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Vorige Pagina Attachment 7.9.1 About the Author

Construction of order 36 Self Orthogonal Composed Latin Diagonal Squares

Construct an order 35 Self Orthogonal Composed Latin Diagonal Square.

The required order 7 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:

    {0, 1 ... 6}, {7, 8 ... 13}, {14, 15, ... 20}, {21}, {22. 23 .... 28} and {29, 30 ... 35}

with respectively the magic constants s7 = 21, 70, 119, 175 and 224

Sqrs7
35 20 6 28 13
6 28 13 35 20
13 35 20 6 28
20 6 28 13 35
28 13 35 20 6

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

Step 1
35 34 32 30 29 33 31
32 30 29 33 31 35 34
29 33 31 35 34 32 30
31 35 34 32 30 29 33
34 32 30 29 33 31 35
30 29 33 31 35 34 32
33 31 35 34 32 30 29
20 19 17 15 14 18 16
17 15 14 18 16 20 19
14 18 16 20 19 17 15
16 20 19 17 15 14 18
19 17 15 14 18 16 20
15 14 18 16 20 19 17
18 16 20 19 17 15 14
6 5 3 1 0 4 2
3 1 0 4 2 6 5
0 4 2 6 5 3 1
2 6 5 3 1 0 4
5 3 1 0 4 2 6
1 0 4 2 6 5 3
4 2 6 5 3 1 0
28 27 25 23 22 26 24
25 23 22 26 24 28 27
22 26 24 28 27 25 23
24 28 27 25 23 22 26
27 25 23 22 26 24 28
23 22 26 24 28 27 25
26 24 28 27 25 23 22
13 12 10 8 7 11 9
10 8 7 11 9 13 12
7 11 9 13 12 10 8
9 13 12 10 8 7 11
12 10 8 7 11 9 13
8 7 11 9 13 12 10
11 9 13 12 10 8 7
6 5 3 1 0 4 2
3 1 0 4 2 6 5
0 4 2 6 5 3 1
2 6 5 3 1 0 4
5 3 1 0 4 2 6
1 0 4 2 6 5 3
4 2 6 5 3 1 0
28 27 25 23 22 26 24
25 23 22 26 24 28 27
22 26 24 28 27 25 23
24 28 27 25 23 22 26
27 25 23 22 26 24 28
23 22 26 24 28 27 25
26 24 28 27 25 23 22
13 12 10 8 7 11 9
10 8 7 11 9 13 12
7 11 9 13 12 10 8
9 13 12 10 8 7 11
12 10 8 7 11 9 13
8 7 11 9 13 12 10
11 9 13 12 10 8 7
35 34 32 30 29 33 31
32 30 29 33 31 35 34
29 33 31 35 34 32 30
31 35 34 32 30 29 33
34 32 30 29 33 31 35
30 29 33 31 35 34 32
33 31 35 34 32 30 29
20 19 17 15 14 18 16
17 15 14 18 16 20 19
14 18 16 20 19 17 15
16 20 19 17 15 14 18
19 17 15 14 18 16 20
15 14 18 16 20 19 17
18 16 20 19 17 15 14
13 12 10 8 7 11 9
10 8 7 11 9 13 12
7 11 9 13 12 10 8
9 13 12 10 8 7 11
12 10 8 7 11 9 13
8 7 11 9 13 12 10
11 9 13 12 10 8 7
35 34 32 30 29 33 31
32 30 29 33 31 35 34
29 33 31 35 34 32 30
31 35 34 32 30 29 33
34 32 30 29 33 31 35
30 29 33 31 35 34 32
33 31 35 34 32 30 29
20 19 17 15 14 18 16
17 15 14 18 16 20 19
14 18 16 20 19 17 15
16 20 19 17 15 14 18
19 17 15 14 18 16 20
15 14 18 16 20 19 17
18 16 20 19 17 15 14
6 5 3 1 0 4 2
3 1 0 4 2 6 5
0 4 2 6 5 3 1
2 6 5 3 1 0 4
5 3 1 0 4 2 6
1 0 4 2 6 5 3
4 2 6 5 3 1 0
28 27 25 23 22 26 24
25 23 22 26 24 28 27
22 26 24 28 27 25 23
24 28 27 25 23 22 26
27 25 23 22 26 24 28
23 22 26 24 28 27 25
26 24 28 27 25 23 22
20 19 17 15 14 18 16
17 15 14 18 16 20 19
14 18 16 20 19 17 15
16 20 19 17 15 14 18
19 17 15 14 18 16 20
15 14 18 16 20 19 17
18 16 20 19 17 15 14
6 5 3 1 0 4 2
3 1 0 4 2 6 5
0 4 2 6 5 3 1
2 6 5 3 1 0 4
5 3 1 0 4 2 6
1 0 4 2 6 5 3
4 2 6 5 3 1 0
28 27 25 23 22 26 24
25 23 22 26 24 28 27
22 26 24 28 27 25 23
24 28 27 25 23 22 26
27 25 23 22 26 24 28
23 22 26 24 28 27 25
26 24 28 27 25 23 22
13 12 10 8 7 11 9
10 8 7 11 9 13 12
7 11 9 13 12 10 8
9 13 12 10 8 7 11
12 10 8 7 11 9 13
8 7 11 9 13 12 10
11 9 13 12 10 8 7
35 34 32 30 29 33 31
32 30 29 33 31 35 34
29 33 31 35 34 32 30
31 35 34 32 30 29 33
34 32 30 29 33 31 35
30 29 33 31 35 34 32
33 31 35 34 32 30 29
28 27 25 23 22 26 24
25 23 22 26 24 28 27
22 26 24 28 27 25 23
24 28 27 25 23 22 26
27 25 23 22 26 24 28
23 22 26 24 28 27 25
26 24 28 27 25 23 22
13 12 10 8 7 11 9
10 8 7 11 9 13 12
7 11 9 13 12 10 8
9 13 12 10 8 7 11
12 10 8 7 11 9 13
8 7 11 9 13 12 10
11 9 13 12 10 8 7
35 34 32 30 29 33 31
32 30 29 33 31 35 34
29 33 31 35 34 32 30
31 35 34 32 30 29 33
34 32 30 29 33 31 35
30 29 33 31 35 34 32
33 31 35 34 32 30 29
20 19 17 15 14 18 16
17 15 14 18 16 20 19
14 18 16 20 19 17 15
16 20 19 17 15 14 18
19 17 15 14 18 16 20
15 14 18 16 20 19 17
18 16 20 19 17 15 14
6 5 3 1 0 4 2
3 1 0 4 2 6 5
0 4 2 6 5 3 1
2 6 5 3 1 0 4
5 3 1 0 4 2 6
1 0 4 2 6 5 3
4 2 6 5 3 1 0

