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7.5 Self Orthogonal Latin Squares (7 x 7)
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 + 7 * T(A) + [1] is shown below:
Each Self Orthogonal Latin Diagonal Square corresponds with 7! = 5040 Self Orthogonal Latin Diagonal Squares,
which can be obtained by permutation of the integers {ai, i = 1 ... 7}.
In addition to the transformations and permutations described above, each Self Orthogonal Latin Diagonal Square A corresponds with 24 transformations, as described below.
The resulting number of transformations, excluding the 180o rotated aspects, is 8/2 * 6 = 24,
which are shown in Attachment 7.5.12
A construction example of a Pan Magic Square M = A + 7 * T(A) + [1] is shown below:
The total number of order 7 Self Orthogonal Pan Magic Latin Diagonal Squares is 20160
and can be generated within 200 seconds
(ref. SelfOrth7b).
Alternatively this sub collection can be filtered from the main collection of 322560
of Self Orthogonal Latin Diagonal Squares
(ref. SelfOrth7c).
7.5.3 Associated Magic Squares
A construction example of a Associated Magic Square M = A + 7 * T(A) + [1] is shown below:
The total number of order 7 Self Orthogonal Associated Magic Latin Diagonal Squares is 3072
and can be generated within 25 seconds
(ref. SelfOrth7a).
Attachment 7.5.3 contains the sub collection of Associated Idempotent Self Orthogonal Latin Squares,
which has been used as a base for the main collection as discussed in Section 7.5.1 above.
A construction example of an Ultra Magic Square M = A + 7 * T(A) + [1] is shown below:
Attachment 7.5.4
contains the 192 order 7 Self Orthogonal Ultra Magic Latin Diagonal Squares, which
could be generated within 20 seconds
(ref. SelfOrth7a).
7.5.5 Associated Magic Squares, Diamond Inlay
The collection of Associated Squares found above contains a sub collection of Inlaid Magic Squares with order 3 Diamond Inlay.
Attachment 7.5.5
contains the 32 order 7 Self Orthogonal Associated Inlaid Latin Diagonal Squares, which
could be generated within 20 seconds
(ref. SelfOrth7c).
7.6 Composed Latin Squares (28 x 28)
Order 7 Self Orthogonal Latin Diagonal Squares can be used to construct order 28 Self Orthogonal Composed Latin Diagonal Squares.
7.6.1 Composed Associated Squares
Order 7 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:
Sqrs7 The order 4 Self Orthogonal Associated Latin Square shown above is based on the first elemnets of the Sub Squares and has been used as a guideline for the construction of square A shown below. |
A, Associated
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
Attachment 7.6.11
illustrates the construction of an order 28 Composed Associated Square based on the Self Orthogonal Composed Associated Latin Square shown above.
7.6.2 Composed Pan Magic Squares (1)
Order 28 Self Orthogonal Composed Pan Magic and Complete Latin Diagonal Squares can be constructed based on Order 28 Self Orthogonal Composed Associated Latin Diagonal Squares as illustrated below (Euler): Sqrs7 The order 4 Self Orthogonal Associated Latin Square shown above is based on the first elemnets 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)
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
8 13 12 7 10 11 9 10 7 9 11 12 8 13 13 10 11 12 7 9 8 9 8 7 10 13 12 11 12 11 13 8 9 10 7 7 12 8 9 11 13 10 11 9 10 13 8 7 12
1 6 5 0 3 4 2 3 0 2 4 5 1 6 6 3 4 5 0 2 1 2 1 0 3 6 5 4 5 4 6 1 2 3 0 0 5 1 2 4 6 3 4 2 3 6 1 0 5
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
15 20 19 14 17 18 16 17 14 16 18 19 15 20 20 17 18 19 14 16 15 16 15 14 17 20 19 18 19 18 20 15 16 17 14 14 19 15 16 18 20 17 18 16 17 20 15 14 19
22 27 26 21 24 25 23 24 21 23 25 26 22 27 27 24 25 26 21 23 22 23 22 21 24 27 26 25 26 25 27 22 23 24 21 21 26 22 23 25 27 24 25 23 24 27 22 21 26
19 14 15 20 17 16 18 17 20 18 16 15 19 14 14 17 16 15 20 18 19 18 19 20 17 14 15 16 15 16 14 19 18 17 20 20 15 19 18 16 14 17 16 18 17 14 19 20 15
26 21 22 27 24 23 25 24 27 25 23 22 26 21 21 24 23 22 27 25 26 25 26 27 24 21 22 23 22 23 21 26 25 24 27 27 22 26 25 23 21 24 23 25 24 21 26 27 22
4 2 3 6 1 0 5 0 5 1 2 4 6 3 5 4 6 1 2 3 0 2 1 0 3 6 5 4 6 3 4 5 0 2 1 3 0 2 4 5 1 6 1 6 5 0 3 4 2
11 9 10 13 8 7 12 7 12 8 9 11 13 10 12 11 13 8 9 10 7 9 8 7 10 13 12 11 13 10 11 12 7 9 8 10 7 9 11 12 8 13 8 13 12 7 10 11 9
12 7 8 13 10 9 11 10 13 11 9 8 12 7 7 10 9 8 13 11 12 11 12 13 10 7 8 9 8 9 7 12 11 10 13 13 8 12 11 9 7 10 9 11 10 7 12 13 8
5 0 1 6 3 2 4 3 6 4 2 1 5 0 0 3 2 1 6 4 5 4 5 6 3 0 1 2 1 2 0 5 4 3 6 6 1 5 4 2 0 3 2 4 3 0 5 6 1
25 23 24 27 22 21 26 21 26 22 23 25 27 24 26 25 27 22 23 24 21 23 22 21 24 27 26 25 27 24 25 26 21 23 22 24 21 23 25 26 22 27 22 27 26 21 24 25 23
18 16 17 20 15 14 19 14 19 15 16 18 20 17 19 18 20 15 16 17 14 16 15 14 17 20 19 18 20 17 18 19 14 16 15 17 14 16 18 19 15 20 15 20 19 14 17 18 16
Attachment 7.6.21
illustrates the construction of an order 28 Composed Pan Magic and Complete Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.
