Office Applications and Entertainment, Latin Squares

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4.5   Self Orthogonal Latin Squares (4 x 4)

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 ... 4} in natural order, the Self Orthogonal Latin Square is called Idempotent.

4.5.1 Simple Magic Squares

Self Orthogonal Latin Diagonal Squares can be generated with routine SelfOrth4. It appeared that all 48 order 4 Latin Diagonal Squares found in Section 4.1 are Self Orthogonal.

A construction example of a Simple Magic Square M = A + 4 * T(A) + [1] is shown below for an Idempotent Square A:

A
0 3 1 2
2 1 3 0
3 0 2 1
1 2 0 3
B = T(A)
0 2 3 1
3 1 0 2
1 3 2 0
2 0 1 3
M = A + 4 * B + 1
1 12 14 7
15 6 4 9
8 13 11 2
10 3 5 16

Each Self Orthogonal Latin Square has 8 orientations which can be reached by means of rotation and/or reflection.

  • Attachment 4.5.11 shows a possible Base of six Unique Self Orthogonal Latin Squares for the whole collection of 48 Self Orthogonal Latin Squares.

  • Attachment 4.5.12, shows the whole collection of 48 Self Orthogonal Latin Squares based on these six Base Squares.

Each Self Orthogonal Latin Diagonal Square corresponds with 4! = 24 Self Orthogonal Latin Diagonal Squares, which can be obtained by permutation of the integers {ai, i = 1 ... 4}.

  • Attachment 4.5.13 page 1, shows this sub collection, based on the Self Orthogonal Latin Square A1 = A shown above.

  • Attachment 4.5.13 page 2, shows the 24 permutations of another Self Orthogonal Latin Square A2 not belonging to the sub collection of page 1.

Consequently the pair Self Orthogonal Latin Squares {A1, A2} shown below, can be considered as a Base for the whole collection of 48 Self Orthogonal Latin Squares.

A1
0 3 1 2
2 1 3 0
3 0 2 1
1 2 0 3
A2 = T(A1)
0 2 3 1
3 1 0 2
1 3 2 0
2 0 1 3

It can be noticed that A2 is the transposed of A1 and that the collection {A2} contains the transposed of all elements of collection {A1}.

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 (5 - n). The possible number of transformations is 22 = 4
    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 provided that the same permutation is applied to the lines 4, 3. The possible number of transformations is 2! = 2.

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

Base
0 3 1 2
2 1 3 0
3 0 2 1
1 2 0 3
Tr1
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0
Sw12(Base)
1 2 0 3
3 0 2 1
2 1 3 0
0 3 1 2
Sw12(Tr1)
1 0 2 3
2 3 1 0
3 2 0 1
0 1 3 2

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.

4.5.2 Pan Magic Squares

Self Orthogonal Pan Magic Latin Diagonal Squares can be generated with routine SelfOrth4.

A construction example of a Pan Magic Square M = A + 4 * T(A) + [1] is shown below:

A
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
T(A)
1 2 0 3
0 3 1 2
3 0 2 1
2 1 3 0
M
6 9 4 15
3 16 5 10
13 2 11 8
12 7 14 1

It appeared that all 16 order 4 Pan Magic Latin Diagonal Squares found in Section 4.2.2 are Self Orthogonal (ref. Attachment 4.5.2).

4.5.3 Associated Magic Squares

Self Orthogonal Associated Latin Diagonal Squares can be generated with routine SelfOrth4.

A construction example of an Associated Magic Square M = A + 4 * T(A) + [1] is shown below:

A
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0
T(A)
3 0 1 2
2 1 0 3
0 3 2 1
1 2 3 0
M
16 3 5 10
9 6 4 15
2 13 11 8
7 12 14 1

It appeared that all 16 order 4 Associated Latin Diagonal Squares found in Section 4.2.3 are Self Orthogonal (ref. Attachment 4.5.3).

4.6   Composed Latin Squares (16 x 16)

Order 4 Self Orthogonal Latin Diagonal Squares can be used to construct order 16 Self Orthogonal Composed Latin Diagonal Squares.

4.6.1 Composed Pan Magic Squares (1)

Order 4 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:

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

each with magic constants s4 = 30.

