Office Applications and Entertainment, Latin Squares

9.5   Self Orthogonal Latin Squares (9 x 9)

A Self Orthogonal Latin Square A is a Latin Square that is Orthogonal to its Transposed T(A). The transposed square T(A) can be obtained by exchanging the rows and columns of A.

If the main diagonal contains the integers {a_i, i = 1 ... 9} in natural order, the Self Orthogonal Latin Square is called Idempotent.

9.5.1 Simple Magic Squares

A construction example of a Simple Magic Square M = A + 9 * T(A) + [1] is shown below:

A 0 3 6 4 7 1 8 2 5 7 1 4 2 5 8 3 6 0 5 8 2 6 0 3 1 4 7 2 5 8 3 6 0 7 1 4 6 0 3 1 4 7 5 8 2 4 7 1 8 2 5 0 3 6 1 4 7 5 8 2 6 0 3 8 2 5 0 3 6 4 7 1 3 6 0 7 1 4 2 5 8

B = T(A) 0 7 5 2 6 4 1 8 3 3 1 8 5 0 7 4 2 6 6 4 2 8 3 1 7 5 0 4 2 6 3 1 8 5 0 7 7 5 0 6 4 2 8 3 1 1 8 3 0 7 5 2 6 4 8 3 1 7 5 0 6 4 2 2 6 4 1 8 3 0 7 5 5 0 7 4 2 6 3 1 8

M = A + 9 * B + 1 1 67 52 23 62 38 18 75 33 35 11 77 48 6 72 40 25 55 60 45 21 79 28 13 65 50 8 39 24 63 31 16 73 53 2 68 70 46 4 56 41 26 78 36 12 14 80 29 9 66 51 19 58 43 74 32 17 69 54 3 61 37 22 27 57 42 10 76 34 5 71 47 49 7 64 44 20 59 30 15 81

Unlike the order 4, 5, 7 and 8 Self Orthogonal Latin Diagonal Squares not all order 9 Self Orthogonal Latin Diagonal Squares are Double Self Orthogonal.

• For Double Self Orthogonal Latin Diagonal Squares each square has eight aspects which can be reached by means of rotation and/or reflection as illustrated in Attachment 9.5.11.
  
  
• For Single Self Orthogonal Latin Diagonal Square each square has only four aspects which can be reached by means of rotation and/or reflection as illustrated in Attachment 9.5.12.

Each Self Orthogonal Latin Diagonal Square corresponds with 9! = 362.880 Self Orthogonal Latin Diagonal Squares, which can be obtained by permutation of the integers {a_i, i = 1 ... 9}.

The total number of Self Orthogonal Latin Diagonal Squares is 224.832 * 362.880 = 81.587.036.160 as previously published by Harry White and confirmed by Eduard Vatutin (ref. OEIS A287762).

A Base of 23232 Associated Idempotent Squares has been found in Section 9.5.2 below.

Interesting Sub Collections, based on this Base, can be generated quite fast with routine SelfOrth9c.

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 (10 - n). The possible number of transformations is 2⁴ = 16
  It should be noted that for each square the 180^o rotated aspect is included in this collection.
  
  
• Any permutation can be applied to the lines 1, 2, 3, 4 provided that the same permutation is applied to the lines 9, 8, 7, 6. The possible number of transformations is 4! = 24.

The resulting number of transformations, excluding the 180^o rotated aspects, is 16/2 * 24 = 192, which are shown in Attachment 9.5.13.

9.5.2 Associated Magic Squares

A construction example of an Associated Magic Square M = A + 9 * T(A) + [1] is shown below:

A 0 8 7 6 5 4 3 2 1 6 1 4 8 2 7 0 5 3 3 5 2 0 7 1 8 4 6 4 2 1 3 0 6 5 8 7 8 7 6 5 4 3 2 1 0 1 0 3 2 8 5 7 6 4 2 4 0 7 1 8 6 3 5 5 3 8 1 6 0 4 7 2 7 6 5 4 3 2 1 0 8

B = T(A) 0 6 3 4 8 1 2 5 7 8 1 5 2 7 0 4 3 6 7 4 2 1 6 3 0 8 5 6 8 0 3 5 2 7 1 4 5 2 7 0 4 8 1 6 3 4 7 1 6 3 5 8 0 2 3 0 8 5 2 7 6 4 1 2 5 4 8 1 6 3 7 0 1 3 6 7 0 4 5 2 8

M = A + 9 * B + 1 1 63 35 43 78 14 22 48 65 79 11 50 27 66 8 37 33 58 67 42 21 10 62 29 9 77 52 59 75 2 31 46 25 69 18 44 54 26 70 6 41 76 12 56 28 38 64 13 57 36 51 80 7 23 30 5 73 53 20 72 61 40 15 24 49 45 74 16 55 32 71 3 17 34 60 68 4 39 47 19 81

A collection of 23232 Associated Idempotent Self Orthogonal Latin Diagonal Squares, could be generated within about two hours (ref. SelfOrth9a).

