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8.5   Self Orthogonal Latin Squares (8 x 8)

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

8.5.1 Simple Magic Squares

A construction example of a Simple Magic Square M = A + 8 * T(A) + [1] is shown below:

A
0 4 6 2 7 1 3 5
5 1 3 7 0 6 4 2
4 0 2 6 1 7 5 3
1 5 7 3 6 0 2 4
2 7 5 0 4 3 1 6
6 3 1 4 2 5 7 0
7 2 0 5 3 4 6 1
3 6 4 1 5 2 0 7
B = T(A)
0 5 4 1 2 6 7 3
4 1 0 5 7 3 2 6
6 3 2 7 5 1 0 4
2 7 6 3 0 4 5 1
7 0 1 6 4 2 3 5
1 6 7 0 3 5 4 2
3 4 5 2 1 7 6 0
5 2 3 4 6 0 1 7
M = A + 8 * B + 1
1 45 39 11 24 50 60 30
38 10 4 48 57 31 21 51
53 25 19 63 42 16 6 36
18 62 56 28 7 33 43 13
59 8 14 49 37 20 26 47
15 52 58 5 27 46 40 17
32 35 41 22 12 61 55 2
44 23 29 34 54 3 9 64

Each (Order 8) Self Orthogonal Latin Diagonal Square is Double Self Orthogonal.

Consequently each Self Orthogonal Latin Diagonal Square has eight orientations (aspects) which can be reached by means of rotation and/or reflection and are shown in Attachment 8.5.11.

Each Self Orthogonal Latin Diagonal Square corresponds with 8! = 40320 Self Orthogonal Latin Diagonal Squares, which can be obtained by permutation of the integers {ai, i = 1 ... 8}.

A Base of 1152 Idempotent Squares has been found in Section 8.5.2 and Section 8.5.5 below.

The total number of Self Orthogonal Latin Diagonal Squares will be 1152 * 40320 = 46.448.640, which can be generated quite fast with routine SelfOrth8c.

In addition to the transformations and permutations described above, each Self Orthogonal Latin Diagonal Square A corresponds with 192 transformations, as described below.

  • Any line n can be interchanged with line (9 - n). The possible number of transformations is 24 = 16
    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, 4 provided that the same permutation is applied to the lines 8, 7, 6, 5. The possible number of transformations is 4! = 24.

The resulting number of transformations, excluding the 180o rotated aspects, is 16/2 * 24 = 192, which are shown in Attachment 8.5.12.

8.5.2 Associated Magic Squares

A construction example of an Associated Magic Square M = A + 8 * T(A) + [1] is shown below:

A
0 7 6 5 1 2 4 3
6 1 0 4 7 3 5 2
3 4 2 1 5 6 7 0
1 6 7 3 0 4 2 5
2 5 3 7 4 0 1 6
7 0 1 2 6 5 3 4
5 2 4 0 3 7 6 1
4 3 5 6 2 1 0 7
B = T(A)
0 6 3 1 2 7 5 4
7 1 4 6 5 0 2 3
6 0 2 7 3 1 4 5
5 4 1 3 7 2 0 6
1 7 5 0 4 6 3 2
2 3 6 4 0 5 7 1
4 5 7 2 1 3 6 0
3 2 0 5 6 4 1 7
M = A + 8 * B + 1
1 56 31 14 18 59 45 36
63 10 33 53 48 4 22 27
52 5 19 58 30 15 40 41
42 39 16 28 57 21 3 54
11 62 44 8 37 49 26 23
24 25 50 35 7 46 60 13
38 43 61 17 12 32 55 2
29 20 6 47 51 34 9 64

Attachment 8.5.3 shows the collection of 384 Associated Idempotent Self Orthogonal Latin Squares, which has been generated within 178 seconds (ref. SelfOrth8a).

This Sub Collection has been incorporated in the Base for the main collection as discussed in Section 8.5.1 above.

The total number of order 8 Self Orthogonal Associated Magic Latin Diagonal Squares is 147456 and can be generated within 1220 seconds (ref. SelfOrth8c).

8.5.3 Ultra Magic Squares

A construction example of an Ultra Magic Square M = A + 8 * T(A) + [1] is shown below:

A
0 7 6 1 4 3 2 5
6 1 0 7 2 5 4 3
5 2 3 4 1 6 7 0
3 4 5 2 7 0 1 6
1 6 7 0 5 2 3 4
7 0 1 6 3 4 5 2
4 3 2 5 0 7 6 1
2 5 4 3 6 1 0 7
B = T(A)
0 6 5 3 1 7 4 2
7 1 2 4 6 0 3 5
6 0 3 5 7 1 2 4
1 7 4 2 0 6 5 3
4 2 1 7 5 3 0 6
3 5 6 0 2 4 7 1
2 4 7 1 3 5 6 0
5 3 0 6 4 2 1 7
M = A + 8 * B + 1
1 56 47 26 13 60 35 22
63 10 17 40 51 6 29 44
54 3 28 45 58 15 24 33
12 61 38 19 8 49 42 31
34 23 16 57 46 27 4 53
32 41 50 7 20 37 62 11
21 36 59 14 25 48 55 2
43 30 5 52 39 18 9 64

Attachment 8.5.4 contains the 768 order 8 Self Orthogonal Ultra Magic Latin Diagonal Squares, which could be generated within 1130 seconds (ref. SelfOrth8c).

8.5.4 Pan Magic Squares

A construction example of a Pan Magic Square M = A + 8 * T(A) + [1] is shown below:

A
0 4 5 6 1 2 3 7
2 1 3 7 0 4 6 5
4 0 2 1 6 5 7 3
5 6 7 3 4 0 1 2
6 5 4 0 7 3 2 1
7 3 1 2 5 6 4 0
1 2 0 4 3 7 5 6
3 7 6 5 2 1 0 4
B = T(A)
0 2 4 5 6 7 1 3
4 1 0 6 5 3 2 7
5 3 2 7 4 1 0 6
6 7 1 3 0 2 4 5
1 0 6 4 7 5 3 2
2 4 5 0 3 6 7 1
3 6 7 1 2 4 5 0
7 5 3 2 1 0 6 4
M = A + 8 * B + 1
1 21 38 47 50 59 12 32
35 10 4 56 41 29 23 62
45 25 19 58 39 14 8 52
54 63 16 28 5 17 34 43
15 6 53 33 64 44 27 18
24 36 42 3 30 55 61 9
26 51 57 13 20 40 46 7
60 48 31 22 11 2 49 37

The total number of order 8 Self Orthogonal Pan Magic Latin Diagonal Squares is 127.488 and can be generated within 3560 seconds (ref. SelfOrth8c).

This collection includes the sub collection of 86016 Pan Magic and Complete Self Orthogonal Latin Diagonal Squares which can be filtered from the main collection (ref. SelfOrth8c).

8.5.5 Non Associated Idempotent Squares

      Introduction

The total number of Idempotent Self Orthogonal Latin Diagonal Squares is 1152 and was calculated by Francis Gaspalou in 2010.

The number of Associated Idempotent Self Orthogonal Latin Square is 384 and can be found in 175 seconds as explained in Section 8.5.2 above.

The generation of the remaining 768 Idempotent Self Orthogonal Latin Diagonal Squares requires a forty (40) parameter procedure which is however quite slow (ref. SelfOrth8a2).

In order to find at least a few of the required squares two columns (1, 2) and two rows (2, 3) can be split.

Attachment 8.5.51 shows 24 Non-Associated Idempotent Self Orthogonal Latin Diagonal Squares which could be generated within 433 seconds.

A few more squares (8), which could be found with a comparable procedure while using some of the known squares as a starting point, have been added to the same attachment.

The 32 squares described above result in 22 unique squares, which are shown in Attachment 8.5.52.

The unique squares can be used as Generators and enable the construction of 768 Non-Associated Idempotent Self Orthogonal Latin Squares.

      Transformations

Any Self Orthogonal Latin Diagonal Square A1 can be transformed to an Idempotent Self orthogonal Latin Diagonal Square A2 by means of substitution of the integers

   {a1(i), i= 1, 10, 19 ... 64) by {0, 1, 2 ... 7} for each element {a1(j), j=1 ... 64}

as illustrated below (ref, SelfOrth8d):

A1
5 3 1 7 2 6 4 0
2 4 6 0 7 3 1 5
3 5 7 1 6 2 0 4
4 2 0 6 3 7 5 1
6 1 3 4 0 5 7 2
0 7 5 2 4 1 3 6
1 6 4 3 5 0 2 7
7 0 2 5 1 4 6 3
A2
0 7 5 2 6 3 1 4
6 1 3 4 2 7 5 0
7 0 2 5 3 6 4 1
1 6 4 3 7 2 0 5
3 5 7 1 4 0 2 6
4 2 0 6 1 5 7 3
5 3 1 7 0 4 6 2
2 4 6 0 5 1 3 7

A few applications of subject transformation are shown in:

It appeared that the number of different results depends from the base square of subject collections as illustrated in Attachment 8.5.55 for the 8 aspects of the 22 Generators.

However when the 264 transformations described above are applied on subject Generators, a collection of 5808 (= 22 * 264) Latin Diagonal Squares will result, which can be transformed to Idempotent Latin diagonal Squares.

After removing the identical squares, a collection of 768 Non Associated Idempotent Self Orthogonal Latin Diagonal Squares remains, which are shown in Attachment 8.5.56.

8.5.6 Magic Squares, Compact (4 x 4)

A construction example of a Compact (4 x 4) Magic Square M = A + 8 * T(A) + [1] is shown below:

A
0 4 5 6 1 2 3 7
2 1 3 7 0 4 6 5
4 0 2 1 6 5 7 3
5 6 7 3 4 0 1 2
6 5 4 0 7 3 2 1
7 3 1 2 5 6 4 0
1 2 0 4 3 7 5 6
3 7 6 5 2 1 0 4
B = T(A)
0 2 4 5 6 7 1 3
4 1 0 6 5 3 2 7
5 3 2 7 4 1 0 6
6 7 1 3 0 2 4 5
1 0 6 4 7 5 3 2
2 4 5 0 3 6 7 1
3 6 7 1 2 4 5 0
7 5 3 2 1 0 6 4
M = A + 8 * B + 1
1 21 38 47 50 59 12 32
35 10 4 56 41 29 23 62
45 25 19 58 39 14 8 52
54 63 16 28 5 17 34 43
15 6 53 33 64 44 27 18
24 36 42 3 30 55 61 9
26 51 57 13 20 40 46 7
60 48 31 22 11 2 49 37

The total number of subject order 8 Self Orthogonal Magic Latin Diagonal Squares is 135.168 and can be generated within 4130 seconds (ref. SelfOrth8c).