7.6.3 Composed Pan Magic Squares (2)
Order 7 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:
Sqrs7 The order 4 Self Orthogonal Pan Magic Latin Square shown above is based on first elemnets of the Sub Squares and has been used as a guideline for the construction of square A shown below. |
A, Pan Magic
11 10 9 8 7 13 12 9 8 7 13 12 11 10 7 13 12 11 10 9 8 12 11 10 9 8 7 13 10 9 8 7 13 12 11 8 7 13 12 11 10 9 13 12 11 10 9 8 7
4 3 2 1 0 6 5 2 1 0 6 5 4 3 0 6 5 4 3 2 1 5 4 3 2 1 0 6 3 2 1 0 6 5 4 1 0 6 5 4 3 2 6 5 4 3 2 1 0
25 24 23 22 21 27 26 23 22 21 27 26 25 24 21 27 26 25 24 23 22 26 25 24 23 22 21 27 24 23 22 21 27 26 25 22 21 27 26 25 24 23 27 26 25 24 23 22 21
18 17 16 15 14 20 19 16 15 14 20 19 18 17 14 20 19 18 17 16 15 19 18 17 16 15 14 20 17 16 15 14 20 19 18 15 14 20 19 18 17 16 20 19 18 17 16 15 14
18 17 16 15 14 20 19 16 15 14 20 19 18 17 14 20 19 18 17 16 15 19 18 17 16 15 14 20 17 16 15 14 20 19 18 15 14 20 19 18 17 16 20 19 18 17 16 15 14
25 24 23 22 21 27 26 23 22 21 27 26 25 24 21 27 26 25 24 23 22 26 25 24 23 22 21 27 24 23 22 21 27 26 25 22 21 27 26 25 24 23 27 26 25 24 23 22 21
4 3 2 1 0 6 5 2 1 0 6 5 4 3 0 6 5 4 3 2 1 5 4 3 2 1 0 6 3 2 1 0 6 5 4 1 0 6 5 4 3 2 6 5 4 3 2 1 0
11 10 9 8 7 13 12 9 8 7 13 12 11 10 7 13 12 11 10 9 8 12 11 10 9 8 7 13 10 9 8 7 13 12 11 8 7 13 12 11 10 9 13 12 11 10 9 8 7
4 3 2 1 0 6 5 2 1 0 6 5 4 3 0 6 5 4 3 2 1 5 4 3 2 1 0 6 3 2 1 0 6 5 4 1 0 6 5 4 3 2 6 5 4 3 2 1 0
11 10 9 8 7 13 12 9 8 7 13 12 11 10 7 13 12 11 10 9 8 12 11 10 9 8 7 13 10 9 8 7 13 12 11 8 7 13 12 11 10 9 13 12 11 10 9 8 7
18 17 16 15 14 20 19 16 15 14 20 19 18 17 14 20 19 18 17 16 15 19 18 17 16 15 14 20 17 16 15 14 20 19 18 15 14 20 19 18 17 16 20 19 18 17 16 15 14
25 24 23 22 21 27 26 23 22 21 27 26 25 24 21 27 26 25 24 23 22 26 25 24 23 22 21 27 24 23 22 21 27 26 25 22 21 27 26 25 24 23 27 26 25 24 23 22 21
25 24 23 22 21 27 26 23 22 21 27 26 25 24 21 27 26 25 24 23 22 26 25 24 23 22 21 27 24 23 22 21 27 26 25 22 21 27 26 25 24 23 27 26 25 24 23 22 21
18 17 16 15 14 20 19 16 15 14 20 19 18 17 14 20 19 18 17 16 15 19 18 17 16 15 14 20 17 16 15 14 20 19 18 15 14 20 19 18 17 16 20 19 18 17 16 15 14
11 10 9 8 7 13 12 9 8 7 13 12 11 10 7 13 12 11 10 9 8 12 11 10 9 8 7 13 10 9 8 7 13 12 11 8 7 13 12 11 10 9 13 12 11 10 9 8 7
4 3 2 1 0 6 5 2 1 0 6 5 4 3 0 6 5 4 3 2 1 5 4 3 2 1 0 6 3 2 1 0 6 5 4 1 0 6 5 4 3 2 6 5 4 3 2 1 0
Attachment 7.6.31
illustrates the construction of an order 28 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.
