Latin Squares (5a)

Office Applications and Entertainment, Latin Squares
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5.5 Self Orthogonal Latin Squares (5 x 5)

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 {a_i, i = 1 ... 5} in natural order, the Self Orthogonal Latin Square is called Idempotent.

5.5.1 Simple Magic Squares

Self Orthogonal Latin Diagonal Squares can be generated with routine SelfOrth5. It appeared that 480 of tht 960 order 5 Latin Diagonal Squares found in Section 5.1 are Self Orthogonal.

A construction example of a Simple Magic Square M = A + 5 * T(A) + [1] is shown below for the first occuring order 5 Self Orthogonal Latin Diagonal Squares A:

A

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

B = T(A)

2 0 3 1 4
1 4 2 0 3
0 3 1 4 2
4 2 0 3 1
3 1 4 2 0

M = A + 5 * B + 1

13 2 16 10 24
6 25 14 3 17
4 18 7 21 15
22 11 5 19 8
20 9 23 12 1

Each Self Orthogonal Latin Square has eight orientations which can be reached by means of rotation and/or reflection.

Attachment 5.5.11 shows the eight orientations (aspects) for the first occuring order 5 Self Orthogonal Latin Diagonal Square.
Attachment 5.5.12 shows a possible Base of 60 Unique Self Orthogonal Latin Squares for the whole collection of 480 Self Orthogonal Latin Squares.

Each Self Orthogonal Latin Diagonal Square corresponds with 5! = 120 Self Orthogonal Latin Diagonal Squares, which can be obtained by permutation of the integers {a_i, i = 1 ... 5}.

The Self Orthogonal Latin Squares {A1, A2, A3, A4} shown below, can be considered as a Base for the whole collection of 480 Self Orthogonal Latin Squares.

A1

0 4 3 2 1
2 1 0 4 3
4 3 2 1 0
1 0 4 3 2
3 2 1 0 4

A2 = T(A1)

0 2 4 1 3
4 1 3 0 2
3 0 2 4 1
2 4 1 3 0
1 3 0 2 4

A3

0 3 4 2 1
2 1 3 4 0
1 4 2 0 3
4 0 1 3 2
3 2 0 1 4

A4 = T(A3)

0 2 1 4 3
3 1 4 0 2
4 3 2 1 0
2 4 0 3 1
1 0 3 2 4

It can be noticed that A2 is the transposed of A1 and A4 is the transposed of A3.

The collection {A2} contains the transposed of all elements of collection {A1}.
The collection {A4} contains the transposed of all elements of collection {A3}.

Attachment 5.5.13 page 1, 2, 3 and 4 show these sub collections, based on respectively the Self Orthogonal Latin Squares A1, A2, A3 and A4 shown above.

In addition to the transformations and permutations described above, each Self Orthogonal Latin Diagonal Square A corresponds with 4 transformations, as described below.

Any line n can be interchanged with line (6 - n). The possible number of transformations is 2² = 4
It should be noted that for each square the 180^o rotated aspect is included in this collection.
Any permutation can be applied to the lines 1, 2 provided that the same permutation is applied to the lines 5, 4. The possible number of transformations is 2! = 2.

The resulting number of transformations, excluding the 180^o rotated aspects, is 4/2 * 2 = 4, which are shown below:

Base

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

Tr1

0 3 2 1 4
1 4 3 2 0
4 2 1 0 3
2 0 4 3 1
3 1 0 4 2

Sw12(Base)

4 0 3 1 2
1 2 0 3 4
2 3 1 4 0
3 4 2 0 1
0 1 4 2 3

Sw12(Tr1)

4 1 3 0 2
3 0 2 4 1
2 4 1 3 0
1 3 0 2 4
0 2 4 1 3

Each transformation shown has again eight orientations which can be reached by means of rotation and/or reflection.

However - due to the nature of Latin Squares - the last two transformations will return the same sixteen squares as the first two.

5.5.2 Pan Magic Squares

Self Orthogonal Pan Magic Latin Diagonal Squares can be generated with routine SelfOrth5.