8.5.7 Magic Squares, V Type ZigZag (4 Way)

A construction example of a V type ZigZag (4 way) Magic Square M = A + 8 * T(A) + [1] is shown below:

A
0 5 4 6 7 2 3 1
3 1 5 0 4 6 2 7
7 2 6 4 0 5 1 3
1 3 2 7 6 4 5 0
5 0 1 3 2 7 6 4
6 4 0 5 1 3 7 2
2 7 3 1 5 0 4 6
4 6 7 2 3 1 0 5
B = T(A)
0 3 7 1 5 6 2 4
5 1 2 3 0 4 7 6
4 5 6 2 1 0 3 7
6 0 4 7 3 5 1 2
7 4 0 6 2 1 5 3
2 6 5 4 7 3 0 1
3 2 1 5 6 7 4 0
1 7 3 0 4 2 6 5
M = A + 8 * B + 1
1 30 61 15 48 51 20 34
44 10 22 25 5 39 59 56
40 43 55 21 9 6 26 60
50 4 35 64 31 45 14 17
62 33 2 52 19 16 47 29
23 53 41 38 58 28 8 11
27 24 12 42 54 57 37 7
13 63 32 3 36 18 49 46

The total number of subject order 8 Self Orthogonal Magic Latin Diagonal Squares is 133.632 and can be generated within 3600 seconds (ref. SelfOrth8c).

8.5.8 Magic Squares, Bent Diagonals (4 Way)

A construction example of a Bent Diagonal (4 way) Magic Square M = A + 8 * T(A) + [1] is shown below:

A
0 4 5 1 2 6 7 3
5 1 0 4 7 3 2 6
3 7 6 2 1 5 4 0
6 2 3 7 4 0 1 5
1 5 4 0 3 7 6 2
4 0 1 5 6 2 3 7
2 6 7 3 0 4 5 1
7 3 2 6 5 1 0 4
B = T(A)
0 5 3 6 1 4 2 7
4 1 7 2 5 0 6 3
5 0 6 3 4 1 7 2
1 4 2 7 0 5 3 6
2 7 1 4 3 6 0 5
6 3 5 0 7 2 4 1
7 2 4 1 6 3 5 0
3 6 0 5 2 7 1 4
M = A + 8 * B + 1
1 45 30 50 11 39 24 60
38 10 57 21 48 4 51 31
44 8 55 27 34 14 61 17
15 35 20 64 5 41 26 54
18 62 13 33 28 56 7 43
53 25 42 6 63 19 36 16
59 23 40 12 49 29 46 2
32 52 3 47 22 58 9 37

The total number of subject order 8 Self Orthogonal Magic Latin Diagonal Squares is 4608 and can be generated within 3600 seconds (ref. SelfOrth8c).

The collection includes several sub collections which can be filtered from the main collection and summarised as follows:

Type n9 Cmpct4 Notes
 Simple 2688  0   Not Pan Magic
 Pan Magic 1636  1152   Not Complete
 Complete 384  384   -

Attachment 8.5.81 shows one example of each of the Self Orthogonal Latin Diagonal Squares listed above.

8.6   Interesting Sub Collections

8.6.1 Bordered Magic Squares

A construction example of a Bordered Magic Square M = A + 8 * T(A) + [1] is shown below:

A
0 5 4 6 3 1 7 2
6 3 2 0 5 7 1 4
1 4 5 7 2 0 6 3
4 1 0 2 7 5 3 6
2 7 6 4 1 3 5 0
7 2 3 1 4 6 0 5
3 6 7 5 0 2 4 1
5 0 1 3 6 4 2 7
B = T(A)
0 6 1 4 2 7 3 5
5 3 4 1 7 2 6 0
4 2 5 0 6 3 7 1
6 0 7 2 4 1 5 3
3 5 2 7 1 4 0 6
1 7 0 5 3 6 2 4
7 1 6 3 5 0 4 2
2 4 3 6 0 5 1 7
M = A + 8 * B + 1
1 54 13 39 20 58 32 43
47 28 35 9 62 24 50 5
34 21 46 8 51 25 63 12
53 2 57 19 40 14 44 31
27 48 23 61 10 36 6 49
16 59 4 42 29 55 17 38
60 15 56 30 41 3 37 18
22 33 26 52 7 45 11 64

The total number of order 8 Self Orthogonal Bordered Latin Diagonal Squares is 49152 and can be generated within 3450 seconds (ref. SelfOrth8c).

The collection includes several sub collections which can be filtered from the main collection and summarised as follows:

Main Type n9 Cntr Simple Cntr Ass Cntr PM Notes
 Simple 41184  26400  6144  8640   Not Ass, Not PM
 Associated 6144  0  6144  0   Including Ultra
 Pan Magic 864  288  0  576   Not Ultra, Not Complete
 Ultra Magic 288  0  288  0  -
 Complete 960  960  0  0  -

Attachment 8.6.1 shows one example of each of the Self Orthogonal Bordered Latin Diagonal Squares listed above.

8.6.2 Magic Squares with Corner Square

A construction example of a Magic Square M = A + 8 * T(A) + [1] with Order 4 Corner Square is shown below:

A
0 7 2 5 1 3 6 4
4 3 6 1 0 2 7 5
7 0 5 2 3 1 4 6
3 4 1 6 2 0 5 7
5 1 4 0 7 6 3 2
2 6 3 7 5 4 1 0
1 5 0 4 6 7 2 3
6 2 7 3 4 5 0 1
B = T(A)
0 4 7 3 5 2 1 6
7 3 0 4 1 6 5 2
2 6 5 1 4 3 0 7
5 1 2 6 0 7 4 3
1 0 3 2 7 5 6 4
3 2 1 0 6 4 7 5
6 7 4 5 3 1 2 0
4 5 6 7 2 0 3 1
M = A + 8 * B + 1
1 40 59 30 42 20 15 53
61 28 7 34 9 51 48 22
24 49 46 11 36 26 5 63
44 13 18 55 3 57 38 32
14 2 29 17 64 47 52 35
27 23 12 8 54 37 58 41
50 62 33 45 31 16 19 4
39 43 56 60 21 6 25 10

The total number of order 8 Self Orthogonal Latin Diagonal Squares with Order 4 Corner Square is 23040 and can be generated within 3330 seconds (ref. SelfOrth8c).

The collection includes several sub collections which can be filtered from the main collection and summarised as follows:

Main Type n9 Crnr Simple Crnr Ass Crnr PM Notes
 Simple 19968  11616  4320  4032   Not Ass, Not PM
 Associated 960  960  0  0  -
 Pan Magic 1152  288  288  576   Not Complete
 Complete 960  960  0  0  -

Attachment 8.6.2 shows one example of each of the Self Orthogonal Latin Diagonal Squares listed above.

8.6.3 Composed Magic Squares

A construction example of a Composed Magic Square M = A + 8 * T(A) + [1] is shown below:

A
0 6 3 5 2 4 1 7
7 1 4 2 5 3 6 0
4 2 7 1 6 0 5 3
3 5 0 6 1 7 2 4
1 7 2 4 3 5 0 6
6 0 5 3 4 2 7 1
5 3 6 0 7 1 4 2
2 4 1 7 0 6 3 5
B = T(A)
0 7 4 3 1 6 5 2
6 1 2 5 7 0 3 4
3 4 7 0 2 5 6 1
5 2 1 6 4 3 0 7
2 5 6 1 3 4 7 0
4 3 0 7 5 2 1 6
1 6 5 2 0 7 4 3
7 0 3 4 6 1 2 5
M = A + 8 * B + 1
1 63 36 30 11 53 42 24
56 10 21 43 62 4 31 33
29 35 64 2 23 41 54 12
44 22 9 55 34 32 3 61
18 48 51 13 28 38 57 7
39 25 6 60 45 19 16 50
14 52 47 17 8 58 37 27
59 5 26 40 49 15 20 46

The total number of order 8 Self Orthogonal Composed Latin Diagonal Squares with Order 4 Sub Squares is 9984 and can be filtered from the collection described in Section 8.6.2 above.

The collection includes several sub collections which can be filtered from the main collection and summarised as follows:

Main Type n9 Crnr Simple Crnr Ass Crnr PM Notes
 Simple 6912  3168  2016  1728   Not Ass, Not PM
 Associated 960  960  0  0  -
 Pan Magic 1152  288  288  576   Not Complete
 Complete 960  960  0  0  -

Attachment 8.6.4 shows one example of each of the Self Orthogonal Composed Latin Diagonal Squares listed above.

8.6.4 Magic Squares with Square Inlay

A construction example of a Magic Square M = A + 8 * T(A) + [1] with Order 4 Square Inlay is shown below:

A
0 2 4 7 1 5 3 6
3 1 7 4 2 6 0 5
2 0 6 5 3 7 1 4
4 6 0 3 5 1 7 2
5 7 1 2 4 0 6 3
7 5 3 0 6 2 4 1
6 4 2 1 7 3 5 0
1 3 5 6 0 4 2 7
B = T(A)
0 3 2 4 5 7 6 1
2 1 0 6 7 5 4 3
4 7 6 0 1 3 2 5
7 4 5 3 2 0 1 6
1 2 3 5 4 6 7 0
5 6 7 1 0 2 3 4
3 0 1 7 6 4 5 2
6 5 4 2 3 1 0 7
M = A + 8 * B + 1
1 27 21 40 42 62 52 15
20 10 8 53 59 47 33 30
35 57 55 6 12 32 18 45
61 39 41 28 22 2 16 51
14 24 26 43 37 49 63 4
48 54 60 9 7 19 29 34
31 5 11 58 56 36 46 17
50 44 38 23 25 13 3 64

The total number of order 8 Self Orthogonal Latin Diagonal Squares with Order 4 Square Inlay is 4608 and can be generated within 3330 seconds (ref. SelfOrth8c).

The collection includes several sub collections which can be filtered from the main collection and summarised as follows:

Main Type n9 Inlay Simple Inlay Ass Inlay PM Notes
 Simple 3840  2112  864  864   Not PM
 Pan Magic 576  0  288  288   Not Complete
 Complete 192  192  0  0  -

Attachment 8.6.3 shows one example of each of the Self Orthogonal Latin Diagonal Squares listed above.