The total number of order 9 Self Orthogonal Associated Magic Latin Diagonal Squares is 384 * 23232 = 8.921.088 and can be generated within 5170 seconds (ref. SelfOrth9c).

9.5.3 Ultra Magic Squares

A construction example of an Ultra Magic Square M = A + 9 * T(A) + [1] is shown below:

A 0 7 4 2 6 3 1 8 5 6 1 5 8 0 4 7 2 3 7 4 2 6 3 1 8 5 0 4 2 8 3 1 7 5 0 6 1 5 6 0 4 8 2 3 7 2 8 3 1 7 5 0 6 4 8 3 0 7 5 2 6 4 1 5 6 1 4 8 0 3 7 2 3 0 7 5 2 6 4 1 8

B = T(A) 0 6 7 4 1 2 8 5 3 7 1 4 2 5 8 3 6 0 4 5 2 8 6 3 0 1 7 2 8 6 3 0 1 7 4 5 6 0 3 1 4 7 5 8 2 3 4 1 7 8 5 2 0 6 1 7 8 5 2 0 6 3 4 8 2 5 0 3 6 4 7 1 5 3 0 6 7 4 1 2 8

M = A + 9 * B + 1 1 62 68 39 16 22 74 54 33 70 11 42 27 46 77 35 57 4 44 50 21 79 58 29 9 15 64 23 75 63 31 2 17 69 37 52 56 6 34 10 41 72 48 76 26 30 45 13 65 80 51 19 7 59 18 67 73 53 24 3 61 32 38 78 25 47 5 36 55 40 71 12 49 28 8 60 66 43 14 20 81

The total number of order 9 Self Orthogonal Ultra Magic Latin Diagonal Squares is 1296 and can be generated within 863 seconds (ref. SelfOrth9c).

9.5.4 Magic Squares, Compact, Pan Magic

The sub collection described in this section can be determined based on the defining properties rather then on permutations of the integers of Self Orthogonal Latin Diagonal Base Squares.

A construction example of a Compact Pan Magic Square M = A + 9 * T(A) + [1] is shown below:

A 1 7 4 0 6 3 2 8 5 0 6 3 2 8 5 1 7 4 2 8 5 1 7 4 0 6 3 4 1 7 3 0 6 5 2 8 3 0 6 5 2 8 4 1 7 5 2 8 4 1 7 3 0 6 7 4 1 6 3 0 8 5 2 6 3 0 8 5 2 7 4 1 8 5 2 7 4 1 6 3 0

B = T(A) 1 0 2 4 3 5 7 6 8 7 6 8 1 0 2 4 3 5 4 3 5 7 6 8 1 0 2 0 2 1 3 5 4 6 8 7 6 8 7 0 2 1 3 5 4 3 5 4 6 8 7 0 2 1 2 1 0 5 4 3 8 7 6 8 7 6 2 1 0 5 4 3 5 4 3 8 7 6 2 1 0

M = A + 9 * B + 1 11 8 23 37 34 49 66 63 78 64 61 76 12 9 24 38 35 50 39 36 51 65 62 77 10 7 22 5 20 17 31 46 43 60 75 72 58 73 70 6 21 18 32 47 44 33 48 45 59 74 71 4 19 16 26 14 2 52 40 28 81 69 57 79 67 55 27 15 3 53 41 29 54 42 30 80 68 56 25 13 1

The total number of order 9 Self Orthogonal Compact Pan Magic Latin Diagonal Squares is 10368 and can be constructed based on the 3456 Ternary Compact Pan Magic Squares as deducted and discussed in Section 9.5.3.

Subject Compact Pan Magic Latin Diagonal Squares could be generated within about 8 hours (ref. SelfOrth9b).