Sqrs4
3 1 7 5
5 7 1 3
1 3 5 7
7 5 3 1
A
3 2 13 12
12 13 2 3
2 3 12 13
13 12 3 2
1 0 15 14
14 15 0 1
0 1 14 15
15 14 1 0
7 6 9 8
8 9 6 7
6 7 8 9
9 8 7 6
5 4 11 10
10 11 4 5
4 5 10 11
11 10 5 4
5 4 11 10
10 11 4 5
4 5 10 11
11 10 5 4
7 6 9 8
8 9 6 7
6 7 8 9
9 8 7 6
1 0 15 14
14 15 0 1
0 1 14 15
15 14 1 0
3 2 13 12
12 13 2 3
2 3 12 13
13 12 3 2
1 0 15 14
14 15 0 1
0 1 14 15
15 14 1 0
3 2 13 12
12 13 2 3
2 3 12 13
13 12 3 2
5 4 11 10
10 11 4 5
4 5 10 11
11 10 5 4
7 6 9 8
8 9 6 7
6 7 8 9
9 8 7 6
7 6 9 8
8 9 6 7
6 7 8 9
9 8 7 6
5 4 11 10
10 11 4 5
4 5 10 11
11 10 5 4
3 2 13 12
12 13 2 3
2 3 12 13
13 12 3 2
1 0 15 14
14 15 0 1
0 1 14 15
15 14 1 0

The order 4 Self Orthogonal Pan Magic 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 4.6.31 illustrates the construction of an order 16 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above (Type 1).

4.6.2 Composed Pan Magic Squares (2)

Order 4 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} and {12, 13, 14, 15}

with respectively the magic constants s4 = 6, 22, 38 and 54.

s4
22 6 54 38
38 54 6 22
6 22 38 54
54 38 22 6
A
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0

The order 4 Self orthogonal Pan Magic Latin Square left is based on the four Magic Constants s4 and has been used as a guideline for the construction shown above.

  • Attachment 4.6.11 illustrates the construction of an order 16 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above (Type 2).

  • Attachment 4.6.12 illustrates the construction of an order 16 Composed Pan Magic and Complete Square based on an alternative Self Orthogonal Composed Pan Magic Latin Square.

4.6.3 Composed Associated Squares

Order 4 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} and {12, 13, 14, 15}

with respectively the magic constants s4 = 6, 22, 38 and 54.

s4
54 38 6 22
6 22 54 38
22 6 38 54
38 54 22 6
A
15 14 12 13
12 13 15 14
13 12 14 15
14 15 13 12
11 10 8 9
8 9 11 10
9 8 10 11
10 11 9 8
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0
7 6 4 5
4 5 7 6
5 4 6 7
6 7 5 4
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0
7 6 4 5
4 5 7 6
5 4 6 7
6 7 5 4
15 14 12 13
12 13 15 14
13 12 14 15
14 15 13 12
11 10 8 9
8 9 11 10
9 8 10 11
10 11 9 8
7 6 4 5
4 5 7 6
5 4 6 7
6 7 5 4
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0
11 10 8 9
8 9 11 10
9 8 10 11
10 11 9 8
15 14 12 13
12 13 15 14
13 12 14 15
14 15 13 12
11 10 8 9
8 9 11 10
9 8 10 11
10 11 9 8
15 14 12 13
12 13 15 14
13 12 14 15
14 15 13 12
7 6 4 5
4 5 7 6
5 4 6 7
6 7 5 4
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0

The order 4 Self orthogonal Associated Latin Square left is based on the four Magic Constants s4 and has been used as a guideline for the construction shown above.

  • Attachment 4.6.21 illustrates the construction of an order 16 Composed Associoated Square based on the Self Orthogonal Composed Associated Latin Square shown above.

  • Attachment 4.6.22 illustrates the construction of an order 16 Composed Pan Magic and Complete Square based on a transformation of the Self Orthogonal Composed Square shown above (Euler's Transformation).

  • Attachment 4.6.23 illustrates alternatively the construction of an order 16 Composed Associoated Square based on a Self Orthogonal Composed Associated Latin Square for the sub series:

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

    with respectively the magic constants s4 = 10, 18, 42 and 50.

4.6.4 Magic Series and Sub Series

Suitable Magic Series, composed of four Balanced Sub Series, for the integers {0 ... 15} can be constructed by means of following procedure:

  • Generate Magic Series for the Magic Sums s4(i), i = 1 ... 4 with Σ s4(i) = 120, within the range {0 ... 15} (ref. MgcLns4).