8.7   Composed Latin Squares (32 x 32)

Order 8 Self Orthogonal Latin Diagonal Squares can be used to construct order 32 Self Orthogonal Composed Latin Diagonal Squares.

8.7.1 Composed Associated Squares

Order 8 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:

    {0, 1 ... 7}, {8, 9 ... 15}, {16, 17 ... 23} and {24, 25 ... 31}

with respectively the magic constants s8 = 28, 92, 156 and 220

Sqrs8
24 16 0 8
0 8 24 16
8 0 16 24
16 24 8 0

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

A, Associated
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7

Attachment 8.7.1 shows the resulting order 32 Composed Associated Square based on the Self Orthogonal Composed Associated Latin Square shown above.

8.7.2 Composed Pan Magic Squares (1)

Order 32 Self Orthogonal Composed Pan Magic and Complete Latin Diagonal Squares can be constructed based on Order 32 Self Orthogonal Composed Associated Latin Diagonal Squares as illustrated below (Euler):

Sqrs8
24 16 0 8
0 8 24 16
8 0 16 24
16 24 8 0

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

A, Pan Magic (Euler)
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
12 11 10 9 13 14 15 8
10 13 12 15 11 8 9 14
8 15 14 13 9 10 11 12
13 10 11 8 12 15 14 9
14 9 8 11 15 12 13 10
11 12 13 14 10 9 8 15
9 14 15 12 8 11 10 13
15 8 9 10 14 13 12 11
4 3 2 1 5 6 7 0
2 5 4 7 3 0 1 6
0 7 6 5 1 2 3 4
5 2 3 0 4 7 6 1
6 1 0 3 7 4 5 2
3 4 5 6 2 1 0 7
1 6 7 4 0 3 2 5
7 0 1 2 6 5 4 3
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
20 19 18 17 21 22 23 16
18 21 20 23 19 16 17 22
16 23 22 21 17 18 19 20
21 18 19 16 20 23 22 17
22 17 16 19 23 20 21 18
19 20 21 22 18 17 16 23
17 22 23 20 16 19 18 21
23 16 17 18 22 21 20 19
28 27 26 25 29 30 31 24
26 29 28 31 27 24 25 30
24 31 30 29 25 26 27 28
29 26 27 24 28 31 30 25
30 25 24 27 31 28 29 26
27 28 29 30 26 25 24 31
25 30 31 28 24 27 26 29
31 24 25 26 30 29 28 27
19 20 21 22 18 17 16 23
21 18 19 16 20 23 22 17
23 16 17 18 22 21 20 19
18 21 20 23 19 16 17 22
17 22 23 20 16 19 18 21
20 19 18 17 21 22 23 16
22 17 16 19 23 20 21 18
16 23 22 21 17 18 19 20
27 28 29 30 26 25 24 31
29 26 27 24 28 31 30 25
31 24 25 26 30 29 28 27
26 29 28 31 27 24 25 30
25 30 31 28 24 27 26 29
28 27 26 25 29 30 31 24
30 25 24 27 31 28 29 26
24 31 30 29 25 26 27 28
7 0 1 2 6 5 4 3
1 6 7 4 0 3 2 5
3 4 5 6 2 1 0 7
6 1 0 3 7 4 5 2
5 2 3 0 4 7 6 1
0 7 6 5 1 2 3 4
2 5 4 7 3 0 1 6
4 3 2 1 5 6 7 0
15 8 9 10 14 13 12 11
9 14 15 12 8 11 10 13
11 12 13 14 10 9 8 15
14 9 8 11 15 12 13 10
13 10 11 8 12 15 14 9
8 15 14 13 9 10 11 12
10 13 12 15 11 8 9 14
12 11 10 9 13 14 15 8
11 12 13 14 10 9 8 15
13 10 11 8 12 15 14 9
15 8 9 10 14 13 12 11
10 13 12 15 11 8 9 14
9 14 15 12 8 11 10 13
12 11 10 9 13 14 15 8
14 9 8 11 15 12 13 10
8 15 14 13 9 10 11 12
3 4 5 6 2 1 0 7
5 2 3 0 4 7 6 1
7 0 1 2 6 5 4 3
2 5 4 7 3 0 1 6
1 6 7 4 0 3 2 5
4 3 2 1 5 6 7 0
6 1 0 3 7 4 5 2
0 7 6 5 1 2 3 4
31 24 25 26 30 29 28 27
25 30 31 28 24 27 26 29
27 28 29 30 26 25 24 31
30 25 24 27 31 28 29 26
29 26 27 24 28 31 30 25
24 31 30 29 25 26 27 28
26 29 28 31 27 24 25 30
28 27 26 25 29 30 31 24
23 16 17 18 22 21 20 19
17 22 23 20 16 19 18 21
19 20 21 22 18 17 16 23
22 17 16 19 23 20 21 18
21 18 19 16 20 23 22 17
16 23 22 21 17 18 19 20
18 21 20 23 19 16 17 22
20 19 18 17 21 22 23 16

Attachment 8.7.2 shows the resulting order 32 Composed Pan Magic and Complete Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.

8.7.3 Composed Pan Magic Squares (2)

Order 8 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:

    {0, 1 ... 7}, {8, 9 ... 15}, {16, 17 ... 23} and {24, 25 ... 31}

with respectively the magic constants s8 = 28, 92, 156 and 220

Sqrs8
9 1 25 17
17 25 1 9
1 9 17 25
25 17 9 1

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

A, Pan Magic
9 12 13 8 11 14 15 10
15 10 11 14 13 8 9 12
12 9 8 13 14 11 10 15
10 15 14 11 8 13 12 9
13 8 9 12 15 10 11 14
11 14 15 10 9 12 13 8
8 13 12 9 10 15 14 11
14 11 10 15 12 9 8 13
1 4 5 0 3 6 7 2
7 2 3 6 5 0 1 4
4 1 0 5 6 3 2 7
2 7 6 3 0 5 4 1
5 0 1 4 7 2 3 6
3 6 7 2 1 4 5 0
0 5 4 1 2 7 6 3
6 3 2 7 4 1 0 5
25 28 29 24 27 30 31 26
31 26 27 30 29 24 25 28
28 25 24 29 30 27 26 31
26 31 30 27 24 29 28 25
29 24 25 28 31 26 27 30
27 30 31 26 25 28 29 24
24 29 28 25 26 31 30 27
30 27 26 31 28 25 24 29
17 20 21 16 19 22 23 18
23 18 19 22 21 16 17 20
20 17 16 21 22 19 18 23
18 23 22 19 16 21 20 17
21 16 17 20 23 18 19 22
19 22 23 18 17 20 21 16
16 21 20 17 18 23 22 19
22 19 18 23 20 17 16 21
17 20 21 16 19 22 23 18
23 18 19 22 21 16 17 20
20 17 16 21 22 19 18 23
18 23 22 19 16 21 20 17
21 16 17 20 23 18 19 22
19 22 23 18 17 20 21 16
16 21 20 17 18 23 22 19
22 19 18 23 20 17 16 21
25 28 29 24 27 30 31 26
31 26 27 30 29 24 25 28
28 25 24 29 30 27 26 31
26 31 30 27 24 29 28 25
29 24 25 28 31 26 27 30
27 30 31 26 25 28 29 24
24 29 28 25 26 31 30 27
30 27 26 31 28 25 24 29
1 4 5 0 3 6 7 2
7 2 3 6 5 0 1 4
4 1 0 5 6 3 2 7
2 7 6 3 0 5 4 1
5 0 1 4 7 2 3 6
3 6 7 2 1 4 5 0
0 5 4 1 2 7 6 3
6 3 2 7 4 1 0 5
9 12 13 8 11 14 15 10
15 10 11 14 13 8 9 12
12 9 8 13 14 11 10 15
10 15 14 11 8 13 12 9
13 8 9 12 15 10 11 14
11 14 15 10 9 12 13 8
8 13 12 9 10 15 14 11
14 11 10 15 12 9 8 13
1 4 5 0 3 6 7 2
7 2 3 6 5 0 1 4
4 1 0 5 6 3 2 7
2 7 6 3 0 5 4 1
5 0 1 4 7 2 3 6
3 6 7 2 1 4 5 0
0 5 4 1 2 7 6 3
6 3 2 7 4 1 0 5
9 12 13 8 11 14 15 10
15 10 11 14 13 8 9 12
12 9 8 13 14 11 10 15
10 15 14 11 8 13 12 9
13 8 9 12 15 10 11 14
11 14 15 10 9 12 13 8
8 13 12 9 10 15 14 11
14 11 10 15 12 9 8 13
17 20 21 16 19 22 23 18
23 18 19 22 21 16 17 20
20 17 16 21 22 19 18 23
18 23 22 19 16 21 20 17
21 16 17 20 23 18 19 22
19 22 23 18 17 20 21 16
16 21 20 17 18 23 22 19
22 19 18 23 20 17 16 21
25 28 29 24 27 30 31 26
31 26 27 30 29 24 25 28
28 25 24 29 30 27 26 31
26 31 30 27 24 29 28 25
29 24 25 28 31 26 27 30
27 30 31 26 25 28 29 24
24 29 28 25 26 31 30 27
30 27 26 31 28 25 24 29
25 28 29 24 27 30 31 26
31 26 27 30 29 24 25 28
28 25 24 29 30 27 26 31
26 31 30 27 24 29 28 25
29 24 25 28 31 26 27 30
27 30 31 26 25 28 29 24
24 29 28 25 26 31 30 27
30 27 26 31 28 25 24 29
17 20 21 16 19 22 23 18
23 18 19 22 21 16 17 20
20 17 16 21 22 19 18 23
18 23 22 19 16 21 20 17
21 16 17 20 23 18 19 22
19 22 23 18 17 20 21 16
16 21 20 17 18 23 22 19
22 19 18 23 20 17 16 21
9 12 13 8 11 14 15 10
15 10 11 14 13 8 9 12
12 9 8 13 14 11 10 15
10 15 14 11 8 13 12 9
13 8 9 12 15 10 11 14
11 14 15 10 9 12 13 8
8 13 12 9 10 15 14 11
14 11 10 15 12 9 8 13
1 4 5 0 3 6 7 2
7 2 3 6 5 0 1 4
4 1 0 5 6 3 2 7
2 7 6 3 0 5 4 1
5 0 1 4 7 2 3 6
3 6 7 2 1 4 5 0
0 5 4 1 2 7 6 3
6 3 2 7 4 1 0 5

Attachment 8.7.3 shows the resulting order 32 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.