A construction example of a Pan Magic Square M = A + 5 * T(A) + [1] is shown below:

A

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

B = T(A)

2 0 3 1 4
1 4 2 0 3
0 3 1 4 2
4 2 0 3 1
3 1 4 2 0

M = A + 5 * B + 1

13 2 16 10 24
6 25 14 3 17
4 18 7 21 15
22 11 5 19 8
20 9 23 12 1

It appeared that all 240 order 5 Pan Magic Latin Diagonal Squares found in Section 5.2.2 are Self Orthogonal (ref. Attachment 5.5.2).

5.5.3 Associated Magic Squares

Self Orthogonal Associated Latin Diagonal Squares can be generated with routine SelfOrth5.

A construction example of an Associated Magic Square M = A + 5 * T(A) + [1] is shown below:

A

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

B = T(A)

4 2 1 0 3
3 1 0 4 2
0 3 2 1 4
2 0 4 3 1
1 4 3 2 0

M = A + 5 * B + 1

25 14 6 3 17
18 7 4 21 15
2 16 13 10 24
11 5 22 19 8
9 23 20 12 1

It appeared that 32 of the 64 order 5 Associated Latin Diagonal Squares found in Section 5.2.4 are Self Orthogonal (ref. Attachment 5.5.3).

5.5.4 Ultra Magic Squares

Self Orthogonal Ultra Latin Diagonal Squares can be generated with routine SelfOrth5.

A construction example of an Ultra Magic Square M = A + 5 * T(A) + [1] is shown below:

A

4 2 0 3 1
0 3 1 4 2
1 4 2 0 3
2 0 3 1 4
3 1 4 2 0

B = T(A)

4 0 1 2 3
2 3 4 0 1
0 1 2 3 4
3 4 0 1 2
1 2 3 4 0

M = A + 5 * B + 1

25 3 6 14 17
11 19 22 5 8
2 10 13 16 24
18 21 4 7 15
9 12 20 23 1

It appeared that all 16 order 5 Ultra Latin Diagonal Squares found in Section 5.2.3 are Self Orthogonal (ref. Attachment 5.5.4).

5.6 Composed Latin Squares (20 x 20)

Order 5 Self Orthogonal Latin Diagonal Squares can be used to construct order 20 Self Orthogonal Composed Latin Diagonal Squares.

5.6.1 Composed Associated Squares

Order 5 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:

{0, 1, 2, 3, 4}, {5, 6, 7, 8, 9}, {10, 11, 12, 13, 14} and {15, 16, 17, 18, 19}

with respectively the magic constants s5 = 10, 35, 60 and 85

Sqrs5

19 14 4 9
4 9 19 14
9 4 14 19
14 19 9 4

A

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

The order 4 Self Orthogonal Associated Latin Square left is based on the first elemnets of the Sub Squares and has been used as a guideline for the construction shown above.

Attachment 5.6.11 illustrates the construction of an order 20 Composed Associated Square based on the Self Orthogonal Composed Associated Latin Square shown above.

5.6.2 Composed Pan Magic Squares (1)

Order 20 Self Orthogonal Composed Pan Magic and Complete Latin Diagonal Squares can be constructed based on Order 20 Self Orthogonal Composed Associated Latin Diagonal Squares as illustrated below (Euler):