  • Construct Generators with four Magic Rows, based on the Magic Series obtained above.

    The Generators can be transformed to the required Magic Series {0 ... 15} within the same procedure (ref. CnstrGen4).

Based on the 252 order 4 Balanced Sub Series 1990 Series, composed of four Balanced Sub Series, could be found.

The 1990 Composed Series are all suitable for the construction of Simple Composed Self Orthogonal Latin Diagonal Squares, composed of either Pan Magic, Associated or Simple Magic Squares.

The following Sub Collections contain (a few) composed Series suitable for either Pan Magic, Pan Magic Complete or Associated Composed Self Orthogonal Latin Diagonal Squares

  • Attachment 4.6.41 Unbalanced Series composed of Balanced Sub Series, s4 = 30
    Suitable for Pan Magic Composed Self Orthogonal Latin Diagonal Squares (Type 1).

  • Attachment 4.6.44 Misc. Series composed of Balanced Sub Series, s4(i), i = 1 ... 4
    Suitable for Pan Magic Composed Self Orthogonal Latin Diagonal Squares (Type 2).

  • Attachment 4.6.42 Balanced Series composed of Balanced Sub Series, s4(i), i = 1 ... 4
    Suitable for Pan Magic Complete or Associated Composed Self Orthogonal Latin Diagonal Squares.

  • Attachment 4.6.43 Balanced Series composed of Unbalanced Sub Series, s4(i), i = 1 ... 4
    Suitable for Pan Magic Complete or Associated Composed Self Orthogonal Latin Diagonal Squares,
    composed of Simple Magic Sub Squares.

The collections listed above are not complete. Numerous additional Series can be obtained by permutation of and/or within the (Balanced) Sub Series.

4.7   Composed Latin Squares (17 x 17)

Order 4 Self orthogonal Latin Diagonal Squares can be used to construct order 17 Self Orthogonal Composed Latin Diagonal Squares.

The required order 4 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:

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

with respectively the magic constants s4 = 6, 22, 42 and 58.

Sqrs4
5 1 14 10
10 14 1 5
1 5 10 14
14 10 5 1
A
5 4 8 7 1 0 3 2 6 14 13 16 15 10 9 12 11
8 7 6 5 2 3 0 1 4 15 16 13 14 11 12 9 10
6 5 4 8 0 1 2 3 7 13 14 15 16 9 10 11 12
4 8 7 6 3 2 1 0 5 16 15 14 13 12 11 10 9
10 9 12 11 14 13 8 16 15 1 0 3 2 5 4 7 6
11 12 9 10 8 16 15 14 13 2 3 0 1 6 7 4 5
9 10 11 12 15 14 13 8 16 0 1 2 3 4 5 6 7
12 11 10 9 13 8 16 15 14 3 2 1 0 7 6 5 4
7 6 5 4 16 15 14 13 8 11 12 9 10 2 3 0 1
1 0 3 2 5 4 7 6 12 9 10 8 11 14 13 16 15
2 3 0 1 6 7 4 5 10 8 11 12 9 15 16 13 14
0 1 2 3 4 5 6 7 11 12 9 10 8 13 14 15 16
3 2 1 0 7 6 5 4 9 10 8 11 12 16 15 14 13
14 13 16 15 10 9 12 11 3 5 4 7 6 0 1 8 2
15 16 13 14 11 12 9 10 1 6 7 4 5 8 2 3 0
13 14 15 16 9 10 11 12 2 4 5 6 7 3 0 1 8
16 15 14 13 12 11 10 9 0 7 6 5 4 1 8 2 3

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

  • Attachment 4.7.1 illustrates and describes the construction of the order 17 Self Orthogonal Composed Latin Diagonal Square shown above.

  • Attachment 4.7.2 illustrates the construction of an order 17 Composed Simple Magic Square based on an order 17 Self Orthogonal Composed Latin Diagonal Square.

4.8   Composed Latin Squares (20 x 20)

Order 4 Self Orthogonal Latin Diagonal Squares can be used to construct order 20 Self Orthogonal Composed Latin Diagonal Squares.

4.8.1 Composed Pan Magic Squares (1)

Order 4 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:

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

each with magic constants s4 = 38.