8.8   Composed Latin Squares (33 x 33)

Order 8 Self orthogonal Latin Diagonal Squares can be used to construct order 33 Self Orthogonal Composed Latin Diagonal Squares.

The required order 8 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:

    {0, 1 ... 7}, {8, 9 ... 15}, {16}, {17, 18 ... 24} and {25, 26 ... 32}

with respectively the magic constants s8 = 28, 92, 164 and 228

Sqrs8
25 17 0 8
0 8 25 17
8 0 17 25
17 25 8 0

The order 4 Self orthogonal Latin Diagonal Square shown above is based on the first elements of the original Sub Squares, and has been used as a guideline for the construction of square A shown below.

A
25 27 29 16 31 32 30 28 17 24 23 22 18 19 20 21 26 0 7 6 5 1 2 3 4 8 15 14 13 9 10 11 12
28 26 32 29 30 31 16 25 23 18 17 20 24 21 22 19 27 6 1 0 3 7 4 5 2 14 9 8 11 15 12 13 10
26 31 27 32 25 16 28 30 21 20 19 18 22 23 24 17 29 4 3 2 1 5 6 7 0 12 11 10 9 13 14 15 8
30 25 16 28 26 27 32 29 18 23 24 21 17 20 19 22 31 1 6 7 4 0 3 2 5 9 14 15 12 8 11 10 13
16 32 31 30 29 28 27 26 19 22 21 24 20 17 18 23 25 2 5 4 7 3 0 1 6 10 13 12 15 11 8 9 14
27 29 26 31 32 30 25 16 24 17 18 19 23 22 21 20 28 7 0 1 2 6 5 4 3 15 8 9 10 14 13 12 11
29 28 30 25 16 26 31 27 22 19 20 17 21 24 23 18 32 5 2 3 0 4 7 6 1 13 10 11 8 12 15 14 9
31 16 25 27 28 29 26 32 20 21 22 23 19 18 17 24 30 3 4 5 6 2 1 0 7 11 12 13 14 10 9 8 15
0 7 6 5 1 2 3 4 8 10 12 16 14 15 13 11 9 25 32 31 30 26 27 28 29 17 24 23 22 18 19 20 21
6 1 0 3 7 4 5 2 11 9 15 12 13 14 16 8 10 31 26 25 28 32 29 30 27 23 18 17 20 24 21 22 19
4 3 2 1 5 6 7 0 9 14 10 15 8 16 11 13 12 29 28 27 26 30 31 32 25 21 20 19 18 22 23 24 17
1 6 7 4 0 3 2 5 13 8 16 11 9 10 15 12 14 26 31 32 29 25 28 27 30 18 23 24 21 17 20 19 22
2 5 4 7 3 0 1 6 16 15 14 13 12 11 10 9 8 27 30 29 32 28 25 26 31 19 22 21 24 20 17 18 23
7 0 1 2 6 5 4 3 10 12 9 14 15 13 8 16 11 32 25 26 27 31 30 29 28 24 17 18 19 23 22 21 20
5 2 3 0 4 7 6 1 12 11 13 8 16 9 14 10 15 30 27 28 25 29 32 31 26 22 19 20 17 21 24 23 18
3 4 5 6 2 1 0 7 14 16 8 10 11 12 9 15 13 28 29 30 31 27 26 25 32 20 21 22 23 19 18 17 24
32 30 28 26 27 25 29 31 15 13 11 9 10 8 12 14 16 18 20 24 22 23 21 19 17 1 3 7 5 6 4 2 0
8 15 14 13 9 10 11 12 0 7 6 5 1 2 3 4 19 17 23 20 21 22 24 16 18 25 32 31 30 26 27 28 29
14 9 8 11 15 12 13 10 6 1 0 3 7 4 5 2 17 22 18 23 16 24 19 21 20 31 26 25 28 32 29 30 27
12 11 10 9 13 14 15 8 4 3 2 1 5 6 7 0 21 16 24 19 17 18 23 20 22 29 28 27 26 30 31 32 25
9 14 15 12 8 11 10 13 1 6 7 4 0 3 2 5 24 23 22 21 20 19 18 17 16 26 31 32 29 25 28 27 30
10 13 12 15 11 8 9 14 2 5 4 7 3 0 1 6 18 20 17 22 23 21 16 24 19 27 30 29 32 28 25 26 31
15 8 9 10 14 13 12 11 7 0 1 2 6 5 4 3 20 19 21 16 24 17 22 18 23 32 25 26 27 31 30 29 28
13 10 11 8 12 15 14 9 5 2 3 0 4 7 6 1 22 24 16 18 19 20 17 23 21 30 27 28 25 29 32 31 26
11 12 13 14 10 9 8 15 3 4 5 6 2 1 0 7 23 21 19 17 18 16 20 22 24 28 29 30 31 27 26 25 32
17 24 23 22 18 19 20 21 25 32 31 30 26 27 28 29 2 8 15 14 13 9 10 11 12 0 6 3 4 5 7 16 1
23 18 17 20 24 21 22 19 31 26 25 28 32 29 30 27 0 14 9 8 11 15 12 13 10 5 1 6 16 7 2 4 3
21 20 19 18 22 23 24 17 29 28 27 26 30 31 32 25 4 12 11 10 9 13 14 15 8 16 7 2 0 1 6 3 5
18 23 24 21 17 20 19 22 26 31 32 29 25 28 27 30 7 9 14 15 12 8 11 10 13 6 5 4 3 2 1 0 16
19 22 21 24 20 17 18 23 27 30 29 32 28 25 26 31 1 10 13 12 15 11 8 9 14 3 0 5 6 4 16 7 2
24 17 18 19 23 22 21 20 32 25 26 27 31 30 29 28 3 15 8 9 10 14 13 12 11 2 4 16 7 0 5 1 6
22 19 20 17 21 24 23 18 30 27 28 25 29 32 31 26 5 13 10 11 8 12 15 14 9 7 16 1 2 3 0 6 4
20 21 22 23 19 18 17 24 28 29 30 31 27 26 25 32 6 11 12 13 14 10 9 8 15 4 2 0 1 16 3 5 7
  • Attachment 8.8.1 illustrates and describes the construction of the order 33 Self Orthogonal Composed Latin Diagonal Square shown above.

  • Attachment 8.8.2 shows the resulting order 33 Composed Simple Magic Square based on the order 33 Self Orthogonal Composed Latin Diagonal Square shown above.

8.9   Composed Latin Squares (40 x 40)

Order 8 Self Orthogonal Latin Diagonal Squares can be used to construct order 40 Self Orthogonal Composed Latin Diagonal Squares.

8.9.1 Composed Associated Squares

Order 8 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:

    {0, 1 ... 7}, {8, 9 ... 15}, {16, 17 ... 23}, {24, 25 ... 31} and {32, 33 ... 39}

with respectively the magic constants s8 = 28, 92, 156, 220 and 284

Sqrs8
32 24 0 16 8
16 8 24 0 32
8 0 16 32 24
0 32 8 24 16
24 16 32 8 0

The order 5 Self Orthogonal Associated Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A shown below.

A, Associated
32 39 38 37 33 34 35 36
38 33 32 35 39 36 37 34
36 35 34 33 37 38 39 32
33 38 39 36 32 35 34 37
34 37 36 39 35 32 33 38
39 32 33 34 38 37 36 35
37 34 35 32 36 39 38 33
35 36 37 38 34 33 32 39
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7
32 39 38 37 33 34 35 36
38 33 32 35 39 36 37 34
36 35 34 33 37 38 39 32
33 38 39 36 32 35 34 37
34 37 36 39 35 32 33 38
39 32 33 34 38 37 36 35
37 34 35 32 36 39 38 33
35 36 37 38 34 33 32 39
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
32 39 38 37 33 34 35 36
38 33 32 35 39 36 37 34
36 35 34 33 37 38 39 32
33 38 39 36 32 35 34 37
34 37 36 39 35 32 33 38
39 32 33 34 38 37 36 35
37 34 35 32 36 39 38 33
35 36 37 38 34 33 32 39
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7
32 39 38 37 33 34 35 36
38 33 32 35 39 36 37 34
36 35 34 33 37 38 39 32
33 38 39 36 32 35 34 37
34 37 36 39 35 32 33 38
39 32 33 34 38 37 36 35
37 34 35 32 36 39 38 33
35 36 37 38 34 33 32 39
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
24 31 30 29 25 26 27 28
30 25 24 27 31 28 29 26
28 27 26 25 29 30 31 24
25 30 31 28 24 27 26 29
26 29 28 31 27 24 25 30
31 24 25 26 30 29 28 27
29 26 27 24 28 31 30 25
27 28 29 30 26 25 24 31
16 23 22 21 17 18 19 20
22 17 16 19 23 20 21 18
20 19 18 17 21 22 23 16
17 22 23 20 16 19 18 21
18 21 20 23 19 16 17 22
23 16 17 18 22 21 20 19
21 18 19 16 20 23 22 17
19 20 21 22 18 17 16 23
32 39 38 37 33 34 35 36
38 33 32 35 39 36 37 34
36 35 34 33 37 38 39 32
33 38 39 36 32 35 34 37
34 37 36 39 35 32 33 38
39 32 33 34 38 37 36 35
37 34 35 32 36 39 38 33
35 36 37 38 34 33 32 39
8 15 14 13 9 10 11 12
14 9 8 11 15 12 13 10
12 11 10 9 13 14 15 8
9 14 15 12 8 11 10 13
10 13 12 15 11 8 9 14
15 8 9 10 14 13 12 11
13 10 11 8 12 15 14 9
11 12 13 14 10 9 8 15
0 7 6 5 1 2 3 4
6 1 0 3 7 4 5 2
4 3 2 1 5 6 7 0
1 6 7 4 0 3 2 5
2 5 4 7 3 0 1 6
7 0 1 2 6 5 4 3
5 2 3 0 4 7 6 1
3 4 5 6 2 1 0 7

Attachment 8.9.1 shows the resulting order 40 Composed Associated Square based on the Self Orthogonal Composed Associated Latin Square shown above.