Sqrs5

19 14 4 9
4 9 19 14
9 4 14 19
14 19 9 4

A

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

6 7 5 8 9
9 5 8 6 7
8 9 7 5 6
7 8 6 9 5
5 6 9 7 8

1 2 0 3 4
4 0 3 1 2
3 4 2 0 1
2 3 1 4 0
0 1 4 2 3

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

11 12 10 13 14
14 10 13 11 12
13 14 12 10 11
12 13 11 14 10
10 11 14 12 13

16 17 15 18 19
19 15 18 16 17
18 19 17 15 16
17 18 16 19 15
15 16 19 17 18

13 12 14 11 10
10 14 11 13 12
11 10 12 14 13
12 11 13 10 14
14 13 10 12 11

18 17 19 16 15
15 19 16 18 17
16 15 17 19 18
17 16 18 15 19
19 18 15 17 16

0 1 4 2 3
2 3 1 4 0
3 4 2 0 1
4 0 3 1 2
1 2 0 3 4

5 6 9 7 8
7 8 6 9 5
8 9 7 5 6
9 5 8 6 7
6 7 5 8 9

8 7 9 6 5
5 9 6 8 7
6 5 7 9 8
7 6 8 5 9
9 8 5 7 6

3 2 4 1 0
0 4 1 3 2
1 0 2 4 3
2 1 3 0 4
4 3 0 2 1

15 16 19 17 18
17 18 16 19 15
18 19 17 15 16
19 15 18 16 17
16 17 15 18 19

10 11 14 12 13
12 13 11 14 10
13 14 12 10 11
14 10 13 11 12
11 12 10 13 14

The order 4 Self Orthogonal Associated Latin Square left is based on the first elemnets of the Sub Squares (before transformation) and has been used as a guideline for the construction shown above.

Attachment 5.6.21 illustrates the construction of an order 20 Composed Pan Magic and Complete Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.

5.6.3 Composed Pan Magic Squares (2)

Order 5 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:

{0, 1, 2, 3, 4}, {5, 6, 7, 8, 9}, {10, 11, 12, 13, 14} and {15, 16, 17, 18, 19}

with respectively the magic constants s5 = 10, 35, 60 and 85

Sqrs5

7 2 17 12
12 17 2 7
2 7 12 17
17 12 7 2

A

7 6 5 9 8
5 9 8 7 6
8 7 6 5 9
6 5 9 8 7
9 8 7 6 5

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15

12 11 10 14 13
10 14 13 12 11
13 12 11 10 14
11 10 14 13 12
14 13 12 11 10

12 11 10 14 13
10 14 13 12 11
13 12 11 10 14
11 10 14 13 12
14 13 12 11 10

17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

7 6 5 9 8
5 9 8 7 6
8 7 6 5 9
6 5 9 8 7
9 8 7 6 5

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

7 6 5 9 8
5 9 8 7 6
8 7 6 5 9
6 5 9 8 7
9 8 7 6 5

12 11 10 14 13
10 14 13 12 11
13 12 11 10 14
11 10 14 13 12
14 13 12 11 10

17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15

17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15

12 11 10 14 13
10 14 13 12 11
13 12 11 10 14
11 10 14 13 12
14 13 12 11 10

7 6 5 9 8
5 9 8 7 6
8 7 6 5 9
6 5 9 8 7
9 8 7 6 5

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

The order 4 Self Orthogonal Pan Magic Latin Square left is based on first elemnets of the Sub Squares and has been used as a guideline for the construction shown above.

Attachment 5.6.31 illustrates the construction of an order 20 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.

5.7 Composed Latin Squares (24 x 24)

A combination of order 4 and 5 (Inlaid) Self Orthogonal Latin Diagonal Squares can be used to construct order 24 Self Orthogonal Composed Latin Diagonal Squares.