7.7 Composed Latin Squares (29 x 29)
Order 7 Self orthogonal Latin Diagonal Squares can be used to construct order 29 Self Orthogonal Composed Latin Diagonal Squares.
The required order 7 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:
Sqrs7 The order 4 Self orthogonal Latin Diagonal Square shown above is based on the first elemnets of the original Sub Squares, and has been used as a guideline for the construction of square A shown below. |
A
22 24 27 28 14 26 25 17 19 18 15 20 21 16 23 2 4 3 0 5 6 1 9 11 10 7 12 13 8 25 23 14 26 27 28 22 21 16 20 19 17 15 18 24 6 1 5 4 2 0 3 13 8 12 11 9 7 10 23 27 24 14 28 25 26 16 17 15 20 19 18 21 22 1 2 0 5 4 3 6 8 9 7 12 11 10 13 26 22 28 25 24 14 23 19 20 21 18 15 16 17 27 4 5 6 3 0 1 2 11 12 13 10 7 8 9 24 28 22 27 26 23 14 15 18 17 16 21 19 20 25 0 3 2 1 6 4 5 7 10 9 8 13 11 12 14 25 26 23 22 27 24 18 21 19 17 16 20 15 28 3 6 4 2 1 5 0 10 13 11 9 8 12 7 27 14 23 24 25 22 28 20 15 16 21 18 17 19 26 5 0 1 6 3 2 4 12 7 8 13 10 9 11 2 4 3 0 5 6 1 7 9 12 13 14 11 10 8 24 26 25 22 27 28 23 17 19 18 15 20 21 16 6 1 5 4 2 0 3 10 8 14 11 12 13 7 9 28 23 27 26 24 22 25 21 16 20 19 17 15 18 1 2 0 5 4 3 6 8 12 9 14 13 10 11 7 23 24 22 27 26 25 28 16 17 15 20 19 18 21 4 5 6 3 0 1 2 11 7 13 10 9 14 8 12 26 27 28 25 22 23 24 19 20 21 18 15 16 17 0 3 2 1 6 4 5 9 13 7 12 11 8 14 10 22 25 24 23 28 26 27 15 18 17 16 21 19 20 3 6 4 2 1 5 0 14 10 11 8 7 12 9 13 25 28 26 24 23 27 22 18 21 19 17 16 20 15 5 0 1 6 3 2 4 12 14 8 9 10 7 13 11 27 22 23 28 25 24 26 20 15 16 21 18 17 19 28 26 25 22 23 24 27 13 11 10 7 8 9 12 14 16 19 20 21 18 17 15 1 4 5 6 3 2 0 9 11 10 7 12 13 8 2 4 3 0 5 6 1 17 15 21 18 19 20 14 16 24 26 25 22 27 28 23 13 8 12 11 9 7 10 6 1 5 4 2 0 3 15 19 16 21 20 17 18 14 28 23 27 26 24 22 25 8 9 7 12 11 10 13 1 2 0 5 4 3 6 18 14 20 17 16 21 15 19 23 24 22 27 26 25 28 11 12 13 10 7 8 9 4 5 6 3 0 1 2 16 20 14 19 18 15 21 17 26 27 28 25 22 23 24 7 10 9 8 13 11 12 0 3 2 1 6 4 5 21 17 18 15 14 19 16 20 22 25 24 23 28 26 27 10 13 11 9 8 12 7 3 6 4 2 1 5 0 19 21 15 16 17 14 20 18 25 28 26 24 23 27 22 12 7 8 13 10 9 11 5 0 1 6 3 2 4 20 18 17 14 15 16 19 21 27 22 23 28 25 24 26 17 19 18 15 20 21 16 24 26 25 22 27 28 23 2 9 11 10 7 12 13 8 0 6 3 4 5 14 1 21 16 20 19 17 15 18 28 23 27 26 24 22 25 0 13 8 12 11 9 7 10 4 1 6 5 2 3 14 16 17 15 20 19 18 21 23 24 22 27 26 25 28 3 8 9 7 12 11 10 13 14 5 2 1 6 0 4 19 20 21 18 15 16 17 26 27 28 25 22 23 24 1 11 12 13 10 7 8 9 5 14 4 3 0 6 2 15 18 17 16 21 19 20 22 25 24 23 28 26 27 6 7 10 9 8 13 11 12 2 3 0 14 4 1 5 18 21 19 17 16 20 15 25 28 26 24 23 27 22 4 10 13 11 9 8 12 7 6 0 1 2 14 5 3 20 15 16 21 18 17 19 27 22 23 28 25 24 26 5 12 7 8 13 10 9 11 3 2 14 0 1 4 6
7.8 Composed Latin Squares (35 x 35)
Order 7 Self Orthogonal Latin Diagonal Squares can be used to construct order 35 Self Orthogonal Composed Latin Diagonal Squares.