Sqrs4
5 3 1 9 7
1 9 7 5 3
7 5 3 1 9
3 1 9 7 5
9 7 5 3 1
A
5 4 15 14
14 15 4 5
4 5 14 15
15 14 5 4
3 2 17 16
16 17 2 3
2 3 16 17
17 16 3 2
1 0 19 18
18 19 0 1
0 1 18 19
19 18 1 0
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
7 6 13 12
12 13 6 7
6 7 12 13
13 12 7 6
1 0 19 18
18 19 0 1
0 1 18 19
19 18 1 0
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
7 6 13 12
12 13 6 7
6 7 12 13
13 12 7 6
5 4 15 14
14 15 4 5
4 5 14 15
15 14 5 4
3 2 17 16
16 17 2 3
2 3 16 17
17 16 3 2
7 6 13 12
12 13 6 7
6 7 12 13
13 12 7 6
5 4 15 14
14 15 4 5
4 5 14 15
15 14 5 4
3 2 17 16
16 17 2 3
2 3 16 17
17 16 3 2
1 0 19 18
18 19 0 1
0 1 18 19
19 18 1 0
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
3 2 17 16
16 17 2 3
2 3 16 17
17 16 3 2
1 0 19 18
18 19 0 1
0 1 18 19
19 18 1 0
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
7 6 13 12
12 13 6 7
6 7 12 13
13 12 7 6
5 4 15 14
14 15 4 5
4 5 14 15
15 14 5 4
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
7 6 13 12
12 13 6 7
6 7 12 13
13 12 7 6
5 4 15 14
14 15 4 5
4 5 14 15
15 14 5 4
3 2 17 16
16 17 2 3
2 3 16 17
17 16 3 2
1 0 19 18
18 19 0 1
0 1 18 19
19 18 1 0

The order 5 Self Orthogonal Pan Magic 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 4.8.11 illustrates the construction of an order 20 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.

4.8.2 Composed Pan Magic Squares (2)

Order 4 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} and {16, 17, 18, 19}

with respectively the magic constants s4 = 6, 22, 38, 54 and 70

s4
38 22 6 70 54
6 70 54 38 22
54 38 22 6 70
22 6 70 54 38
70 54 38 22 6
A
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0

The order 5 Self Orthogonal Pan Magic Latin Square left is based on the five Magic Constants s4 and has been used as a guideline for the construction shown above.

Attachment 4.8.21 illustrates the construction of an order 20 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.

4.8.3 Composed Associated Squares

Order 4 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} and {16, 17, 18, 19}

with respectively the magic constants s4 = 6, 22, 38, 54 and 70

s4
70 54 6 38 22
38 22 54 6 70
22 6 38 70 54
6 70 22 54 38
54 38 70 22 6
A
19 18 16 17
16 17 19 18
17 16 18 19
18 19 17 16
15 14 12 13
12 13 15 14
13 12 14 15
14 15 13 12
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0
11 10 8 9
8 9 11 10
9 8 10 11
10 11 9 8
7 6 4 5
4 5 7 6
5 4 6 7
6 7 5 4
11 10 8 9
8 9 11 10
9 8 10 11
10 11 9 8
7 6 4 5
4 5 7 6
5 4 6 7
6 7 5 4
15 14 12 13
12 13 15 14
13 12 14 15
14 15 13 12
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0
19 18 16 17
16 17 19 18
17 16 18 19
18 19 17 16
7 6 4 5
4 5 7 6
5 4 6 7
6 7 5 4
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0
11 10 8 9
8 9 11 10
9 8 10 11
10 11 9 8
19 18 16 17
16 17 19 18
17 16 18 19
18 19 17 16
15 14 12 13
12 13 15 14
13 12 14 15
14 15 13 12
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0
19 18 16 17
16 17 19 18
17 16 18 19
18 19 17 16
7 6 4 5
4 5 7 6
5 4 6 7
6 7 5 4
15 14 12 13
12 13 15 14
13 12 14 15
14 15 13 12
11 10 8 9
8 9 11 10
9 8 10 11
10 11 9 8
15 14 12 13
12 13 15 14
13 12 14 15
14 15 13 12
11 10 8 9
8 9 11 10
9 8 10 11
10 11 9 8
19 18 16 17
16 17 19 18
17 16 18 19
18 19 17 16
7 6 4 5
4 5 7 6
5 4 6 7
6 7 5 4
3 2 0 1
0 1 3 2
1 0 2 3
2 3 1 0

The order 5 Self Orthogonal Associated Latin Square left is based on the five Magic Constants s4 and has been used as a guideline for the construction shown above.

