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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.

If the main diagonal contains the integers {ai, i = 1 ... 7} in natural order, the Self Orthogonal Latin Square is called Idempotent.

7.5.1 Simple Magic Squares

A construction example of a Simple Magic Square M = A + 7 * T(A) + [1] is shown below:

A
0 5 6 4 2 3 1
2 1 3 5 6 4 0
4 6 2 1 0 5 3
1 4 0 3 5 2 6
3 0 5 6 4 1 2
5 2 1 0 3 6 4
6 3 4 2 1 0 5
B = T(A)
0 2 4 1 3 5 6
5 1 6 4 0 2 3
6 3 2 0 5 1 4
4 5 1 3 6 0 2
2 6 0 5 4 3 1
3 4 5 2 1 6 0
1 0 3 6 2 4 5
M = A + 7 * B + 1
1 20 35 12 24 39 44
38 9 46 34 7 19 22
47 28 17 2 36 13 32
30 40 8 25 48 3 21
18 43 6 42 33 23 10
27 31 37 15 11 49 5
14 4 26 45 16 29 41

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}.

A Base of 64 Idempotent Squares has been found in Section 7.5.3 below. The total number of Self Orthogonal Latin Diagonal Squares will be 64 * 5040 = 322560, which can be generated quite fast with routine SelfOrth7c.

In addition to the transformations and permutations described above, each Self Orthogonal Latin Diagonal Square A corresponds with 24 transformations, as described below.

  • Any line n can be interchanged with line (8 - n). The possible number of transformations is 23 = 8
    It should be noted that for each square the 180o rotated aspect is included in this collection.

  • Any permutation can be applied to the lines 1, 2, 3 provided that the same permutation is applied to the lines 7, 6, 5. The possible number of transformations is 3! = 6.

The resulting number of transformations, excluding the 180o rotated aspects, is 8/2 * 6 = 24, which are shown in Attachment 7.5.12

7.5.2 Pan Magic Squares

A construction example of a Pan Magic Square M = A + 7 * T(A) + [1] is shown below:

A
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
B = T(A)
4 2 0 5 3 1 6
3 1 6 4 2 0 5
2 0 5 3 1 6 4
1 6 4 2 0 5 3
0 5 3 1 6 4 2
6 4 2 0 5 3 1
5 3 1 6 4 2 0
M = A + 7 * B + 1
33 18 3 37 22 14 48
24 9 43 35 20 5 39
15 7 41 26 11 45 30
13 47 32 17 2 36 28
4 38 23 8 49 34 19
44 29 21 6 40 25 10
42 27 12 46 31 16 1

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 an Associated Magic Square M = A + 7 * T(A) + [1] is shown below:

A
0 6 5 4 3 2 1
2 1 0 6 5 4 3
4 3 2 1 0 6 5
6 5 4 3 2 1 0
1 0 6 5 4 3 2
3 2 1 0 6 5 4
5 4 3 2 1 0 6
B = T(A)
0 2 4 6 1 3 5
6 1 3 5 0 2 4
5 0 2 4 6 1 3
4 6 1 3 5 0 2
3 5 0 2 4 6 1
2 4 6 1 3 5 0
1 3 5 0 2 4 6
M = A + 7 * B + 1
1 21 34 47 11 24 37
45 9 22 42 6 19 32
40 4 17 30 43 14 27
35 48 12 25 38 2 15
23 36 7 20 33 46 10
18 31 44 8 28 41 5
13 26 39 3 16 29 49

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.

7.5.4 Ultra Magic Squares

A construction example of an Ultra Magic Square M = A + 7 * T(A) + [1] is shown below:

A
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
B = T(A)
6 3 0 2 5 1 4
5 1 4 6 3 0 2
3 0 2 5 1 4 6
1 4 6 3 0 2 5
0 2 5 1 4 6 3
4 6 3 0 2 5 1
2 5 1 4 6 3 0
M = A + 7 * B + 1
49 27 4 16 36 12 31
39 9 29 47 24 7 20
22 5 17 42 13 32 44
10 35 48 25 2 15 40
6 18 37 8 33 45 28
30 43 26 3 21 41 11
19 38 14 34 46 23 1

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.

A construction example of an Inlaid Magic Square M = A + 7 * T(A) + [1] is shown below:

A
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
B = T(A)
2 6 1 4 0 3 5
4 1 2 5 3 6 0
3 5 0 6 2 4 1
0 4 5 3 1 2 6
5 2 4 0 6 1 3
6 0 3 1 4 5 2
1 3 6 2 5 0 4
M = A + 7 * B + 1
17 47 11 29 6 28 37
35 9 20 40 24 43 4
23 38 1 48 19 32 14
5 34 42 25 8 16 45
36 18 31 2 49 12 27
46 7 26 10 30 41 15
13 22 44 21 39 3 33

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:

    {0, 1 ... 6}, {7, 8 ... 13}, {14, 15 ... 20} and {21, 22 ... 27}

with respectively the magic constants s7 = 21, 70, 119 and 168

Sqrs7
23 16 2 9
2 9 23 16
9 2 16 23
16 23 9 2

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
23 16 2 9
2 9 23 16
9 2 16 23
16 23 9 2

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:

    {0, 1 ... 6}, {7, 8 ... 13}, {14, 15 ... 20} and {21, 22 ... 27}

with respectively the magic constants s7 = 21, 70, 119 and 168

Sqrs7
11 4 25 18
18 25 4 11
4 11 18 25
25 18 11 4

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:

    {0, 1 ... 6}, {7, 8 ... 13}, {14}, {15, 16 ... 21} and {22, 23 ... 28}

with respectively the magic constants s7 = 21, 70, 126 and 175

Sqrs7
24 17 2 9
2 9 24 17
9 2 17 24
17 24 9 2

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
  • Attachment 7.7.1 illustrates and describes the construction of the order 29 Self Orthogonal Composed Latin Diagonal Square shown above.