8.9.2 Composed Pan Magic Squares

Order 8 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:

    {0, 1 ... 7}, {8, 9 ... 15}, {16, 17 ... 23}, {24, 25 ... 31} and {32, 33 ... 39}

with respectively the magic constants s8 = 28, 92, 156, 220 and 284

Sqrs8
17 9 1 33 25
1 33 25 17 9
25 17 9 1 33
9 1 33 25 17
33 25 17 9 1

The order 5 Self Orthogonal Pan Magic Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A shown below.

A, Pan Magic
17 20 21 16 19 22 23 18
23 18 19 22 21 16 17 20
20 17 16 21 22 19 18 23
18 23 22 19 16 21 20 17
21 16 17 20 23 18 19 22
19 22 23 18 17 20 21 16
16 21 20 17 18 23 22 19
22 19 18 23 20 17 16 21
9 12 13 8 11 14 15 10
15 10 11 14 13 8 9 12
12 9 8 13 14 11 10 15
10 15 14 11 8 13 12 9
13 8 9 12 15 10 11 14
11 14 15 10 9 12 13 8
8 13 12 9 10 15 14 11
14 11 10 15 12 9 8 13
1 4 5 0 3 6 7 2
7 2 3 6 5 0 1 4
4 1 0 5 6 3 2 7
2 7 6 3 0 5 4 1
5 0 1 4 7 2 3 6
3 6 7 2 1 4 5 0
0 5 4 1 2 7 6 3
6 3 2 7 4 1 0 5
33 36 37 32 35 38 39 34
39 34 35 38 37 32 33 36
36 33 32 37 38 35 34 39
34 39 38 35 32 37 36 33
37 32 33 36 39 34 35 38
35 38 39 34 33 36 37 32
32 37 36 33 34 39 38 35
38 35 34 39 36 33 32 37
25 28 29 24 27 30 31 26
31 26 27 30 29 24 25 28
28 25 24 29 30 27 26 31
26 31 30 27 24 29 28 25
29 24 25 28 31 26 27 30
27 30 31 26 25 28 29 24
24 29 28 25 26 31 30 27
30 27 26 31 28 25 24 29
1 4 5 0 3 6 7 2
7 2 3 6 5 0 1 4
4 1 0 5 6 3 2 7
2 7 6 3 0 5 4 1
5 0 1 4 7 2 3 6
3 6 7 2 1 4 5 0
0 5 4 1 2 7 6 3
6 3 2 7 4 1 0 5
33 36 37 32 35 38 39 34
39 34 35 38 37 32 33 36
36 33 32 37 38 35 34 39
34 39 38 35 32 37 36 33
37 32 33 36 39 34 35 38
35 38 39 34 33 36 37 32
32 37 36 33 34 39 38 35
38 35 34 39 36 33 32 37
25 28 29 24 27 30 31 26
31 26 27 30 29 24 25 28
28 25 24 29 30 27 26 31
26 31 30 27 24 29 28 25
29 24 25 28 31 26 27 30
27 30 31 26 25 28 29 24
24 29 28 25 26 31 30 27
30 27 26 31 28 25 24 29
17 20 21 16 19 22 23 18
23 18 19 22 21 16 17 20
20 17 16 21 22 19 18 23
18 23 22 19 16 21 20 17
21 16 17 20 23 18 19 22
19 22 23 18 17 20 21 16
16 21 20 17 18 23 22 19
22 19 18 23 20 17 16 21
9 12 13 8 11 14 15 10
15 10 11 14 13 8 9 12
12 9 8 13 14 11 10 15
10 15 14 11 8 13 12 9
13 8 9 12 15 10 11 14
11 14 15 10 9 12 13 8
8 13 12 9 10 15 14 11
14 11 10 15 12 9 8 13
25 28 29 24 27 30 31 26
31 26 27 30 29 24 25 28
28 25 24 29 30 27 26 31
26 31 30 27 24 29 28 25
29 24 25 28 31 26 27 30
27 30 31 26 25 28 29 24
24 29 28 25 26 31 30 27
30 27 26 31 28 25 24 29
17 20 21 16 19 22 23 18
23 18 19 22 21 16 17 20
20 17 16 21 22 19 18 23
18 23 22 19 16 21 20 17
21 16 17 20 23 18 19 22
19 22 23 18 17 20 21 16
16 21 20 17 18 23 22 19
22 19 18 23 20 17 16 21
9 12 13 8 11 14 15 10
15 10 11 14 13 8 9 12
12 9 8 13 14 11 10 15
10 15 14 11 8 13 12 9
13 8 9 12 15 10 11 14
11 14 15 10 9 12 13 8
8 13 12 9 10 15 14 11
14 11 10 15 12 9 8 13
1 4 5 0 3 6 7 2
7 2 3 6 5 0 1 4
4 1 0 5 6 3 2 7
2 7 6 3 0 5 4 1
5 0 1 4 7 2 3 6
3 6 7 2 1 4 5 0
0 5 4 1 2 7 6 3
6 3 2 7 4 1 0 5
33 36 37 32 35 38 39 34
39 34 35 38 37 32 33 36
36 33 32 37 38 35 34 39
34 39 38 35 32 37 36 33
37 32 33 36 39 34 35 38
35 38 39 34 33 36 37 32
32 37 36 33 34 39 38 35
38 35 34 39 36 33 32 37
9 12 13 8 11 14 15 10
15 10 11 14 13 8 9 12
12 9 8 13 14 11 10 15
10 15 14 11 8 13 12 9
13 8 9 12 15 10 11 14
11 14 15 10 9 12 13 8
8 13 12 9 10 15 14 11
14 11 10 15 12 9 8 13
1 4 5 0 3 6 7 2
7 2 3 6 5 0 1 4
4 1 0 5 6 3 2 7
2 7 6 3 0 5 4 1
5 0 1 4 7 2 3 6
3 6 7 2 1 4 5 0
0 5 4 1 2 7 6 3
6 3 2 7 4 1 0 5
33 36 37 32 35 38 39 34
39 34 35 38 37 32 33 36
36 33 32 37 38 35 34 39
34 39 38 35 32 37 36 33
37 32 33 36 39 34 35 38
35 38 39 34 33 36 37 32
32 37 36 33 34 39 38 35
38 35 34 39 36 33 32 37
25 28 29 24 27 30 31 26
31 26 27 30 29 24 25 28
28 25 24 29 30 27 26 31
26 31 30 27 24 29 28 25
29 24 25 28 31 26 27 30
27 30 31 26 25 28 29 24
24 29 28 25 26 31 30 27
30 27 26 31 28 25 24 29
17 20 21 16 19 22 23 18
23 18 19 22 21 16 17 20
20 17 16 21 22 19 18 23
18 23 22 19 16 21 20 17
21 16 17 20 23 18 19 22
19 22 23 18 17 20 21 16
16 21 20 17 18 23 22 19
22 19 18 23 20 17 16 21
33 36 37 32 35 38 39 34
39 34 35 38 37 32 33 36
36 33 32 37 38 35 34 39
34 39 38 35 32 37 36 33
37 32 33 36 39 34 35 38
35 38 39 34 33 36 37 32
32 37 36 33 34 39 38 35
38 35 34 39 36 33 32 37
25 28 29 24 27 30 31 26
31 26 27 30 29 24 25 28
28 25 24 29 30 27 26 31
26 31 30 27 24 29 28 25
29 24 25 28 31 26 27 30
27 30 31 26 25 28 29 24
24 29 28 25 26 31 30 27
30 27 26 31 28 25 24 29
17 20 21 16 19 22 23 18
23 18 19 22 21 16 17 20
20 17 16 21 22 19 18 23
18 23 22 19 16 21 20 17
21 16 17 20 23 18 19 22
19 22 23 18 17 20 21 16
16 21 20 17 18 23 22 19
22 19 18 23 20 17 16 21
9 12 13 8 11 14 15 10
15 10 11 14 13 8 9 12
12 9 8 13 14 11 10 15
10 15 14 11 8 13 12 9
13 8 9 12 15 10 11 14
11 14 15 10 9 12 13 8
8 13 12 9 10 15 14 11
14 11 10 15 12 9 8 13
1 4 5 0 3 6 7 2
7 2 3 6 5 0 1 4
4 1 0 5 6 3 2 7
2 7 6 3 0 5 4 1
5 0 1 4 7 2 3 6
3 6 7 2 1 4 5 0
0 5 4 1 2 7 6 3
6 3 2 7 4 1 0 5

Attachment 8.9.2 shows the resulting order 40 Composed Pan Magic Square based on the Self Orthogonal Composed Pan Magic Latin Square shown above.

8.9.3 Composed Ultra Magic Squares

Order 8 Self Orthogonal Ultra Magic Latin Sub Squares can be constructed based on the sub series:

    {0, 1 ... 7}, {8, 9 ... 15}, {16, 17 ... 23}, {24, 25 ... 31} and {32, 33 ... 39}

with respectively the magic constants s8 = 28, 92, 156, 220 and 284

Sqrs8
32 16 0 24 8
0 24 8 32 16
8 32 16 0 24
16 0 24 8 32
24 8 32 16 0

The order 5 Self Orthogonal Ultra Magic Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A shown below.