Construct an intermediate order 36 square by adding by adding a row and a column, to the order 35 Self Orthogonal Composed Latin Diagonal Square as shown below:

Step 2
35 34 32 30 29 33 31 20 19 17 15 14 18 16 6 5 3 1 0 4 2 0 28 27 25 23 22 26 24 13 12 10 8 7 11 9
32 30 29 33 31 35 34 17 15 14 18 16 20 19 3 1 0 4 2 6 5 0 25 23 22 26 24 28 27 10 8 7 11 9 13 12
29 33 31 35 34 32 30 14 18 16 20 19 17 15 0 4 2 6 5 3 1 0 22 26 24 28 27 25 23 7 11 9 13 12 10 8
31 35 34 32 30 29 33 16 20 19 17 15 14 18 2 6 5 3 1 0 4 0 24 28 27 25 23 22 26 9 13 12 10 8 7 11
34 32 30 29 33 31 35 19 17 15 14 18 16 20 5 3 1 0 4 2 6 0 27 25 23 22 26 24 28 12 10 8 7 11 9 13
30 29 33 31 35 34 32 15 14 18 16 20 19 17 1 0 4 2 6 5 3 0 23 22 26 24 28 27 25 8 7 11 9 13 12 10
33 31 35 34 32 30 29 18 16 20 19 17 15 14 4 2 6 5 3 1 0 0 26 24 28 27 25 23 22 11 9 13 12 10 8 7
6 5 3 1 0 4 2 28 27 25 23 22 26 24 13 12 10 8 7 11 9 0 35 34 32 30 29 33 31 20 19 17 15 14 18 16
3 1 0 4 2 6 5 25 23 22 26 24 28 27 10 8 7 11 9 13 12 0 32 30 29 33 31 35 34 17 15 14 18 16 20 19
0 4 2 6 5 3 1 22 26 24 28 27 25 23 7 11 9 13 12 10 8 0 29 33 31 35 34 32 30 14 18 16 20 19 17 15
2 6 5 3 1 0 4 24 28 27 25 23 22 26 9 13 12 10 8 7 11 0 31 35 34 32 30 29 33 16 20 19 17 15 14 18
5 3 1 0 4 2 6 27 25 23 22 26 24 28 12 10 8 7 11 9 13 0 34 32 30 29 33 31 35 19 17 15 14 18 16 20
1 0 4 2 6 5 3 23 22 26 24 28 27 25 8 7 11 9 13 12 10 0 30 29 33 31 35 34 32 15 14 18 16 20 19 17
4 2 6 5 3 1 0 26 24 28 27 25 23 22 11 9 13 12 10 8 7 0 33 31 35 34 32 30 29 18 16 20 19 17 15 14
13 12 10 8 7 11 9 35 34 32 30 29 33 31 20 19 17 15 14 18 16 0 6 5 3 1 0 4 2 28 27 25 23 22 26 24
10 8 7 11 9 13 12 32 30 29 33 31 35 34 17 15 14 18 16 20 19 0 3 1 0 4 2 6 5 25 23 22 26 24 28 27
7 11 9 13 12 10 8 29 33 31 35 34 32 30 14 18 16 20 19 17 15 0 0 4 2 6 5 3 1 22 26 24 28 27 25 23
9 13 12 10 8 7 11 31 35 34 32 30 29 33 16 20 19 17 15 14 18 0 2 6 5 3 1 0 4 24 28 27 25 23 22 26
12 10 8 7 11 9 13 34 32 30 29 33 31 35 19 17 15 14 18 16 20 0 5 3 1 0 4 2 6 27 25 23 22 26 24 28
8 7 11 9 13 12 10 30 29 33 31 35 34 32 15 14 18 16 20 19 17 0 1 0 4 2 6 5 3 23 22 26 24 28 27 25
11 9 13 12 10 8 7 33 31 35 34 32 30 29 18 16 20 19 17 15 14 0 4 2 6 5 3 1 0 26 24 28 27 25 23 22
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
20 19 17 15 14 18 16 6 5 3 1 0 4 2 28 27 25 23 22 26 24 0 13 12 10 8 7 11 9 35 34 32 30 29 33 31
17 15 14 18 16 20 19 3 1 0 4 2 6 5 25 23 22 26 24 28 27 0 10 8 7 11 9 13 12 32 30 29 33 31 35 34
14 18 16 20 19 17 15 0 4 2 6 5 3 1 22 26 24 28 27 25 23 0 7 11 9 13 12 10 8 29 33 31 35 34 32 30
16 20 19 17 15 14 18 2 6 5 3 1 0 4 24 28 27 25 23 22 26 0 9 13 12 10 8 7 11 31 35 34 32 30 29 33
19 17 15 14 18 16 20 5 3 1 0 4 2 6 27 25 23 22 26 24 28 0 12 10 8 7 11 9 13 34 32 30 29 33 31 35
15 14 18 16 20 19 17 1 0 4 2 6 5 3 23 22 26 24 28 27 25 0 8 7 11 9 13 12 10 30 29 33 31 35 34 32
18 16 20 19 17 15 14 4 2 6 5 3 1 0 26 24 28 27 25 23 22 0 11 9 13 12 10 8 7 33 31 35 34 32 30 29
28 27 25 23 22 26 24 13 12 10 8 7 11 9 35 34 32 30 29 33 31 0 20 19 17 15 14 18 16 6 5 3 1 0 4 2
25 23 22 26 24 28 27 10 8 7 11 9 13 12 32 30 29 33 31 35 34 0 17 15 14 18 16 20 19 3 1 0 4 2 6 5
22 26 24 28 27 25 23 7 11 9 13 12 10 8 29 33 31 35 34 32 30 0 14 18 16 20 19 17 15 0 4 2 6 5 3 1
24 28 27 25 23 22 26 9 13 12 10 8 7 11 31 35 34 32 30 29 33 0 16 20 19 17 15 14 18 2 6 5 3 1 0 4
27 25 23 22 26 24 28 12 10 8 7 11 9 13 34 32 30 29 33 31 35 0 19 17 15 14 18 16 20 5 3 1 0 4 2 6
23 22 26 24 28 27 25 8 7 11 9 13 12 10 30 29 33 31 35 34 32 0 15 14 18 16 20 19 17 1 0 4 2 6 5 3
26 24 28 27 25 23 22 11 9 13 12 10 8 7 33 31 35 34 32 30 29 0 18 16 20 19 17 15 14 4 2 6 5 3 1 0