7.8.1 Composed Associated Squares
Order 7 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:
Sqrs7 The order 5 Self Orthogonal Associated Latin Square shown above is based on the first elemnets of the Sub Squares and has been used as a guideline for the construction of square A shown below. |
A, Associated
30 32 31 28 33 34 29 34 29 33 32 30 28 31 29 30 28 33 32 31 34 32 33 34 31 28 29 30 28 31 30 29 34 32 33 31 34 32 30 29 33 28 33 28 29 34 31 30 32
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
30 32 31 28 33 34 29 34 29 33 32 30 28 31 29 30 28 33 32 31 34 32 33 34 31 28 29 30 28 31 30 29 34 32 33 31 34 32 30 29 33 28 33 28 29 34 31 30 32
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
30 32 31 28 33 34 29 34 29 33 32 30 28 31 29 30 28 33 32 31 34 32 33 34 31 28 29 30 28 31 30 29 34 32 33 31 34 32 30 29 33 28 33 28 29 34 31 30 32
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
30 32 31 28 33 34 29 34 29 33 32 30 28 31 29 30 28 33 32 31 34 32 33 34 31 28 29 30 28 31 30 29 34 32 33 31 34 32 30 29 33 28 33 28 29 34 31 30 32
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
23 25 24 21 26 27 22 27 22 26 25 23 21 24 22 23 21 26 25 24 27 25 26 27 24 21 22 23 21 24 23 22 27 25 26 24 27 25 23 22 26 21 26 21 22 27 24 23 25
16 18 17 14 19 20 15 20 15 19 18 16 14 17 15 16 14 19 18 17 20 18 19 20 17 14 15 16 14 17 16 15 20 18 19 17 20 18 16 15 19 14 19 14 15 20 17 16 18
30 32 31 28 33 34 29 34 29 33 32 30 28 31 29 30 28 33 32 31 34 32 33 34 31 28 29 30 28 31 30 29 34 32 33 31 34 32 30 29 33 28 33 28 29 34 31 30 32
9 11 10 7 12 13 8 13 8 12 11 9 7 10 8 9 7 12 11 10 13 11 12 13 10 7 8 9 7 10 9 8 13 11 12 10 13 11 9 8 12 7 12 7 8 13 10 9 11
2 4 3 0 5 6 1 6 1 5 4 2 0 3 1 2 0 5 4 3 6 4 5 6 3 0 1 2 0 3 2 1 6 4 5 3 6 4 2 1 5 0 5 0 1 6 3 2 4
Attachment 7.8.11
shows the resulting order 35 Composed Associated Square based on the Self Orthogonal Composed Associated Latin Square shown above.