A, Ultra Magic
32 39 36 35 33 38 37 34
38 33 34 37 39 32 35 36
39 32 35 36 38 33 34 37
33 38 37 34 32 39 36 35
36 35 32 39 37 34 33 38
34 37 38 33 35 36 39 32
35 36 39 32 34 37 38 33
37 34 33 38 36 35 32 39
16 23 20 19 17 22 21 18
22 17 18 21 23 16 19 20
23 16 19 20 22 17 18 21
17 22 21 18 16 23 20 19
20 19 16 23 21 18 17 22
18 21 22 17 19 20 23 16
19 20 23 16 18 21 22 17
21 18 17 22 20 19 16 23
0 7 4 3 1 6 5 2
6 1 2 5 7 0 3 4
7 0 3 4 6 1 2 5
1 6 5 2 0 7 4 3
4 3 0 7 5 2 1 6
2 5 6 1 3 4 7 0
3 4 7 0 2 5 6 1
5 2 1 6 4 3 0 7
24 31 28 27 25 30 29 26
30 25 26 29 31 24 27 28
31 24 27 28 30 25 26 29
25 30 29 26 24 31 28 27
28 27 24 31 29 26 25 30
26 29 30 25 27 28 31 24
27 28 31 24 26 29 30 25
29 26 25 30 28 27 24 31
8 15 12 11 9 14 13 10
14 9 10 13 15 8 11 12
15 8 11 12 14 9 10 13
9 14 13 10 8 15 12 11
12 11 8 15 13 10 9 14
10 13 14 9 11 12 15 8
11 12 15 8 10 13 14 9
13 10 9 14 12 11 8 15
0 7 4 3 1 6 5 2
6 1 2 5 7 0 3 4
7 0 3 4 6 1 2 5
1 6 5 2 0 7 4 3
4 3 0 7 5 2 1 6
2 5 6 1 3 4 7 0
3 4 7 0 2 5 6 1
5 2 1 6 4 3 0 7
24 31 28 27 25 30 29 26
30 25 26 29 31 24 27 28
31 24 27 28 30 25 26 29
25 30 29 26 24 31 28 27
28 27 24 31 29 26 25 30
26 29 30 25 27 28 31 24
27 28 31 24 26 29 30 25
29 26 25 30 28 27 24 31
8 15 12 11 9 14 13 10
14 9 10 13 15 8 11 12
15 8 11 12 14 9 10 13
9 14 13 10 8 15 12 11
12 11 8 15 13 10 9 14
10 13 14 9 11 12 15 8
11 12 15 8 10 13 14 9
13 10 9 14 12 11 8 15
32 39 36 35 33 38 37 34
38 33 34 37 39 32 35 36
39 32 35 36 38 33 34 37
33 38 37 34 32 39 36 35
36 35 32 39 37 34 33 38
34 37 38 33 35 36 39 32
35 36 39 32 34 37 38 33
37 34 33 38 36 35 32 39
16 23 20 19 17 22 21 18
22 17 18 21 23 16 19 20
23 16 19 20 22 17 18 21
17 22 21 18 16 23 20 19
20 19 16 23 21 18 17 22
18 21 22 17 19 20 23 16
19 20 23 16 18 21 22 17
21 18 17 22 20 19 16 23
8 15 12 11 9 14 13 10
14 9 10 13 15 8 11 12
15 8 11 12 14 9 10 13
9 14 13 10 8 15 12 11
12 11 8 15 13 10 9 14
10 13 14 9 11 12 15 8
11 12 15 8 10 13 14 9
13 10 9 14 12 11 8 15
32 39 36 35 33 38 37 34
38 33 34 37 39 32 35 36
39 32 35 36 38 33 34 37
33 38 37 34 32 39 36 35
36 35 32 39 37 34 33 38
34 37 38 33 35 36 39 32
35 36 39 32 34 37 38 33
37 34 33 38 36 35 32 39
16 23 20 19 17 22 21 18
22 17 18 21 23 16 19 20
23 16 19 20 22 17 18 21
17 22 21 18 16 23 20 19
20 19 16 23 21 18 17 22
18 21 22 17 19 20 23 16
19 20 23 16 18 21 22 17
21 18 17 22 20 19 16 23
0 7 4 3 1 6 5 2
6 1 2 5 7 0 3 4
7 0 3 4 6 1 2 5
1 6 5 2 0 7 4 3
4 3 0 7 5 2 1 6
2 5 6 1 3 4 7 0
3 4 7 0 2 5 6 1
5 2 1 6 4 3 0 7
24 31 28 27 25 30 29 26
30 25 26 29 31 24 27 28
31 24 27 28 30 25 26 29
25 30 29 26 24 31 28 27
28 27 24 31 29 26 25 30
26 29 30 25 27 28 31 24
27 28 31 24 26 29 30 25
29 26 25 30 28 27 24 31
16 23 20 19 17 22 21 18
22 17 18 21 23 16 19 20
23 16 19 20 22 17 18 21
17 22 21 18 16 23 20 19
20 19 16 23 21 18 17 22
18 21 22 17 19 20 23 16
19 20 23 16 18 21 22 17
21 18 17 22 20 19 16 23
0 7 4 3 1 6 5 2
6 1 2 5 7 0 3 4
7 0 3 4 6 1 2 5
1 6 5 2 0 7 4 3
4 3 0 7 5 2 1 6
2 5 6 1 3 4 7 0
3 4 7 0 2 5 6 1
5 2 1 6 4 3 0 7
24 31 28 27 25 30 29 26
30 25 26 29 31 24 27 28
31 24 27 28 30 25 26 29
25 30 29 26 24 31 28 27
28 27 24 31 29 26 25 30
26 29 30 25 27 28 31 24
27 28 31 24 26 29 30 25
29 26 25 30 28 27 24 31
8 15 12 11 9 14 13 10
14 9 10 13 15 8 11 12
15 8 11 12 14 9 10 13
9 14 13 10 8 15 12 11
12 11 8 15 13 10 9 14
10 13 14 9 11 12 15 8
11 12 15 8 10 13 14 9
13 10 9 14 12 11 8 15
32 39 36 35 33 38 37 34
38 33 34 37 39 32 35 36
39 32 35 36 38 33 34 37
33 38 37 34 32 39 36 35
36 35 32 39 37 34 33 38
34 37 38 33 35 36 39 32
35 36 39 32 34 37 38 33
37 34 33 38 36 35 32 39
24 31 28 27 25 30 29 26
30 25 26 29 31 24 27 28
31 24 27 28 30 25 26 29
25 30 29 26 24 31 28 27
28 27 24 31 29 26 25 30
26 29 30 25 27 28 31 24
27 28 31 24 26 29 30 25
29 26 25 30 28 27 24 31
8 15 12 11 9 14 13 10
14 9 10 13 15 8 11 12
15 8 11 12 14 9 10 13
9 14 13 10 8 15 12 11
12 11 8 15 13 10 9 14
10 13 14 9 11 12 15 8
11 12 15 8 10 13 14 9
13 10 9 14 12 11 8 15
32 39 36 35 33 38 37 34
38 33 34 37 39 32 35 36
39 32 35 36 38 33 34 37
33 38 37 34 32 39 36 35
36 35 32 39 37 34 33 38
34 37 38 33 35 36 39 32
35 36 39 32 34 37 38 33
37 34 33 38 36 35 32 39
16 23 20 19 17 22 21 18
22 17 18 21 23 16 19 20
23 16 19 20 22 17 18 21
17 22 21 18 16 23 20 19
20 19 16 23 21 18 17 22
18 21 22 17 19 20 23 16
19 20 23 16 18 21 22 17
21 18 17 22 20 19 16 23
0 7 4 3 1 6 5 2
6 1 2 5 7 0 3 4
7 0 3 4 6 1 2 5
1 6 5 2 0 7 4 3
4 3 0 7 5 2 1 6
2 5 6 1 3 4 7 0
3 4 7 0 2 5 6 1
5 2 1 6 4 3 0 7

Attachment 8.9.3 shows the resulting order 40 Composed Ultra Magic Square based on the Self Orthogonal Composed Ultra Magic Latin Square shown above.

8.10   Composed Latin Squares (41 x 41)

Order 8 Self orthogonal Latin Diagonal Squares can be used to construct order 41 Self Orthogonal Composed Latin Diagonal Squares.

The required order 8 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:

    {0, 1 ... 7}, {8, 9 ... 15}, {16, 17 ... 23}, {24}, {25, 26 ... 32} and {33, 34 ... 40}

with respectively the magic constants s8 = 28, 92, 156, 228 and 292

Sqrs8
33 25 0 16 8
16 8 25 0 33
8 0 16 33 25
0 33 8 25 16
25 16 33 8 0

The order 5 Self orthogonal Latin Diagonal Square shown above is based on the first elements of the Sub Squares, and has been used as a guideline for the construction of an order 41 square A as shown below.