9.5.5 Magic Squares, Compact, Pan Magic
      Third-rows and Third-columns Summing to s1/3

A construction example of a Compact Pan Magic Square M = A + 9 * T(A) + [1] for which the elements of every third-row and third-column sum to s1/3 is shown below:

A 2 7 3 1 6 5 0 8 4 6 5 1 8 4 0 7 3 2 4 0 8 3 2 7 5 1 6 8 4 0 7 3 2 6 5 1 3 2 7 5 1 6 4 0 8 1 6 5 0 8 4 2 7 3 5 1 6 4 0 8 3 2 7 0 8 4 2 7 3 1 6 5 7 3 2 6 5 1 8 4 0

B = T(A) 2 6 4 8 3 1 5 0 7 7 5 0 4 2 6 1 8 3 3 1 8 0 7 5 6 4 2 1 8 3 7 5 0 4 2 6 6 4 2 3 1 8 0 7 5 5 0 7 2 6 4 8 3 1 0 7 5 6 4 2 3 1 8 8 3 1 5 0 7 2 6 4 4 2 6 1 8 3 7 5 0

M = A + 9 * B + 1 21 62 40 74 34 15 46 9 68 70 51 2 45 23 55 17 76 30 32 10 81 4 66 53 60 38 25 18 77 28 71 49 3 43 24 56 58 39 26 33 11 79 5 64 54 47 7 69 19 63 41 75 35 13 6 65 52 59 37 27 31 12 80 73 36 14 48 8 67 20 61 42 44 22 57 16 78 29 72 50 1

The total number of subject order 9 Self Orthogonal Magic Squares is 576 and can be filtered from the collection of 10368 Compact Pan Magic Self Orthogonal Latin Diagonal Squares as deducted in Section 9.5.4 above.

Alternatively this sub collection can be determined based on the defining properties and generated within 12 seconds (ref. SelfOrth9e).

Attachment 9.6.6 shows a construction example of a Compact Ultra Magic Square for which the elements of every third-row and third-column sum to s1/3.

9.6   Interesting Sub Collections (Associated)

The following Sub Collections of Associated Magic Squares can be found based on the Associated Idempotent Self Orthogonal Latin Diagonal Squares found in Section 9.5.2 above.

9.6.1 Bordered Magic Squares

A construction example of a Bordered Magic Square M = A + 9 * T(A) + [1] is shown below:

A 0 8 5 1 6 3 7 4 2 4 1 8 2 7 0 6 3 5 2 6 3 7 5 4 1 8 0 5 3 4 6 0 8 2 1 7 7 2 0 3 4 5 8 6 1 1 7 6 0 8 2 4 5 3 8 0 7 4 3 1 5 2 6 3 5 2 8 1 6 0 7 4 6 4 1 5 2 7 3 0 8

B = T(A) 0 4 2 5 7 1 8 3 6 8 1 6 3 2 7 0 5 4 5 8 3 4 0 6 7 2 1 1 2 7 6 3 0 4 8 5 6 7 5 0 4 8 3 1 2 3 0 4 8 5 2 1 6 7 7 6 1 2 8 4 5 0 3 4 3 8 1 6 5 2 7 0 2 5 0 7 1 3 6 4 8

M = A + 9 * B + 1 1 45 24 47 70 13 80 32 57 77 11 63 30 26 64 7 49 42 48 79 31 44 6 59 65 27 10 15 22 68 61 28 9 39 74 53 62 66 46 4 41 78 36 16 20 29 8 43 73 54 21 14 60 67 72 55 17 23 76 38 51 3 34 40 33 75 18 56 52 19 71 5 25 50 2 69 12 35 58 37 81

The total number of order 9 Self Orthogonal Bordered Latin Diagonal Squares is 5808 and can be generated within 860 seconds (ref. SelfOrth9c).

Attachment 9.6.1 shows a construction example of a Bordered Magic Square with Diamond Inlay (3 x 3).

9.6.2 Inlaid Magic Squares, Diamond (3 x 3)

A construction example of an Inlaid Magic Square M = A + 9 * T(A) + [1] with 3 x 3 Diamond Inlay is shown below:

A 3 5 2 8 7 4 0 1 6 7 6 3 0 5 1 4 8 2 4 8 0 3 6 2 1 5 7 5 2 6 1 8 3 7 4 0 0 1 5 2 4 6 3 7 8 8 4 1 5 0 7 2 6 3 1 3 7 6 2 5 8 0 4 6 0 4 7 3 8 5 2 1 2 7 8 4 1 0 6 3 5