Attachment 4.8.31 illustrates the construction of an order 20 Composed Associated Square based on the Self Orthogonal Composed Associated Latin Square shown above.

4.9   Composed Latin Squares (21 x 21)

Order 4 Self orthogonal Latin Diagonal Squares can be used to construct order 21 Self Orthogonal Composed Latin Diagonal Squares.

The required order 4 Self orthogonal Latin Diagonal 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} and {17, 18, 19, 20}

with respectively the magic constants s4 = 6, 22, 38, 58 and 74.

Sqrs4
9 5 1 18 14
1 18 14 9 5
14 9 5 1 18
5 1 18 14 9
18 14 9 5 1
A
9 8 12 11 5 4 7 6 1 0 3 2 10 18 17 20 19 14 13 16 15
12 11 10 9 6 7 4 5 2 3 0 1 8 19 20 17 18 15 16 13 14
10 9 8 12 4 5 6 7 0 1 2 3 11 17 18 19 20 13 14 15 16
8 12 11 10 7 6 5 4 3 2 1 0 9 20 19 18 17 16 15 14 13
1 0 3 2 18 17 12 20 14 13 16 15 19 9 8 11 10 5 4 7 6
2 3 0 1 12 20 19 18 15 16 13 14 17 10 11 8 9 6 7 4 5
0 1 2 3 19 18 17 12 13 14 15 16 20 8 9 10 11 4 5 6 7
3 2 1 0 17 12 20 19 16 15 14 13 18 11 10 9 8 7 6 5 4
14 13 16 15 9 8 11 10 5 4 12 7 6 1 0 3 2 18 17 20 19
15 16 13 14 10 11 8 9 12 7 6 5 4 2 3 0 1 19 20 17 18
13 14 15 16 8 9 10 11 6 5 4 12 7 0 1 2 3 17 18 19 20
16 15 14 13 11 10 9 8 4 12 7 6 5 3 2 1 0 20 19 18 17
11 10 9 8 20 19 18 17 7 6 5 4 12 14 13 16 15 1 0 3 2
5 4 7 6 1 0 3 2 18 17 20 19 13 16 15 12 14 9 8 11 10
6 7 4 5 2 3 0 1 19 20 17 18 15 12 14 13 16 10 11 8 9
4 5 6 7 0 1 2 3 17 18 19 20 14 13 16 15 12 8 9 10 11
7 6 5 4 3 2 1 0 20 19 18 17 16 15 12 14 13 11 10 9 8
18 17 20 19 14 13 16 15 9 8 11 10 0 5 4 7 6 3 2 12 1
19 20 17 18 15 16 13 14 10 11 8 9 2 6 7 4 5 12 1 0 3
17 18 19 20 13 14 15 16 8 9 10 11 1 4 5 6 7 0 3 2 12
20 19 18 17 16 15 14 13 11 10 9 8 3 7 6 5 4 2 12 1 0

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

  • Attachment 4.9.1 illustrates and describes the construction of the order 21 Self Orthogonal Composed Latin Diagonal Square shown above.

  • Attachment 4.9.2 illustrates the construction of an order 21 Composed Simple Magic Square based on an order 21 Self Orthogonal Composed Latin Diagonal Square.

4.10   Miscellaneous

4.10.1 Composed Semi Latin Squares

The construction of Self Orthogonal Composed Semi-Latin (Diagonal) Squares based on Order 4 Self orthogonal Latin Diagonal Squares has been deducted and discussed in:

Comparable results for Order 5 Self orthogonal Latin Diagonal Squares has been deducted and discussed in Section 15.2.7.

4.10.2 Summary

The obtained results regarding the order 4 Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment

Subject

Subroutine

-

-

-

Attachment 4.5.1

Self Orth Simple Magic Squares

SelfOrth4

Attachment 4.5.2

Self Orth Pan Magic Squares

Attachment 4.5.3

Self Orth Associated Magic Squares

-

-

-

Comparable methods as described above, can be applied to construct order 5 Self Orthogonal Latin Squares, which will be described in following sections.

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