A
33 35 37 24 39 40 38 36 25 32 31 30 26 27 28 29 0 7 6 5 1 2 3 4 34 16 23 22 21 17 18 19 20 8 15 14 13 9 10 11 12
36 34 40 37 38 39 24 33 31 26 25 28 32 29 30 27 6 1 0 3 7 4 5 2 35 22 17 16 19 23 20 21 18 14 9 8 11 15 12 13 10
34 39 35 40 33 24 36 38 29 28 27 26 30 31 32 25 4 3 2 1 5 6 7 0 37 20 19 18 17 21 22 23 16 12 11 10 9 13 14 15 8
38 33 24 36 34 35 40 37 26 31 32 29 25 28 27 30 1 6 7 4 0 3 2 5 39 17 22 23 20 16 19 18 21 9 14 15 12 8 11 10 13
24 40 39 38 37 36 35 34 27 30 29 32 28 25 26 31 2 5 4 7 3 0 1 6 33 18 21 20 23 19 16 17 22 10 13 12 15 11 8 9 14
35 37 34 39 40 38 33 24 32 25 26 27 31 30 29 28 7 0 1 2 6 5 4 3 36 23 16 17 18 22 21 20 19 15 8 9 10 14 13 12 11
37 36 38 33 24 34 39 35 30 27 28 25 29 32 31 26 5 2 3 0 4 7 6 1 40 21 18 19 16 20 23 22 17 13 10 11 8 12 15 14 9
39 24 33 35 36 37 34 40 28 29 30 31 27 26 25 32 3 4 5 6 2 1 0 7 38 19 20 21 22 18 17 16 23 11 12 13 14 10 9 8 15
16 23 22 21 17 18 19 20 8 10 12 24 14 15 13 11 25 32 31 30 26 27 28 29 9 0 7 6 5 1 2 3 4 33 40 39 38 34 35 36 37
22 17 16 19 23 20 21 18 11 9 15 12 13 14 24 8 31 26 25 28 32 29 30 27 10 6 1 0 3 7 4 5 2 39 34 33 36 40 37 38 35
20 19 18 17 21 22 23 16 9 14 10 15 8 24 11 13 29 28 27 26 30 31 32 25 12 4 3 2 1 5 6 7 0 37 36 35 34 38 39 40 33
17 22 23 20 16 19 18 21 13 8 24 11 9 10 15 12 26 31 32 29 25 28 27 30 14 1 6 7 4 0 3 2 5 34 39 40 37 33 36 35 38
18 21 20 23 19 16 17 22 24 15 14 13 12 11 10 9 27 30 29 32 28 25 26 31 8 2 5 4 7 3 0 1 6 35 38 37 40 36 33 34 39
23 16 17 18 22 21 20 19 10 12 9 14 15 13 8 24 32 25 26 27 31 30 29 28 11 7 0 1 2 6 5 4 3 40 33 34 35 39 38 37 36
21 18 19 16 20 23 22 17 12 11 13 8 24 9 14 10 30 27 28 25 29 32 31 26 15 5 2 3 0 4 7 6 1 38 35 36 33 37 40 39 34
19 20 21 22 18 17 16 23 14 24 8 10 11 12 9 15 28 29 30 31 27 26 25 32 13 3 4 5 6 2 1 0 7 36 37 38 39 35 34 33 40
8 15 14 13 9 10 11 12 0 7 6 5 1 2 3 4 16 18 20 24 22 23 21 19 17 33 40 39 38 34 35 36 37 25 32 31 30 26 27 28 29
14 9 8 11 15 12 13 10 6 1 0 3 7 4 5 2 19 17 23 20 21 22 24 16 18 39 34 33 36 40 37 38 35 31 26 25 28 32 29 30 27
12 11 10 9 13 14 15 8 4 3 2 1 5 6 7 0 17 22 18 23 16 24 19 21 20 37 36 35 34 38 39 40 33 29 28 27 26 30 31 32 25
9 14 15 12 8 11 10 13 1 6 7 4 0 3 2 5 21 16 24 19 17 18 23 20 22 34 39 40 37 33 36 35 38 26 31 32 29 25 28 27 30
10 13 12 15 11 8 9 14 2 5 4 7 3 0 1 6 24 23 22 21 20 19 18 17 16 35 38 37 40 36 33 34 39 27 30 29 32 28 25 26 31
15 8 9 10 14 13 12 11 7 0 1 2 6 5 4 3 18 20 17 22 23 21 16 24 19 40 33 34 35 39 38 37 36 32 25 26 27 31 30 29 28
13 10 11 8 12 15 14 9 5 2 3 0 4 7 6 1 20 19 21 16 24 17 22 18 23 38 35 36 33 37 40 39 34 30 27 28 25 29 32 31 26
11 12 13 14 10 9 8 15 3 4 5 6 2 1 0 7 22 24 16 18 19 20 17 23 21 36 37 38 39 35 34 33 40 28 29 30 31 27 26 25 32
40 38 36 34 35 33 37 39 15 13 11 9 10 8 12 14 23 21 19 17 18 16 20 22 24 26 28 32 30 31 29 27 25 1 3 7 5 6 4 2 0
0 7 6 5 1 2 3 4 33 40 39 38 34 35 36 37 8 15 14 13 9 10 11 12 27 25 31 28 29 30 32 24 26 16 23 22 21 17 18 19 20
6 1 0 3 7 4 5 2 39 34 33 36 40 37 38 35 14 9 8 11 15 12 13 10 25 30 26 31 24 32 27 29 28 22 17 16 19 23 20 21 18
4 3 2 1 5 6 7 0 37 36 35 34 38 39 40 33 12 11 10 9 13 14 15 8 29 24 32 27 25 26 31 28 30 20 19 18 17 21 22 23 16
1 6 7 4 0 3 2 5 34 39 40 37 33 36 35 38 9 14 15 12 8 11 10 13 32 31 30 29 28 27 26 25 24 17 22 23 20 16 19 18 21
2 5 4 7 3 0 1 6 35 38 37 40 36 33 34 39 10 13 12 15 11 8 9 14 26 28 25 30 31 29 24 32 27 18 21 20 23 19 16 17 22
7 0 1 2 6 5 4 3 40 33 34 35 39 38 37 36 15 8 9 10 14 13 12 11 28 27 29 24 32 25 30 26 31 23 16 17 18 22 21 20 19
5 2 3 0 4 7 6 1 38 35 36 33 37 40 39 34 13 10 11 8 12 15 14 9 30 32 24 26 27 28 25 31 29 21 18 19 16 20 23 22 17
3 4 5 6 2 1 0 7 36 37 38 39 35 34 33 40 11 12 13 14 10 9 8 15 31 29 27 25 26 24 28 30 32 19 20 21 22 18 17 16 23
25 32 31 30 26 27 28 29 16 23 22 21 17 18 19 20 33 40 39 38 34 35 36 37 2 8 15 14 13 9 10 11 12 0 6 3 4 5 7 24 1
31 26 25 28 32 29 30 27 22 17 16 19 23 20 21 18 39 34 33 36 40 37 38 35 0 14 9 8 11 15 12 13 10 5 1 6 24 7 2 4 3
29 28 27 26 30 31 32 25 20 19 18 17 21 22 23 16 37 36 35 34 38 39 40 33 4 12 11 10 9 13 14 15 8 24 7 2 0 1 6 3 5
26 31 32 29 25 28 27 30 17 22 23 20 16 19 18 21 34 39 40 37 33 36 35 38 7 9 14 15 12 8 11 10 13 6 5 4 3 2 1 0 24
27 30 29 32 28 25 26 31 18 21 20 23 19 16 17 22 35 38 37 40 36 33 34 39 1 10 13 12 15 11 8 9 14 3 0 5 6 4 24 7 2
32 25 26 27 31 30 29 28 23 16 17 18 22 21 20 19 40 33 34 35 39 38 37 36 3 15 8 9 10 14 13 12 11 2 4 24 7 0 5 1 6
30 27 28 25 29 32 31 26 21 18 19 16 20 23 22 17 38 35 36 33 37 40 39 34 5 13 10 11 8 12 15 14 9 7 24 1 2 3 0 6 4
28 29 30 31 27 26 25 32 19 20 21 22 18 17 16 23 36 37 38 39 35 34 33 40 6 11 12 13 14 10 9 8 15 4 2 0 1 24 3 5 7
  • Attachment 8.10.1 illustrates and describes the construction of the order 41 Self Orthogonal Composed Latin Diagonal Square shown above.

  • Attachment 8.10.2 shows the resulting order 41 Composed Simple Magic Square based on subject order 36 Self Orthogonal Composed Latin Diagonal Square.

8.11   Composed Latin Squares (44 x 44)

A combination of order 8 and 9 (Inlaid) Self Orthogonal Latin Diagonal Squares can be used to construct order 44 Self Orthogonal Composed Latin Diagonal Squares.

Order 8 Self Orthogonal Latin Sub Squares can be constructed based on the sub series:

    {0 ... 7}, {8 ... 15}, {16 ... 23}, {28 ... 35} and (36 ... 43)

with respectively the magic constants s8 = 28, 92, 156, 252, and 316

Sqrs8
36 16 0 28 8
0 28 8 36 16
8 36 16 0 28
16 0 28 8 36
28 8 36 16 0
Aux4
24 26 27 25
27 25 24 26
25 27 26 24
26 24 25 27

The order 5 Self orthogonal Latin Diagonal Square left above (Sqrs8), is based on the first elements of the order 8 Sub Squares.

The order 4 Self orthogonal Latin Diagonal Square right above (Aux4), is based on the sub series {24, 25, 26, 27}.

The construction of an order 44 Self Orthogonal Composed Latin Diagonal Squares A2 can be summarised as follows:

  • Construct an order 40 Self Orthogonal Composed Latin Diagonal Square A1, using the order 5 Self orthogonal Latin Diagonal Square Sqrs8 as a guideline for the construction, as shown in Attachment 8.11.1.

  • Construct an intermediate order 44 square by adding the Auxilliary Square Aux4 and the related rows and columns, to the order 40 Self Orthogonal Composed Latin Diagonal Square as shown in Attachment 8.11.2.

  • The Intermediate Square has to be completed and transformed to a Self Orthogonal Latin Diagonal Square, which can be achieved by means of a set of twenty order 9 Auxiliary Latin Diagonal Squares shown in Attachment 8.11.3.

    The twenty Auxiliary Squares are based on the five sub series defined above and the series {24, 25, 26, 27}.

  • Replace the applicable Sub Squares (of the Intermediate Square) together with the corresponding sections of the 'Crosses' by the contents of these Auxiliary Squares as shown in Attachment 8.11.4.

    The order 44 Self Orthogonal Composed Latin Diagonal Square A2 is ready to be used for the construction of an order 44 Composed Simple Magic Square.

Attachment 8.11.5 shows the resulting order 44 Composed Simple Magic Square based on subject order 44 Composed Self Orthogonal Latin Diagonal Square.

8.12   Composed Latin Squares (56 x 56)

Order 8 Self Orthogonal Latin Diagonal Squares can be used to construct order 56 Self Orthogonal Composed Latin Diagonal Squares.

8.12.1 Composed Associated Squares

Order 8 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:

    {0 ... 7}, {8 ... 15}, {16 ... 23}, {24 ... 31}, {32 ... 39}, (40 ... 47) and (48 ... 55)

with respectively the magic constants s8 = 28, 92, 156, 220, 284, 348 and 412

Sqrs8
16 32 24 0 40 48 8
48 8 40 32 16 0 24
8 16 0 40 32 24 48
32 40 48 24 0 8 16
0 24 16 8 48 32 40
24 48 32 16 8 40 0
40 0 8 48 24 16 32

The order 8 Self Orthogonal Associated Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A as shown in Attachment 8.12.11

Attachment 8.12.12 shows the resulting order 56 Composed Associated Magic Square based on subject order 56 Composed Associated Self Orthogonal Latin Diagonal Square.

8.12.2 Composed Pan Magic Squares

Order 8 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:

    {0 ... 7}, {8 ... 15}, {16 ... 23}, {24 ... 31}, {32 ... 39}, (40 ... 47) and (48 ... 55)

with respectively the magic constants s8 = 28, 92, 156, 220, 284, 348 and 412

Sqrs8
33 25 17 9 1 49 41
17 9 1 49 41 33 25
1 49 41 33 25 17 9
41 33 25 17 9 1 49
25 17 9 1 49 41 33
9 1 49 41 33 25 17
49 41 33 25 17 9 1

The order 8 Self Orthogonal Pan Magic Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A as shown in Attachment 8.12.21

Attachment 8.12.22 shows the resulting order 56 Composed Pan Magic Square based on subject order 56 Composed Pan Magic Self Orthogonal Latin Diagonal Square.

8.12.3 Composed Ultra Magic Squares

Order 8 Self Orthogonal Ultra Magic Latin Sub Squares can be constructed based on the sub series:

    {0 ... 7}, {8 ... 15}, {16 ... 23}, {24 ... 31}, {32 ... 39}, (40 ... 47) and (48 ... 55)

with respectively the magic constants s8 = 28, 92, 156, 220, 284, 348 and 412

Sqrs8
48 40 24 8 0 32 16
24 8 0 32 16 48 40
0 32 16 48 40 24 8
16 48 40 24 8 0 32
40 24 8 0 32 16 48
8 0 32 16 48 40 24
32 16 48 40 24 8 0

The order 8 Self Orthogonal Ultra Magic Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A as shown in Attachment 8.12.31

Attachment 8.12.32 shows the resulting order 56 Composed Ultra Magic Square based on subject order 56 Composed Ultra Magic Self Orthogonal Latin Diagonal Square.

