0 7 8 3 1 2 6 4 6 4 2 0 7 5 3 1 8 2 6 4 5 0 7 8 3 1 3 1 8 6 4 2 0 7 5 7 5 0 1 8 3 4 2 6 0 7 5 3 1 8 6 4 2 4 2 6 7 5 0 1 8 3 8 3 1 2 6 4 5 0 7
Attachment 9.7.3
shows the resulting
order 36 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.
9.8 Composed Latin Squares (37 x 37)
Order 9 Self orthogonal Latin Diagonal Squares can be used to construct order 37 Self Orthogonal Composed Latin Diagonal Squares.
The required order 9 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:
Sqrs9 The order 4 Self orthogonal Latin Diagonal Square shown above is based on the first elements of the original Sub Squares, and has been used as a guideline for the construction of square A shown below. |
A
28 33 29 32 31 36 18 30 34 21 25 27 22 24 19 23 26 20 35 2 6 8 3 5 0 4 7 1 11 15 17 12 14 9 13 16 10 36 29 28 35 30 34 33 31 18 27 22 20 23 26 25 21 19 24 32 8 3 1 4 7 6 2 0 5 17 12 10 13 16 15 11 9 14 35 36 30 33 34 18 31 32 28 23 21 19 24 25 27 22 20 26 29 4 2 0 5 6 8 3 1 7 13 11 9 14 15 17 12 10 16 30 34 18 31 35 29 36 33 32 19 24 26 20 27 21 25 23 22 28 0 5 7 1 8 2 6 4 3 9 14 16 10 17 11 15 13 12 34 35 36 28 32 31 29 18 30 25 19 22 26 23 20 24 27 21 33 6 0 3 7 4 1 5 8 2 15 9 12 16 13 10 14 17 11 32 18 35 36 29 33 30 28 31 24 23 21 25 19 26 20 22 27 34 5 4 2 6 0 7 1 3 8 14 13 11 15 9 16 10 12 17 31 32 33 30 18 28 34 29 35 20 26 24 19 21 22 27 25 23 36 1 7 5 0 2 3 8 6 4 10 16 14 9 11 12 17 15 13 18 30 34 29 36 32 28 35 33 22 27 25 21 20 23 26 24 19 31 3 8 6 2 1 4 7 5 0 12 17 15 11 10 13 16 14 9 29 28 31 18 33 35 32 34 36 26 20 23 27 22 24 19 21 25 30 7 1 4 8 3 5 0 2 6 16 10 13 17 12 14 9 11 15 2 6 8 3 5 0 4 7 1 9 14 10 13 12 17 18 11 15 16 30 34 36 31 33 28 32 35 29 21 25 27 22 24 19 23 26 20 8 3 1 4 7 6 2 0 5 17 10 9 16 11 15 14 12 18 13 36 31 29 32 35 34 30 28 33 27 22 20 23 26 25 21 19 24 4 2 0 5 6 8 3 1 7 16 17 11 14 15 18 12 13 9 10 32 30 28 33 34 36 31 29 35 23 21 19 24 25 27 22 20 26 0 5 7 1 8 2 6 4 3 11 15 18 12 16 10 17 14 13 9 28 33 35 29 36 30 34 32 31 19 24 26 20 27 21 25 23 22 6 0 3 7 4 1 5 8 2 15 16 17 9 13 12 10 18 11 14 34 28 31 35 32 29 33 36 30 25 19 22 26 23 20 24 27 21 5 4 2 6 0 7 1 3 8 13 18 16 17 10 14 11 9 12 15 33 32 30 34 28 35 29 31 36 24 23 21 25 19 26 20 22 27 1 7 5 0 2 3 8 6 4 12 13 14 11 18 9 15 10 16 17 29 35 33 28 30 31 36 34 32 20 26 24 19 21 22 27 25 23 3 8 6 2 1 4 7 5 0 18 11 15 10 17 13 9 16 14 12 31 36 34 30 29 32 35 33 28 22 27 25 21 20 23 26 24 19 7 1 4 8 3 5 0 2 6 10 9 12 18 14 16 13 15 17 11 35 29 32 36 31 33 28 30 34 26 20 23 27 22 24 19 21 25 33 31 32 34 28 30 35 36 