  • Attachment 7.7.2 illustrates the construction of an order 29 Composed Simple Magic Square based on an order 29 Self Orthogonal Composed Latin Diagonal Square.

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:

    {0, 1 ... 6}, {7, 8 ... 13}, {14, 15 ... 20}, {21, 22 ... 27} and {28, 29 ... 34}

with respectively the magic constants s7 = 21, 70, 119, 168 and 217

Sqrs7
30 23 2 16 9
16 9 23 2 30
9 2 16 30 23
2 30 9 23 16
23 16 30 9 2

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:

    {0, 1 ... 6}, {7, 8 ... 13}, {14, 15 ... 20}, {21, 22 ... 27} and {28, 29 ... 34}

with respectively the magic constants s7 = 21, 70, 119, 168 and 217

Sqrs7
18 11 4 32 25
4 32 25 18 11
25 18 11 4 32
11 4 32 25 18
32 25 18 11 4

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:

    {0, 1 ... 6}, {7, 8 ... 13}, {14, 15 ... 20}, {21, 22 ... 27} and {28, 29 ... 34}

with respectively the magic constants s7 = 21, 70, 119, 168 and 217

Sqrs7
34 20 6 27 13
6 27 13 34 20
13 34 20 6 27
20 6 27 13 34
27 13 34 20 6

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:

    {0, 1 ... 6}, {7, 8 ... 13}, {14, 15 ... 20}, {21}, {22, 23 ... 28} and {29, 30 ... 35}

with respectively the magic constants s7 = 21, 70, 119, 175 and 224

Sqrs7
35 20 6 28 13
6 28 13 35 20
13 35 20 6 28
20 6 28 13 35
28 13 35 20 6

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
  • Attachment 7.9.1 illustrates and describes the construction of the order 36 Self Orthogonal Composed Latin Diagonal Square shown above.

  • Attachment 7.9.2 shows the resulting order 36 Composed Simple Magic Square based on subject order 36 Self Orthogonal Composed Latin Diagonal Square.

7.10   Composed Latin Squares (49 x 49)

Order 7 Self Orthogonal Latin Diagonal Squares can be used to construct order 49 Self Orthogonal Composed Latin Diagonal Squares.

7.10.1 Composed Associated Squares

Order 7 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:

    {0 ... 6}, {7 ... 13}, {14 ... 20}, {21 ... 27}, {28 ... 34}, (35 ... 41) and (42 ... 48)

with respectively the magic constants s7 = 21, 70, 119, 168, 217, 266 and 315

Sqrs7
16 30 23 2 37 44 9
44 9 37 30 16 2 23
9 16 2 37 30 23 44
30 37 44 23 2 9 16
2 23 16 9 44 30 37
23 44 30 16 9 37 2
37 2 9 44 23 16 30

The order 7 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 as shown in Attachment 7.10.11

Attachment 7.10.12 shows the resulting order 49 Composed Associated Magic Square based on subject order 49 Composed Associated Self Orthogonal Latin Diagonal Square.

7.10.2 Composed Pan Magic Squares

Order 7 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:

    {0 ... 6}, {7 ... 13}, {14 ... 20}, {21 ... 27}, {28 ... 34}, (35 ... 41) and (42 ... 48)

with respectively the magic constants s7 = 21, 70, 119, 168, 217, 266 and 315

Sqrs7
32 25 18 11 4 46 39
18 11 4 46 39 32 25
4 46 39 32 25 18 11
39 32 25 18 11 4 46
25 18 11 4 46 39 32
11 4 46 39 32 25 18
46 39 32 25 18 11 4

The order 7 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 as shown in Attachment 7.10.21

Attachment 7.10.22 shows the resulting order 49 Composed Pan Magic Magic Square based on subject order 49 Composed Pan Magic Self Orthogonal Latin Diagonal Square.

7.10.3 Composed Ultra Magic Squares

Order 7 Self Orthogonal Ultra Magic Latin Sub Squares can be constructed based on the sub series:

    {0 ... 6}, {7 ... 13}, {14 ... 20}, {21 ... 27}, {28 ... 34}, (35 ... 41) and (42 ... 48)

with respectively the magic constants s7 = 21, 70, 119, 168, 217, 266 and 315

Sqrs7
48 41 27 13 6 34 20
27 13 6 34 20 48 41
6 34 20 48 41 27 13
20 48 41 27 13 6 34
41 27 13 6 34 20 48
13 6 34 20 48 41 27
34 20 48 41 27 13 6

The order 7 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 as shown in Attachment 7.10.31

Attachment 7.10.32 shows the resulting order 49 Composed Ultra Magic Magic Square based on subject order 49 Composed Ultra Magic Self Orthogonal Latin Diagonal Square.

7.11   Miscellaneous

7.11.1 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.

7.11.2 Summary

The obtained results regarding the order 7 Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment

Subject

n9

Subroutine

-

-

-

-

-

Self Orthogonal, Simple

322560 

SelfOrth7c

-

Self Orthogonal, Pan Magic

20160 

SelfOrth7b

-

Self Orthogonal, Associated

3072 

SelfOrth7a

Attachment 7.5.4

Self Orthogonal, Ultra Magic

192 

Attachment 7.5.3

Self Orthogonal, Idempotent

64 

Attachment 7.5.5

Self Orthogonal, Diamond Inlay

32 

SelfOrth7c

-

-

-

-

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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