7.8.2 Composed Pan Magic Squares
Order 7 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:
Sqrs7 The order 5 Self Orthogonal Pan Magic Latin Square shown above is based on the first elemnets of the Sub Squares and has been used as a guideline for the construction of square A shown below. |
A, Pan Magic
18 17 16 15 14 20 19 16 15 14 20 19 18 17 14 20 19 18 17 16 15 19 18 17 16 15 14 20 17 16 15 14 20 19 18 15 14 20 19 18 17 16 20 19 18 17 16 15 14
11 10 9 8 7 13 12 9 8 7 13 12 11 10 7 13 12 11 10 9 8 12 11 10 9 8 7 13 10 9 8 7 13 12 11 8 7 13 12 11 10 9 13 12 11 10 9 8 7
4 3 2 1 0 6 5 2 1 0 6 5 4 3 0 6 5 4 3 2 1 5 4 3 2 1 0 6 3 2 1 0 6 5 4 1 0 6 5 4 3 2 6 5 4 3 2 1 0
32 31 30 29 28 34 33 30 29 28 34 33 32 31 28 34 33 32 31 30 29 33 32 31 30 29 28 34 31 30 29 28 34 33 32 29 28 34 33 32 31 30 34 33 32 31 30 29 28
25 24 23 22 21 27 26 23 22 21 27 26 25 24 21 27 26 25 24 23 22 26 25 24 23 22 21 27 24 23 22 21 27 26 25 22 21 27 26 25 24 23 27 26 25 24 23 22 21
4 3 2 1 0 6 5 2 1 0 6 5 4 3 0 6 5 4 3 2 1 5 4 3 2 1 0 6 3 2 1 0 6 5 4 1 0 6 5 4 3 2 6 5 4 3 2 1 0
32 31 30 29 28 34 33 30 29 28 34 33 32 31 28 34 33 32 31 30 29 33 32 31 30 29 28 34 31 30 29 28 34 33 32 29 28 34 33 32 31 30 34 33 32 31 30 29 28
25 24 23 22 21 27 26 23 22 21 27 26 25 24 21 27 26 25 24 23 22 26 25 24 23 22 21 27 24 23 22 21 27 26 25 22 21 27 26 25 24 23 27 26 25 24 23 22 21
18 17 16 15 14 20 19 16 15 14 20 19 18 17 14 20 19 18 17 16 15 19 18 17 16 15 14 20 17 16 15 14 20 19 18 15 14 20 19 18 17 16 20 19 18 17 16 15 14
11 10 9 8 7 13 12 9 8 7 13 12 11 10 7 13 12 11 10 9 8 12 11 10 9 8 7 13 10 9 8 7 13 12 11 8 7 13 12 11 10 9 13 12 11 10 9 8 7
25 24 23 22 21 27 26 23 22 21 27 26 25 24 21 27 26 25 24 23 22 26 25 24 23 22 21 27 24 23 22 21 27 26 25 22 21 27 26 25 24 23 27 26 25 24 23 22 21
18 17 16 15 14 20 19 16 15 14 20 19 18 17 14 20 19 18 17 16 15 19 18 17 16 15 14 20 17 16 15 14 20 19 18 15 14 20 19 18 17 16 20 19 18 17 16 15 14
11 10 9 8 7 13 12 9 8 7 13 12 11 10 7 13 12 11 10 9 8 12 11 10 9 8 7 13 10 9 8 7 13 12 11 8 7 13 12 11 10 9 13 12 11 10 9 8 7
4 3 2 1 0 6 5 2 1 0 6 5 4 3 0 6 5 4 3 2 1 5 4 3 2 1 0 6 3 2 1 0 6 5 4 1 0 6 5 4 3 2 6 5 4 3 2 1 0
32 31 30 29 28 34 33 30 29 28 34 33 32 31 28 34 33 32 31 30 29 33 32 31 30 29 28 34 31 30 29 28 34 33 32 29 28 34 33 32 31 30 34 33 32 31 30 29 28
11 10 9 8 7 13 12 9 8 7 13 12 11 10 7 13 12 11 10 9 8 12 11 10 9 8 7 13 10 9 8 7 13 12 11 8 7 13 12 11 10 9 13 12 11 10 9 8 7
4 3 2 1 0 6 5 2 1 0 6 5 4 3 0 6 5 4 3 2 1 5 4 3 2 1 0 6 3 2 1 0 6 5 4 1 0 6 5 4 3 2 6 5 4 3 2 1 0
32 31 30 29 28 34 33 30 29 28 34 33 32 31 28 34 33 32 31 30 29 33 32 31 30 29 28 34 31 30 29 28 34 33 32 29 28 34 33 32 31 30 34 33 32 31 30 29 28
25 24 23 22 21 27 26 23 22 21 27 26 25 24 21 27 26 25 24 23 22 26 25 24 23 22 21 27 24 23 22 21 27 26 25 22 21 27 26 25 24 23 27 26 25 24 23 22 21
18 17 16 15 14 20 19 16 15 14 20 19 18 17 14 20 19 18 17 16 15 19 18 17 16 15 14 20 17 16 15 14 20 19 18 15 14 20 19 18 17 16 20 19 18 17 16 15 14
32 31 30 29 28 34 33 30 29 28 34 33 32 31 28 34 33 32 31 30 29 33 32 31 30 29 28 34 31 30 29 28 34 33 32 29 28 34 33 32 31 30 34 33 32 31 30 29 28
25 24 23 22 21 27 26 23 22 21 27 26 25 24 21 27 26 25 24 23 22 26 25 24 23 22 21 27 24 23 22 21 27 26 25 22 21 27 26 25 24 23 27 26 25 24 23 22 21
18 17 16 15 14 20 19 16 15 14 20 19 18 17 14 20 19 18 17 16 15 19 18 17 16 15 14 20 17 16 15 14 20 19 18 15 14 20 19 18 17 16 20 19 18 17 16 15 14
11 10 9 8 7 13 12 9 8 7 13 12 11 10 7 13 12 11 10 9 8 12 11 10 9 8 7 13 10 9 8 7 13 12 11 8 7 13 12 11 10 9 13 12 11 10 9 8 7
4 3 2 1 0 6 5 2 1 0 6 5 4 3 0 6 5 4 3 2 1 5 4 3 2 1 0 6 3 2 1 0 6 5 4 1 0 6 5 4 3 2 6 5 4 3 2 1 0
Attachment 7.8.21
shows the resulting order 35 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.