B = T(A) 3 7 4 5 0 8 1 6 2 5 6 8 2 1 4 3 0 7 2 3 0 6 5 1 7 4 8 8 0 3 1 2 5 6 7 4 7 5 6 8 4 0 2 3 1 4 1 2 3 6 7 5 8 0 0 4 1 7 3 2 8 5 6 1 8 5 4 7 6 0 2 3 6 2 7 0 8 3 4 1 5

M = A + 9 * B + 1 31 69 39 54 8 77 10 56 25 53 61 76 19 15 38 32 9 66 23 36 1 58 52 12 65 42 80 78 3 34 11 27 49 62 68 37 64 47 60 75 41 7 22 35 18 45 14 20 33 55 71 48 79 4 2 40 17 70 30 24 81 46 59 16 73 50 44 67 63 6 21 29 57 26 72 5 74 28 43 13 51

The total number of order 9 Self Orthogonal Inlaid Latin Diagonal Squares with 3 x 3 Diamond Inlay is 56576 and can be generated within 860 seconds (ref. SelfOrth9c).

9.6.3 Inlaid Magic Squares, Diamond (4 x 4)

A construction example of an Inlaid Magic Square M = A + 9 * T(A) + [1] with 4 x 4 Diamond Inlay is shown below:

A 1 8 0 2 7 3 4 6 5 6 0 8 7 2 5 3 1 4 5 1 3 4 0 7 6 2 8 8 5 7 6 3 0 2 4 1 2 3 1 0 4 8 7 5 6 7 4 6 8 5 2 1 3 0 0 6 2 1 8 4 5 7 3 4 7 5 3 6 1 0 8 2 3 2 4 5 1 6 8 0 7

B = T(A) 1 6 5 8 2 7 0 4 3 8 0 1 5 3 4 6 7 2 0 8 3 7 1 6 2 5 4 2 7 4 6 0 8 1 3 5 7 2 0 3 4 5 8 6 1 3 5 7 0 8 2 4 1 6 4 3 6 2 7 1 5 0 8 6 1 2 4 5 3 7 8 0 5 4 8 1 6 0 3 2 7

M = A + 9 * B + 1 11 63 46 75 26 67 5 43 33 79 1 18 53 30 42 58 65 23 6 74 31 68 10 62 25 48 45 27 69 44 61 4 73 12 32 47 66 22 2 28 41 54 80 60 16 35 50 70 9 78 21 38 13 55 37 34 57 20 72 14 51 8 76 59 17 24 40 52 29 64 81 3 49 39 77 15 56 7 36 19 71

The total number of order 9 Self Orthogonal Inlaid Latin Diagonal Squares with 4 x 4 Diamond Inlay is 96 and can be generated within 845 seconds (ref. SelfOrth9c).

9.6.4 Inlaid Magic Squares, Diamond (5 x 5)

A construction example of an Inlaid Magic Square M = A + 9 * T(A) + [1] with 5 x 5 Diamond Inlay is shown below:

A 2 6 8 3 5 0 4 7 1 8 3 1 4 7 6 2 0 5 4 2 0 5 6 8 3 1 7 0 5 7 1 8 2 6 4 3 6 0 3 7 4 1 5 8 2 5 4 2 6 0 7 1 3 8 1 7 5 0 2 3 8 6 4 3 8 6 2 1 4 7 5 0 7 1 4 8 3 5 0 2 6

B = T(A) 2 8 4 0 6 5 1 3 7 6 3 2 5 0 4 7 8 1 8 1 0 7 3 2 5 6 4 3 4 5 1 7 6 0 2 8 5 7 6 8 4 0 2 1 3 0 6 8 2 1 7 3 4 5 4 2 3 6 5 1 8 7 0 7 0 1 4 8 3 6 5 2 1 5 7 3 2 8 4 0 6

M = A + 9 * B + 1 21 79 45 4 60 46 14 35 65 63 31 20 50 8 43 66 73 15 77 12 1 69 34 27 49 56 44 28 42 53 11 72 57 7 23 76 52 64 58 80 41 2 24 18 30 6 59 75 25 10 71 29 40 54 38 26 33 55 48 13 81 70 5 67 9 16 39 74 32 62 51 19 17 47 68 36 22 78 37 3 61

The total number of order 9 Self Orthogonal Inlaid Latin Diagonal Squares with 5 x 5 Diamond Inlay is 128 and can be generated within 615 seconds (ref. SelfOrth9c).