Sqrs4

9 5 1 21 17
1 21 17 9 5
17 9 5 1 21
5 1 21 17 9
21 17 9 5 1

A

9 8 12 11 5 4 13 7 10 6 1 0 14 3 2 22 21 20 15 23 17 16 19 18
12 11 10 9 13 7 6 5 8 4 14 3 2 1 0 20 15 23 22 21 18 19 16 17
10 9 8 12 6 5 4 13 11 7 2 1 0 14 3 23 22 21 20 15 16 17 18 19
8 12 11 10 4 13 7 6 9 5 0 14 3 2 1 21 20 15 23 22 19 18 17 16
1 0 13 3 21 20 14 23 6 2 17 16 15 19 22 18 9 8 11 10 5 4 12 7
13 3 2 1 14 23 22 21 4 0 15 19 18 17 20 16 10 11 8 9 12 7 6 5
2 1 0 13 22 21 20 14 7 3 18 17 16 15 23 19 8 9 10 11 6 5 4 12
0 13 3 2 20 14 23 22 5 1 16 15 19 18 21 17 11 10 9 8 4 12 7 6
11 10 9 8 18 17 16 19 12 14 21 20 23 22 15 13 1 0 3 2 7 6 5 4
3 2 1 0 7 6 5 4 15 13 9 8 11 10 12 14 17 16 19 18 21 20 23 22
17 16 14 19 9 8 15 11 0 20 5 4 7 6 18 10 3 2 12 1 23 22 13 21
14 19 18 17 15 11 10 9 2 22 6 7 4 5 16 8 12 1 0 3 13 21 20 23
18 17 16 14 10 9 8 15 1 21 4 5 6 7 19 11 0 3 2 12 20 23 22 13
16 14 19 18 8 15 11 10 3 23 7 6 5 4 17 9 2 12 1 0 22 13 21 20
19 18 17 16 23 22 21 20 13 15 3 2 1 0 14 12 5 4 7 6 9 8 11 10
6 5 4 7 11 10 9 8 14 12 19 18 17 16 13 15 23 22 21 20 1 0 3 2
4 7 15 6 1 0 3 2 20 16 23 22 12 21 8 5 19 18 13 17 11 10 14 9
15 6 5 4 2 3 0 1 22 18 12 21 20 23 10 7 13 17 16 19 14 9 8 11
5 4 7 15 0 1 2 3 21 17 20 23 22 12 9 6 16 19 18 13 8 11 10 14
7 15 6 5 3 2 1 0 23 19 22 12 21 20 11 4 18 13 17 16 10 14 9 8
21 20 23 22 16 19 12 18 17 8 11 10 13 9 4 0 7 6 14 5 3 2 15 1
22 23 20 21 12 18 17 16 19 10 13 9 8 11 6 2 14 5 4 7 15 1 0 3
20 21 22 23 17 16 19 12 18 9 8 11 10 13 5 1 4 7 6 14 0 3 2 15
23 22 21 20 19 12 18 17 16 11 10 13 9 8 7 3 6 14 5 4 2 15 1 0

Aux4

12 14 15 13
15 13 12 14
13 15 14 12
14 12 13 15

The order 5 Self orthogonal Latin Diagonal Square left/top (Sqrs4) is based on the first elemnets of the original order 4 Sub Squares, and has been used as a guideline for the construction.

The order 4 Self orthogonal Latin Diagonal Square left/bottom (Aux4) is based on the sub series {12, 13, 14, 15}.

Attachment 5.7.1 illustrates and describes the construction of the order 24 Self Orthogonal Composed Latin Diagonal Square shown above.
Attachment 5.7.2 illustrates the construction of an order 24 Composed Simple Magic Square based on an order 24 Self Orthogonal Composed Latin Diagonal Square.

5.8 Composed Latin Squares (25 x 25)

Order 5 Self Orthogonal Latin Diagonal Squares can be used to construct order 25 Self Orthogonal Composed Latin Diagonal Squares.

5.8.1 Composed Associated Squares

Order 5 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:

{0, 1, 2, 3, 4}, {5, 6, 7, 8, 9}, {10, 11, 12, 13, 14}, {15, 16, 17, 18, 19} and {20, 21, 22, 23, 24}

with respectively the magic constants s5 = 10, 35, 60, 85 and 110

Sqrs5

24 19 4 14 9
14 9 19 4 24
9 4 14 24 19
4 24 9 19 14
19 14 24 9 4

A

24 23 20 22 21
22 21 23 20 24
21 20 22 24 23
20 24 21 23 22
23 22 24 21 20

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

24 23 20 22 21
22 21 23 20 24
21 20 22 24 23
20 24 21 23 22
23 22 24 21 20

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

24 23 20 22 21
22 21 23 20 24
21 20 22 24 23
20 24 21 23 22
23 22 24 21 20

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

24 23 20 22 21
22 21 23 20 24
21 20 22 24 23
20 24 21 23 22
23 22 24 21 20

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

19 18 15 17 16
17 16 18 15 19
16 15 17 19 18
15 19 16 18 17
18 17 19 16 15

14 13 10 12 11
12 11 13 10 14
11 10 12 14 13
10 14 11 13 12
13 12 14 11 10

24 23 20 22 21
22 21 23 20 24
21 20 22 24 23
20 24 21 23 22
23 22 24 21 20

9 8 5 7 6
7 6 8 5 9
6 5 7 9 8
5 9 6 8 7
8 7 9 6 5

4 3 0 2 1
2 1 3 0 4
1 0 2 4 3
0 4 1 3 2
3 2 4 1 0

The order 5 Self Orthogonal Latin Square left is based on first elemnets of the Sub Squares and has been used as a guideline for the construction shown above.