7.8.3 Composed Ultra Magic Squares
Order 7 Self Orthogonal Ultra Magic Latin Sub Squares can be constructed based on the sub series:
Sqrs7 The order 5 Self Orthogonal Ultra Magic Latin Square shown above is based on the first elemnets of the Sub Squares and has been used as a guideline for the construction of square A shown below. |
A, Ultra Magic
34 33 31 29 28 32 30 31 29 28 32 30 34 33 28 32 30 34 33 31 29 30 34 33 31 29 28 32 33 31 29 28 32 30 34 29 28 32 30 34 33 31 32 30 34 33 31 29 28
20 19 17 15 14 18 16 17 15 14 18 16 20 19 14 18 16 20 19 17 15 16 20 19 17 15 14 18 19 17 15 14 18 16 20 15 14 18 16 20 19 17 18 16 20 19 17 15 14
6 5 3 1 0 4 2 3 1 0 4 2 6 5 0 4 2 6 5 3 1 2 6 5 3 1 0 4 5 3 1 0 4 2 6 1 0 4 2 6 5 3 4 2 6 5 3 1 0
27 26 24 22 21 25 23 24 22 21 25 23 27 26 21 25 23 27 26 24 22 23 27 26 24 22 21 25 26 24 22 21 25 23 27 22 21 25 23 27 26 24 25 23 27 26 24 22 21
13 12 10 8 7 11 9 10 8 7 11 9 13 12 7 11 9 13 12 10 8 9 13 12 10 8 7 11 12 10 8 7 11 9 13 8 7 11 9 13 12 10 11 9 13 12 10 8 7
6 5 3 1 0 4 2 3 1 0 4 2 6 5 0 4 2 6 5 3 1 2 6 5 3 1 0 4 5 3 1 0 4 2 6 1 0 4 2 6 5 3 4 2 6 5 3 1 0
27 26 24 22 21 25 23 24 22 21 25 23 27 26 21 25 23 27 26 24 22 23 27 26 24 22 21 25 26 24 22 21 25 23 27 22 21 25 23 27 26 24 25 23 27 26 24 22 21
13 12 10 8 7 11 9 10 8 7 11 9 13 12 7 11 9 13 12 10 8 9 13 12 10 8 7 11 12 10 8 7 11 9 13 8 7 11 9 13 12 10 11 9 13 12 10 8 7
34 33 31 29 28 32 30 31 29 28 32 30 34 33 28 32 30 34 33 31 29 30 34 33 31 29 28 32 33 31 29 28 32 30 34 29 28 32 30 34 33 31 32 30 34 33 31 29 28
20 19 17 15 14 18 16 17 15 14 18 16 20 19 14 18 16 20 19 17 15 16 20 19 17 15 14 18 19 17 15 14 18 16 20 15 14 18 16 20 19 17 18 16 20 19 17 15 14
13 12 10 8 7 11 9 10 8 7 11 9 13 12 7 11 9 13 12 10 8 9 13 12 10 8 7 11 12 10 8 7 11 9 13 8 7 11 9 13 12 10 11 9 13 12 10 8 7
34 33 31 29 28 32 30 31 29 28 32 30 34 33 28 32 30 34 33 31 29 30 34 33 31 29 28 32 33 31 29 28 32 30 34 29 28 32 30 34 33 31 32 30 34 33 31 29 28
20 19 17 15 14 18 16 17 15 14 18 16 20 19 14 18 16 20 19 17 15 16 20 19 17 15 14 18 19 17 15 14 18 16 20 15 14 18 16 20 19 17 18 16 20 19 17 15 14
6 5 3 1 0 4 2 3 1 0 4 2 6 5 0 4 2 6 5 3 1 2 6 5 3 1 0 4 5 3 1 0 4 2 6 1 0 4 2 6 5 3 4 2 6 5 3 1 0
27 26 24 22 21 25 23 24 22 21 25 23 27 26 21 25 23 27 26 24 22 23 27 26 24 22 21 25 26 24 22 21 25 23 27 22 21 25 23 27 26 24 25 23 27 26 24 22 21
20 19 17 15 14 18 16 17 15 14 18 16 20 19 14 18 16 20 19 17 15 16 20 19 17 15 14 18 19 17 15 14 18 16 20 15 14 18 16 20 19 17 18 16 20 19 17 15 14
6 5 3 1 0 4 2 3 1 0 4 2 6 5 0 4 2 6 5 3 1 2 6 5 3 1 0 4 5 3 1 0 4 2 6 1 0 4 2 6 5 3 4 2 6 5 3 1 0
27 26 24 22 21 25 23 24 22 21 25 23 27 26 21 25 23 27 26 24 22 23 27 26 24 22 21 25 26 24 22 21 25 23 27 22 21 25 23 27 26 24 25 23 27 26 24 22 21