9.6.5 Magic Squares, Corner Squares (4 x 4)

A construction example of a Magic Square M = A + 9 * T(A) + [1] with 4 x 4 Corner Squares is shown below:

A 0 8 2 6 5 4 3 1 7 5 3 7 1 8 0 4 6 2 8 0 6 2 1 3 5 7 4 3 5 1 7 2 8 0 4 6 7 6 0 3 4 5 8 2 1 2 4 8 0 6 1 7 3 5 4 1 3 5 7 6 2 8 0 6 2 4 8 0 7 1 5 3 1 7 5 4 3 2 6 0 8

B = T(A) 0 5 8 3 7 2 4 6 1 8 3 0 5 6 4 1 2 7 2 7 6 1 0 8 3 4 5 6 1 2 7 3 0 5 8 4 5 8 1 2 4 6 7 0 3 4 0 3 8 5 1 6 7 2 3 4 5 0 8 7 2 1 6 1 6 7 4 2 3 8 5 0 7 2 4 6 1 5 0 3 8

M = A + 9 * B + 1 1 54 75 34 69 23 40 56 17 78 31 8 47 63 37 14 25 66 27 64 61 12 2 76 33 44 50 58 15 20 71 30 9 46 77 43 53 79 10 22 41 60 72 3 29 39 5 36 73 52 11 62 67 24 32 38 49 6 80 70 21 18 55 16 57 68 45 19 35 74 51 4 65 26 42 59 13 48 7 28 81

The total number of order 9 Self Orthogonal Latin Diagonal Squares with 4 x 4 Corner Squares is 3072 and can be generated within 810 seconds (ref. SelfOrth9c).

9.6.6 Magic Squares, Corner Squares (5 x 5)

A construction example of a Magic Square M = A + 9 * T(A) + [1] with each other overlapping 5 x 5 Corner Squares is shown below:

A 0 8 2 4 6 5 3 7 1 5 3 4 1 7 0 8 2 6 4 5 6 2 3 8 0 1 7 8 4 1 7 0 3 5 6 2 3 0 7 6 4 2 1 8 5 6 2 3 5 8 1 7 4 0 1 7 8 0 5 6 2 3 4 2 6 0 8 1 7 4 5 3 7 1 5 3 2 4 6 0 8

B = T(A) 0 5 4 8 3 6 1 2 7 8 3 5 4 0 2 7 6 1 2 4 6 1 7 3 8 0 5 4 1 2 7 6 5 0 8 3 6 7 3 0 4 8 5 1 2 5 0 8 3 2 1 6 7 4 3 8 0 5 1 7 2 4 6 7 2 1 6 8 4 3 5 0 1 6 7 2 5 0 4 3 8

M = A + 9 * B + 1 1 54 39 77 34 60 13 26 65 78 31 50 38 8 19 72 57 16 23 42 61 12 67 36 73 2 53 45 14 20 71 55 49 6 79 30 58 64 35 7 41 75 47 18 24 52 3 76 33 27 11 62 68 37 29 80 9 46 15 70 21 40 59 66 25 10 63 74 44 32 51 4 17 56 69 22 48 5 43 28 81

The total number of order 9 Self Orthogonal Latin Diagonal Squares with each other overlapping 5 x 5 Corner Squares is 448 and can be generated within 810 seconds (ref. SelfOrth9c).

9.6.7 Magic Squares, Compact Ultra Magic

A construction example of a Compact Ultra Magic Square M = A + 9 * T(A) + [1] is shown below:

A 1 7 3 2 8 4 0 6 5 5 6 1 3 7 2 4 8 0 0 8 5 1 6 3 2 7 4 7 4 0 8 5 1 6 3 2 2 3 7 0 4 8 1 5 6 6 5 2 7 3 0 8 4 1 4 1 6 5 2 7 3 0 8 8 0 4 6 1 5 7 2 3 3 2 8 4 0 6 5 1 7

B = T(A) 1 5 0 7 2 6 4 8 3 7 6 8 4 3 5 1 0 2 3 1 5 0 7 2 6 4 8 2 3 1 8 0 7 5 6 4 8 7 6 5 4 3 2 1 0 4 2 3 1 8 0 7 5 6 0 4 2 6 1 8 3 7 5 6 8 7 3 5 4 0 2 1 5 0 4 2 6 1 8 3 7

M = A + 9 * B + 1 11 53 4 66 27 59 37 79 33 69 61 74 40 35 48 14 9 19 28 18 51 2 70 22 57 44 77 26 32 10 81 6 65 52 58 39 75 67 62 46 41 36 20 15 7 43 24 30 17 76 1 72 50 56 5 38 25 60 12 80 31 64 54 63 73 68 34 47 42 8 21 13 49 3 45 23 55 16 78 29 71

The total number of order 9 Self Orthogonal Compact Ultra Magic Squares is 576 and can be generated within 975 seconds (ref. SelfOrth9c).