29 14 12 13 15 9 11 16 17 10 18 23 19 22 21 26 27 20 24 25 4 0 3 2 7 8 1 5 6 11 15 17 12 14 9 13 16 10 2 6 8 3 5 0 4 7 1 26 19 18 25 20 24 23 21 27 22 30 34 36 31 33 28 32 35 29 17 12 10 13 16 15 11 9 14 8 3 1 4 7 6 2 0 5 25 26 20 23 24 27 21 22 18 19 36 31 29 32 35 34 30 28 33 13 11 9 14 15 17 12 10 16 4 2 0 5 6 8 3 1 7 20 24 27 21 25 19 26 23 22 18 32 30 28 33 34 36 31 29 35 9 14 16 10 17 11 15 13 12 0 5 7 1 8 2 6 4 3 24 25 26 18 22 21 19 27 20 23 28 33 35 29 36 30 34 32 31 15 9 12 16 13 10 14 17 11 6 0 3 7 4 1 5 8 2 22 27 25 26 19 23 20 18 21 24 34 28 31 35 32 29 33 36 30 14 13 11 15 9 16 10 12 17 5 4 2 6 0 7 1 3 8 21 22 23 20 27 18 24 19 25 26 33 32 30 34 28 35 29 31 36 10 16 14 9 11 12 17 15 13 1 7 5 0 2 3 8 6 4 27 20 24 19 26 22 18 25 23 21 29 35 33 28 30 31 36 34 32 12 17 15 11 10 13 16 14 9 3 8 6 2 1 4 7 5 0 19 18 21 27 23 25 22 24 26 20 31 36 34 30 29 32 35 33 28 16 10 13 17 12 14 9 11 15 7 1 4 8 3 5 0 2 6 23 21 22 24 18 20 25 26 19 27 35 29 32 36 31 33 28 30 34 21 25 27 22 24 19 23 26 20 30 34 36 31 33 28 32 35 29 7 11 15 17 12 14 9 13 16 10 0 18 6 1 5 4 2 8 3 27 22 20 23 26 25 21 19 24 36 31 29 32 35 34 30 28 33 6 17 12 10 13 16 15 11 9 14 7 1 4 5 8 2 3 18 0 23 21 19 24 25 27 22 20 26 32 30 28 33 34 36 31 29 35 1 13 11 9 14 15 17 12 10 16 5 8 2 6 0 7 4 3 18 19 24 26 20 27 21 25 23 22 28 33 35 29 36 30 34 32 31 5 9 14 16 10 17 11 15 13 12 6 7 18 3 2 0 8 1 4 25 19 22 26 23 20 24 27 21 34 28 31 35 32 29 33 36 30 3 15 9 12 16 13 10 14 17 11 8 6 7 0 4 1 18 2 5 24 23 21 25 19 26 20 22 27 33 32 30 34 28 35 29 31 36 2 14 13 11 15 9 16 10 12 17 3 4 1 8 18 5 0 6 7 20 26 24 19 21 22 27 25 23 29 35 33 28 30 31 36 34 32 8 10 16 14 9 11 12 17 15 13 1 5 0 7 3 18 6 4 2 22 27 25 21 20 23 26 24 19 31 36 34 30 29 32 35 33 28 0 12 17 15 11 10 13 16 14 9 18 2 8 4 6 3 5 7 1 26 20 23 27 22 24 19 21 25 35 29 32 36 31 33 28 30 34 4 16 10 13 17 12 14 9 11 15 2 3 5 18 1 6 7 0 8
9.10.1 Mutual Orthogonal Latin Squares (9 x 9)
The construction of 0rder 9 Magic Squares based on following Mutual Orthogonal Latin (Diagonal) Squares
has been deducted and discussed in Section 9.5.
9.10.2 Semi Latin Squares (9 x 9)
The construction of 0rder 9 Mutual Orthogonal Semi-Latin (Diagonal) Squares has been deducted and discussed in:
Order 9 Mutual Orthogonal Semi-Latin Inlaid Magic Squares with Diamond Inlays have been deducted and discussed in
Section 9.2.8.
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 applied to construct higher order Self Orthogonal Latin Squares.
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