13 12 10 8 7 11 9 10 8 7 11 9 13 12 7 11 9 13 12 10 8 9 13 12 10 8 7 11 12 10 8 7 11 9 13 8 7 11 9 13 12 10 11 9 13 12 10 8 7
34 33 31 29 28 32 30 31 29 28 32 30 34 33 28 32 30 34 33 31 29 30 34 33 31 29 28 32 33 31 29 28 32 30 34 29 28 32 30 34 33 31 32 30 34 33 31 29 28
27 26 24 22 21 25 23 24 22 21 25 23 27 26 21 25 23 27 26 24 22 23 27 26 24 22 21 25 26 24 22 21 25 23 27 22 21 25 23 27 26 24 25 23 27 26 24 22 21
13 12 10 8 7 11 9 10 8 7 11 9 13 12 7 11 9 13 12 10 8 9 13 12 10 8 7 11 12 10 8 7 11 9 13 8 7 11 9 13 12 10 11 9 13 12 10 8 7
34 33 31 29 28 32 30 31 29 28 32 30 34 33 28 32 30 34 33 31 29 30 34 33 31 29 28 32 33 31 29 28 32 30 34 29 28 32 30 34 33 31 32 30 34 33 31 29 28
20 19 17 15 14 18 16 17 15 14 18 16 20 19 14 18 16 20 19 17 15 16 20 19 17 15 14 18 19 17 15 14 18 16 20 15 14 18 16 20 19 17 18 16 20 19 17 15 14
6 5 3 1 0 4 2 3 1 0 4 2 6 5 0 4 2 6 5 3 1 2 6 5 3 1 0 4 5 3 1 0 4 2 6 1 0 4 2 6 5 3 4 2 6 5 3 1 0
Attachment 7.8.31
shows the resulting order 35 Composed Ultra Magic Square based on the Self Orthogonal Composed Ultra Magic Latin Square shown above.
7.9 Composed Latin Squares (36 x 36)
Order 7 Self orthogonal Latin Diagonal Squares can be used to construct order 36 Self Orthogonal Composed Latin Diagonal Squares.
The required order 7 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:
Sqrs7 The order 5 Self orthogonal Latin Diagonal Square shown above is based on the first elemnets of the original Sub Squares, and has been used as a guideline for the construction of square A shown below. |
A
29 31 34 35 21 33 32 20 19 17 15 14 18 16 6 5 3 1 0 4 2 30 28 27 25 23 22 26 24 13 12 10 8 7 11 9 32 30 21 33 34 35 29 17 15 14 18 16 20 19 3 1 0 4 2 6 5 31 25 23 22 26 24 28 27 10 8 7 11 9 13 12 30 34 31 21 35 32 33 14 18 16 20 19 17 15 0 4 2 6 5 3 1 29 22 26 24 28 27 25 23 7 11 9 13 12 10 8 33 29 35 32 31 21 30 16 20 19 17 15 14 18 2 6 5 3 1 0 4 34 24 28 27 25 23 22 26 9 13 12 10 8 7 11 31 35 29 34 33 30 21 19 17 15 14 18 16 20 5 3 1 0 4 2 6 32 27 25 23 22 26 24 28 12 10 8 7 11 9 13 21 32 33 30 29 34 31 15 14 18 16 20 19 17 1 0 4 2 6 5 3 35 23 22 26 24 28 27 25 8 7 11 9 13 12 10 34 21 30 31 32 29 35 18 16 20 19 17 15 14 4 2 6 5 3 1 0 33 26 24 28 27 25 23 22 11 9 13 12 10 8 7 6 5 3 1 0 4 2 22 24 27 28 21 26 25 13 12 10 8 7 11 9 23 35 34 32 30 29 33 31 20 19 17 15 14 18 16 3 1 0 4 2 6 5 25 23 21 26 27 28 22 10 8 7 11 9 13 12 24 32 30 29 33 31 35 34 17 15 14 18 16 20 19 0 4 2 6 5 3 1 23 27 24 21 28 25 26 7 11 9 13 12 10 8 22 29 33 31 35 34 32 30 14 18 16 20 19 17 15 2 6 5 3 1 0 4 26 22 28 25 24 21 23 9 13 12 10 8 7 11 27 31 35 34 32 30 29 33 16 20 19 17 15 14 18 5 3 1 0 4 2 6 24 28 22 27 26 23 21 12 10 8 7 11 9 13 25 34 32 30 29 33 31 35 19 17 15 14 18 16 20 1 0 4 2 6 5 3 21 25 26 23 22 27 24 8 7 11 9 13 12 10 28 30 29 33 31 35 34 32 15 14 18 16 