Attachment 9.6.6 shows a construction example of a Compact Ultra Magic Square for which the elements of every third-row and third-column sum to s1/3.

9.6.8 Magic Squares
      Third-rows and Third-columns Summing to s1/3

A construction example of a Magic Square M = A + 9 * T(A) + [1] for which the elements of every third-row and third-column sum to s1/3. is shown below:

A (Row/3 and Clmn/3) 0 8 4 7 3 2 5 1 6 5 1 6 0 8 4 7 3 2 7 3 2 5 1 6 0 8 4 8 4 0 3 2 7 1 6 5 1 6 5 8 4 0 3 2 7 3 2 7 1 6 5 8 4 0 4 0 8 2 7 3 6 5 1 6 5 1 4 0 8 2 7 3 2 7 3 6 5 1 4 0 8

B = T(A) 0 5 7 8 1 3 4 6 2 8 1 3 4 6 2 0 5 7 4 6 2 0 5 7 8 1 3 7 0 5 3 8 1 2 4 6 3 8 1 2 4 6 7 0 5 2 4 6 7 0 5 3 8 1 5 7 0 1 3 8 6 2 4 1 3 8 6 2 4 5 7 0 6 2 4 5 7 0 1 3 8

M = A + 9 * B + 1 1 54 68 80 13 30 42 56 25 78 11 34 37 63 23 8 49 66 44 58 21 6 47 70 73 18 32 72 5 46 31 75 17 20 43 60 29 79 15 27 41 55 67 3 53 22 39 62 65 7 51 36 77 10 50 64 9 12 35 76 61 24 38 16 33 74 59 19 45 48 71 4 57 26 40 52 69 2 14 28 81

The total number of subject order 9 Self Orthogonal Magic Squares is 576 and can be generated within 865 seconds (ref. SelfOrth9c).

Attachment 9.6.6 shows a construction example of a Compact Ultra Magic Square for which the elements of every third-row and third-column sum to s1/3.

9.10   Miscellaneous

9.10.1 Mutual Orthogonal Latin Squares (9 x 9)

The construction of 0rder 9 Magic Squares based on following Mutual Orthogonal Latin (Diagonal) Squares

• Associated, Compact Pan Magic
• Associated, Regular Sub Squares summing to s1
• Associated, every third-row and third-column summing to s1/3

has been deducted and discussed in Section 9.5.

The construction of 0rder 9 Bimagic Squares based on Mutual Orthogonal Latin (Diagonal) Squares has been discussed in Section 15.2, Section 15.4 and Section 15.5

9.10.2 Semi Latin Squares (9 x 9)

The construction of 0rder 9 Mutual Orthogonal Semi-Latin (Diagonal) Squares has been deducted and discussed in:

• Section 9.2.2  Concentric (Bordered) Magic Squares
• Section 9.2.5  Composed   Magic Squares, Associated Border
• Section 9.2.6  Composed   Magic Squares, Sub Squares order 4 and 5 (Pan Magic)
• Section 9.2.7  Associated Magic Squares, Sub Squares order 4 and 5
• Section 9.2.9  Composed   Magic Squares, Sub Squares Order 3 and 6
• Section 9.2.10 Composed   Magic Squares, Sub Squares Order 3 and 7 (Overlapping)

Order 9 Mutual Orthogonal Semi-Latin Inlaid Magic Squares with Diamond Inlays have been deducted and discussed in Section 9.2.8.

8.16.3 Summary

The obtained results regarding the order 9 Latin - and related Magic Squares, as deducted and discussed in previous sections, are summarized in following table:

Attachment	Subject	n9	Subroutine
-	-	-	-
-	Self Orth, Associated	8.921.088	SelfOrth9c
-	Self Orth, Ultra Magic	1296
-	Self Orth, Compact, Pan Magic	10368	SelfOrth9b
-	Self Orth, Compact, Pan Magic Third-rows and Third-columns	576	SelfOrth9e
-	Self Orth, Idempotent (Associated)	23232	SelfOrth9a
-	-	-	-

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.