Office Applications and Entertainment, Latin Squares

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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 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.

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

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.

8.6   Interesting Sub Collections

8.6.1 Bodered 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 with Order 4 Corner Square M = A + 8 * T(A) + [1] 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 Bordered Latin Diagonal Squares listed above,

8.11   Miscellaneous

8.11.1 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 Secion 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.

8.11.2 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.

8.11.3 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, Pan Magic

127.488 

-

Self Orth, Pan Magic, Complete

86.016 

-

Self Orth, Associated

147.456 

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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