Attachment 5.8.11 illustrates the construction of an order 25 Composed Associated Magic Square based on the Self Orthogonal Composed Latin Square shown above.

5.8.2 Composed Pan Magic Squares

Order 5 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:

{0, 1, 2, 3, 4}, {5, 6, 7, 8, 9}, {10, 11, 12, 13, 14}, {15, 16, 17, 18, 19} and {20, 21, 22, 23, 24}

with respectively the magic constants s5 = 10, 35, 60, 85 and 110

Sqrs5

12 7 2 22 17
2 22 17 12 7
17 12 7 2 22
7 2 22 17 12
22 17 12 7 2

A

12 11 10 14 13
10 14 13 12 11
13 12 11 10 14
11 10 14 13 12
14 13 12 11 10

7 6 5 9 8
5 9 8 7 6
8 7 6 5 9
6 5 9 8 7
9 8 7 6 5

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

22 21 20 24 23
20 24 23 22 21
23 22 21 20 24
21 20 24 23 22
24 23 22 21 20

17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

22 21 20 24 23
20 24 23 22 21
23 22 21 20 24
21 20 24 23 22
24 23 22 21 20

17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15

12 11 10 14 13
10 14 13 12 11
13 12 11 10 14
11 10 14 13 12
14 13 12 11 10

7 6 5 9 8
5 9 8 7 6
8 7 6 5 9
6 5 9 8 7
9 8 7 6 5

17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15

12 11 10 14 13
10 14 13 12 11
13 12 11 10 14
11 10 14 13 12
14 13 12 11 10

7 6 5 9 8
5 9 8 7 6
8 7 6 5 9
6 5 9 8 7
9 8 7 6 5

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

22 21 20 24 23
20 24 23 22 21
23 22 21 20 24
21 20 24 23 22
24 23 22 21 20

7 6 5 9 8
5 9 8 7 6
8 7 6 5 9
6 5 9 8 7
9 8 7 6 5

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

22 21 20 24 23
20 24 23 22 21
23 22 21 20 24
21 20 24 23 22
24 23 22 21 20

17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15

12 11 10 14 13
10 14 13 12 11
13 12 11 10 14
11 10 14 13 12
14 13 12 11 10

22 21 20 24 23
20 24 23 22 21
23 22 21 20 24
21 20 24 23 22
24 23 22 21 20

17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15

12 11 10 14 13
10 14 13 12 11
13 12 11 10 14
11 10 14 13 12
14 13 12 11 10

7 6 5 9 8
5 9 8 7 6
8 7 6 5 9
6 5 9 8 7
9 8 7 6 5

2 1 0 4 3
0 4 3 2 1
3 2 1 0 4
1 0 4 3 2
4 3 2 1 0

The order 5 Self Orthogonal Latin Square left is based on first elemnets of the Sub Squares and has been used as a guideline for the construction shown above.

Attachment 5.8.21 illustrates the construction of an order 25 Composed Pan Magic Square based on the Self Orthogonal Composed Latin Square shown above.

5.9 Miscellaneous

5.9.1 Composed Semi Latin Squares

The construction of Self Orthogonal Composed Semi-Latin (Diagonal) Squares based on Order 5 Self orthogonal Latin Diagonal Squares has been deducted and discussed in Section 15.2.7.

5.9.2 Summary

The obtained results regarding the order 5 Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment

Subject

Subroutine

-

-

-

Attachment 5.5.1

Self Orth Simple Magic Squares

SelfOrth5

Attachment 5.5.2

Self Orth Pan Magic Squares

Attachment 5.5.3

Self Orth Associated Magic Squares

Attachment 5.5.4

Self Orth Ultra Magic Squares

-

-

-

Comparable methods as described above, can be applied to construct higher order Self Orthogonal Latin Squares, of which a few examples will be described in following sections.

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