8.13   Composed Latin Squares (57 x 57)

Order 8 Self orthogonal Latin Diagonal Squares can be used to construct order 57 Self Orthogonal Composed Latin Diagonal Squares.

Order 8 Self Orthogonal Latin Sub Squares can be constructed based on the sub series:

    {0 ... 7}, {8 ... 15}, {16 ... 23}, {24 ... 31}, {32}, {33 ... 40}, (41 ... 48) and (49 ... 56)

with respectively the magic constants s8 = 28, 92, 156, 220, 292, 356 and 420

Sqrs8
49 41 24 8 0 33 16
24 8 0 33 16 49 41
0 33 16 49 41 24 8
16 49 41 24 8 0 33
41 24 8 0 33 16 49
8 0 33 16 49 41 24
33 16 49 41 24 8 0

The order 8 Self Orthogonal Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of an order 56 square A1 as shown in Attachment 8.13.1

The construction of an order 57 Self Orthogonal Composed Latin Diagonal Squares A2 can be summarised as follows:

  • Construct an intermediate order 57 square by adding a row (33) and a column (33), to the order 56 Self Orthogonal Composed Latin Diagonal Square A1.

  • The Intermediate Square has to be transformed to a Self Orthogonal Latin Diagonal Square, by means of a set of seven order 9 Auxiliary Latin Diagonal Squares shown in Attachment 8.13.2.

    The seven Auxiliary Squares are based on the seven sub series defined above and the number 33.

  • Replace the Diagonal Sub Squares (of the Intermediate Square) together with the corresponding sections of the 'Cross' by the contents of these Auxiliary Squares as shown in Attachment 8.13.3.

    The order 57 Self Orthogonal Composed Latin Diagonal Square A2 is ready to be used for the construction of an order 57 Composed Simple Magic Square.

Attachment 8.13.4 shows the resulting order 57 Composed Simple Magic Square based on subject order 57 Composed Self Orthogonal Latin Diagonal Square.

8.14   Composed Latin Squares (64 x 64)

Order 8 Self Orthogonal Latin Diagonal Squares can be used to construct order 64 Self Orthogonal Composed Latin Diagonal Squares.

8.14.1 Composed Associated Squares

Order 8 Self Orthogonal Associated Latin Sub Squares can be constructed based on the sub series:

    {0 ... 7}, {8 ... 15}, {16 ... 23}, {24 ... 31}, {32 ... 39}, (40 ... 47), (48 ... 55) and {56 ... 63}

with respectively the magic constants s8 = 28, 92, 156, 220, 284, 348, 412 and 476

Sqrs8
0 56 48 40 8 16 24 32
48 8 0 24 56 32 40 16
32 24 16 8 40 48 56 0
8 48 56 32 0 24 16 40
16 40 32 56 24 0 8 48
56 0 8 16 48 40 32 24
40 16 24 0 32 56 48 8
24 32 40 48 16 8 0 56

The order 8 Self Orthogonal Associated Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A as shown in Attachment 8.14.11

Attachment 8.14.12 shows the resulting order 64 Composed Associated Magic Square based on subject order 64 Composed Associated Self Orthogonal Latin Diagonal Square.

8.14.2 Composed Pan Magic Squares (1)

Order 64 Self Orthogonal Composed Pan Magic and Complete Latin Diagonal Squares can be constructed based on Order 64 Self Orthogonal Composed Associated Latin Diagonal Squares as illustrated in Attachment 8.14.21 (Eulre Square A).

Sqrs8
0 56 48 40 8 16 24 32
48 8 0 24 56 32 40 16
32 24 16 8 40 48 56 0
8 48 56 32 0 24 16 40
16 40 32 56 24 0 8 48
56 0 8 16 48 40 32 24
40 16 24 0 32 56 48 8
24 32 40 48 16 8 0 56

The order 8 Self Orthogonal Associated Latin Square shown above is based on the first elements of the Sub Squares (before transformation) and has been used as a guideline for the construction of Euler Suare A.

Attachment 8.14.22 shows the resulting order 64 Composed Pan Magic and Complete Square based on subject order 64 Composed Pan Magic and Complete Self Orthogonal Latin Diagonal Suare A.

8.14.3 Composed Pan Magic Squares (2)

Order 8 Self Orthogonal Pan Magic Latin Sub Squares can be constructed based on the sub series:

    {0 ... 7}, {8 ... 15}, {16 ... 23}, {24 ... 31}, {32 ... 39}, (40 ... 47), (48 ... 55) and {56 ... 63}

with respectively the magic constants s8 = 28, 92, 156, 220, 284, 348, 412 and 476

Sqrs8
9 33 41 1 25 49 57 17
57 17 25 49 41 1 9 33
33 9 1 41 49 25 17 57
17 57 49 25 1 41 33 9
41 1 9 33 57 17 25 49
25 49 57 17 9 33 41 1
1 41 33 9 17 57 49 25
49 25 17 57 33 9 1 41

The order 8 Self Orthogonal Pan Magic Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A as shown in Attachment 8.14.31

Attachment 8.14.32 shows the resulting order 64 Composed Pan Magic Square based on subject order 64 Composed Pan Magic Self Orthogonal Latin Diagonal Square.

8.14.4 Composed Ultra Magic Squares

Order 8 Self Orthogonal Ultra Magic Latin Sub Squares can be constructed based on the sub series:

    {0 ... 7}, {8 ... 15}, {16 ... 23}, {24 ... 31}, {32 ... 39}, (40 ... 47), (48 ... 55) and {56 ... 63}

with respectively the magic constants s8 = 28, 92, 156, 220, 284, 348, 412 and 476

Sqrs8
0 56 32 24 8 48 40 16
48 8 16 40 56 0 24 32
56 0 24 32 48 8 16 40
8 48 40 16 0 56 32 24
32 24 0 56 40 16 8 48
16 40 48 8 24 32 56 0
24 32 56 0 16 40 48 8
40 16 8 48 32 24 0 56

The order 8 Self Orthogonal Ultra Magic Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of square A as shown in Attachment 8.14.41

Attachment 8.14.42 shows the resulting order 64 Composed Ultra Magic Square based on subject order 64 Composed Ultra Magic Self Orthogonal Latin Diagonal Square.

8.15   Composed Latin Squares (65 x 65)

Order 8 Self orthogonal Latin Diagonal Squares can be used to construct order 65 Self Orthogonal Composed Latin Diagonal Squares.

Order 8 Self Orthogonal Latin Sub Squares can be constructed based on the sub series:

    {0 .. 7}, {8 .. 15}, {16 .. 23}, {24 .. 31}, {32}, {33 .. 40}, (41 .. 48), (49 .. 56) and {57 .. 64}

with respectively the magic constants s8 = 28, 92, 156, 220, 292, 356, 420 and 484

Sqrs8
0 57 33 24 8 49 41 16
49 8 16 41 57 0 24 33
57 0 24 33 49 8 16 41
8 49 41 16 0 57 33 24
33 24 0 57 41 16 8 49
16 41 49 8 24 33 57 0
24 33 57 0 16 41 49 8
41 16 8 49 33 24 0 57

The order 8 Self Orthogonal Latin Square shown above is based on the first elements of the Sub Squares and has been used as a guideline for the construction of an order 64 square A1 as shown in Attachment 8.15.1

The construction of an order 65 Self Orthogonal Composed Latin Diagonal Squares A2 can be summarised as follows:

  • Construct an intermediate order 65 square by adding a row (33) and a column (33), to the order 64 Self Orthogonal Composed Latin Diagonal Square A1.

  • The Intermediate Square has to be transformed to a Self Orthogonal Latin Diagonal Square, by means of a set of eight order 9 Auxiliary Latin Diagonal Squares shown in Attachment 8.15.2.

    The eight Auxiliary Squares are based on the eight sub series defined above and the number 32.

  • Replace the Diagonal Sub Squares (of the Intermediate Square) together with the corresponding sections of the 'Center Cross' by the contents of these Auxiliary Squares as shown in Attachment 8.15.3.

    The order 65 Self Orthogonal Composed Latin Diagonal Square A2 is ready to be used for the construction of an order 65 Composed Simple Magic Square.

Attachment 8.15.4 shows the resulting order 65 Composed Simple Magic Square based on subject order 65 Composed Self Orthogonal Latin Diagonal Square.

8.16   Miscellaneous

8.16.1 Mutual Orthogonal Latin Squares (8 x 8)

The construction of 0rder 8 Magic Squares based on following Mutual Orthogonal Latin (Diagonal) Squares

  • Associated Pan Magic, Non Overlapping Sub Squares (2 x 2)
  • Complete   Pan Magic, Non Overlapping Sub Squares (2 x 2)
  • Complete   Pan Magic, Rectangular Compact
  • Associated, Partly Rectangular Compact
  • Complete  , Partly Rectangular Compact

has been deducted and discussed in Section 8.7.3.

The construction of 0rder 8 Bimagic Squares based on Mutual Orthogonal Latin (Diagonal) Squares has been discussed in Section 15.1.1 and Section 15.1.2.

8.16.2 Semi Latin Squares (8 x 8)

The construction of 0rder 8 Mutual Orthogonal Semi-Latin (Diagonal) Squares has been deducted and discussed in:

Order 8 Mutual Orthogonal Semi-Latin Composed Magic Squares have been deducted and discussed in Section 8.2.3 and Section 8.2.4.

8.16.3 Summary

The obtained results regarding the order 8 Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment

Subject

n9

Subroutine

-

-

-

-

-

Self Orth, Simple

46.448.640 

SelfOrth8c

-

Self Orth, Associated

147.456 

-

Self Orth, Pan Magic

127.488 

-

Self Orth, Pan Magic, Complete

86.016 

-

Self Orth, Compact (4 x 4)

135.168 

-

Self Orth, V type ZigZag (4-Way)

133.632 

-

Self Orth, Bent Dia (4-Way)

4608 

Attachment 8.5.4

Self Orth, Ultra Magic

768 

Attachment 8.5.3

Self Orth, Idempotent (Associated)

384 

SelfOrth8a

Attachment 8.5.56

Self Orth, Idempotent (Non-Ass)

768 

SelfOrth8a2

-

-

-

-

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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