20 19 17 4 2 6 5 3 1 0 27 21 23 24 25 22 28 11 9 13 12 10 8 7 26 33 31 35 34 32 30 29 18 16 20 19 17 15 14 13 12 10 8 7 11 9 35 34 32 30 29 33 31 14 16 19 20 21 18 17 15 6 5 3 1 0 4 2 28 27 25 23 22 26 24 10 8 7 11 9 13 12 32 30 29 33 31 35 34 17 15 21 18 19 20 14 16 3 1 0 4 2 6 5 25 23 22 26 24 28 27 7 11 9 13 12 10 8 29 33 31 35 34 32 30 15 19 16 21 20 17 18 14 0 4 2 6 5 3 1 22 26 24 28 27 25 23 9 13 12 10 8 7 11 31 35 34 32 30 29 33 18 14 20 17 16 21 15 19 2 6 5 3 1 0 4 24 28 27 25 23 22 26 12 10 8 7 11 9 13 34 32 30 29 33 31 35 16 20 14 19 18 15 21 17 5 3 1 0 4 2 6 27 25 23 22 26 24 28 8 7 11 9 13 12 10 30 29 33 31 35 34 32 21 17 18 15 14 19 16 20 1 0 4 2 6 5 3 23 22 26 24 28 27 25 11 9 13 12 10 8 7 33 31 35 34 32 30 29 19 21 15 16 17 14 20 18 4 2 6 5 3 1 0 26 24 28 27 25 23 22 35 33 32 29 30 31 34 28 26 25 22 23 24 27 20 18 17 14 15 16 19 21 8 11 12 13 10 9 7 1 4 5 6 3 2 0 20 19 17 15 14 18 16 6 5 3 1 0 4 2 28 27 25 23 22 26 24 9 7 13 10 11 12 21 8 35 34 32 30 29 33 31 17 15 14 18 16 20 19 3 1 0 4 2 6 5 25 23 22 26 24 28 27 7 11 8 13 12 9 10 21 32 30 29 33 31 35 34 14 18 16 20 19 17 15 0 4 2 6 5 3 1 22 26 24 28 27 25 23 10 21 12 9 8 13 7 11 29 33 31 35 34 32 30 16 20 19 17 15 14 18 2 6 5 3 1 0 4 24 28 27 25 23 22 26 8 12 21 11 10 7 13 9 31 35 34 32 30 29 33 19 17 15 14 18 16 20 5 3 1 0 4 2 6 27 25 23 22 26 24 28 13 9 10 7 21 11 8 12 34 32 30 29 33 31 35 15 14 18 16 20 19 17 1 0 4 2 6 5 3 23 22 26 24 28 27 25 11 13 7 8 9 21 12 10 30 29 33 31 35 34 32 18 16 20 19 17 15 14 4 2 6 5 3 1 0 26 24 28 27 25 23 22 12 10 9 21 7 8 11 13 33 31 35 34 32 30 29 28 27 25 23 22 26 24 13 12 10 8 7 11 9 35 34 32 30 29 33 31 2 20 19 17 15 14 18 16 0 6 3 4 5 21 1 25 23 22 26 24 28 27 10 8 7 11 9 13 12 32 30 29 33 31 35 34 0 17 15 14 18 16 20 19 4 1 6 5 2 3 21 22 26 24 28 27 25 23 7 11 9 13 12 10 8 29 33 31 35 34 32 30 3 14 18 16 20 19 17 15 21 5 2 1 6 0 4 24 28 27 25 23 22 26 9 13 12 10 8 7 11 31 35 34 32 30 29 33 1 16 20 19 17 15 14 18 5 21 4 3 0 6 2 27 25 23 22 26 24 28 12 10 8 7 11 9 13 34 32 30 29 33 31 35 6 19 17 15 14 18 16 20 2 3 0 21 4 1 5 23 22 26 24 28 27 25 8 7 11 9 13 12 10 30 29 33 31 35 34 32 4 15 14 18 16 20 19 17 6 0 1 2 21 5 3 26 24 28 27 25 23 22 11 9 13 12 10 8 7 33 31 35 34 32 30 29 5 18 16 20 19 17 15 14 3 2 21 0 1 4 6
7.10.1 Semi Latin Squares (7 x 7)
The construction of 0rder 7 Mutual Orthogonal Semi-Latin (Diagonal) Squares has been deducted and discussed in:
Order 7 Mutual Orthogonal Semi-Latin (Diagonal) Squares with Diamond Inlays
have been deducted and discussed in Section 7.2.7.
The obtained results regarding the order 7 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 order 8 Self Orthogonal Latin Squares,
which will be described in following sections.
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