The Intermediate Square has to be transformed to a Self Orthogonal Latin Diagonal Square, which can be achieved by means of a set of five order 9 Auxiliary Latin Diagonal Squares:

A91
29 31 34 35 21 33 32 30
32 30 21 33 34 35 29 31
30 34 31 21 35 32 33 29
33 29 35 32 31 21 30 34
31 35 29 34 33 30 21 32
21 32 33 30 29 34 31 35
34 21 30 31 32 29 35 33
35 33 32 29 30 31 34 21
A92
22 24 27 28 21 26 25 23
25 23 21 26 27 28 22 24
23 27 24 21 28 25 26 22
26 22 28 25 24 21 23 27
24 28 22 27 26 23 21 25
21 25 26 23 22 27 24 28
27 21 23 24 25 22 28 26
28 26 25 22 23 24 27 21
A93
14 16 19 20 21 18 17 15
17 15 21 18 19 20 14 16
15 19 16 21 20 17 18 14
18 14 20 17 16 21 15 19
16 20 14 19 18 15 21 17
21 17 18 15 14 19 16 20
19 21 15 16 17 14 20 18
20 18 17 14 15 16 19 21
A94
21 8 11 12 13 10 9 7
9 7 13 10 11 12 21 8
7 11 8 13 12 9 10 21
10 21 12 9 8 13 7 11
8 12 21 11 10 7 13 9
13 9 10 7 21 11 8 12
11 13 7 8 9 21 12 10
12 10 9 21 7 8 11 13
A95
21 1 4 5 6 3 2 0
2 0 6 3 4 5 21 1
0 4 1 6 5 2 3 21
3 21 5 2 1 6 0 4
1 5 21 4 3 0 6 2
6 2 3 0 21 4 1 5
4 6 0 1 2 21 5 3
5 3 2 21 0 1 4 6

The five Auxiliary Squares are based on the five sub series defined above and the number 21.

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 below:

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

The order 36 Self Orthogonal Composed Latin Diagonal Square shown above is ready to be used for the construction of an order 36 Composed Simple Magic Square.

Vorige Pagina About the Author