Office Applications and Entertainment, Latin Squares
Vorige Pagina Attachment 9.8.1 About the Author

Construction of order 37 Self Orthogonal Composed Latin Diagonal Squares

Construct an order 36 Self Orthogonal Composed Latin Diagonal Square.

The required order 9 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:

    {0, 1 ... 8}, {9, 10 ... 17}, {18}, {19, 20 ... 27} and {28, 29 ... 36}

with respectively the magic constants s9 = 36, 117, 207 and 288

Sqrs9
30 21 2 11
2 11 30 21
11 2 21 30
21 30 11 2

The order 4 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 the square shown below.

Step 1
30 34 36 31 33 28 32 35 29
36 31 29 32 35 34 30 28 33
32 30 28 33 34 36 31 29 35
28 33 35 29 36 30 34 32 31
34 28 31 35 32 29 33 36 30
33 32 30 34 28 35 29 31 36
29 35 33 28 30 31 36 34 32
31 36 34 30 29 32 35 33 28
35 29 32 36 31 33 28 30 34
21 25 27 22 24 19 23 26 20
27 22 20 23 26 25 21 19 24
23 21 19 24 25 27 22 20 26
19 24 26 20 27 21 25 23 22
25 19 22 26 23 20 24 27 21
24 23 21 25 19 26 20 22 27
20 26 24 19 21 22 27 25 23
22 27 25 21 20 23 26 24 19
26 20 23 27 22 24 19 21 25
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
11 15 17 12 14 9 13 16 10
17 12 10 13 16 15 11 9 14
13 11 9 14 15 17 12 10 16
9 14 16 10 17 11 15 13 12
15 9 12 16 13 10 14 17 11
14 13 11 15 9 16 10 12 17
10 16 14 9 11 12 17 15 13
12 17 15 11 10 13 16 14 9
16 10 13 17 12 14 9 11 15
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
11 15 17 12 14 9 13 16 10
17 12 10 13 16 15 11 9 14
13 11 9 14 15 17 12 10 16
9 14 16 10 17 11 15 13 12
15 9 12 16 13 10 14 17 11
14 13 11 15 9 16 10 12 17
10 16 14 9 11 12 17 15 13
12 17 15 11 10 13 16 14 9
16 10 13 17 12 14 9 11 15
30 34 36 31 33 28 32 35 29
36 31 29 32 35 34 30 28 33
32 30 28 33 34 36 31 29 35
28 33 35 29 36 30 34 32 31
34 28 31 35 32 29 33 36 30
33 32 30 34 28 35 29 31 36
29 35 33 28 30 31 36 34 32
31 36 34 30 29 32 35 33 28
35 29 32 36 31 33 28 30 34
21 25 27 22 24 19 23 26 20
27 22 20 23 26 25 21 19 24
23 21 19 24 25 27 22 20 26
19 24 26 20 27 21 25 23 22
25 19 22 26 23 20 24 27 21
24 23 21 25 19 26 20 22 27
20 26 24 19 21 22 27 25 23
22 27 25 21 20 23 26 24 19
26 20 23 27 22 24 19 21 25
11 15 17 12 14 9 13 16 10
17 12 10 13 16 15 11 9 14
13 11 9 14 15 17 12 10 16
9 14 16 10 17 11 15 13 12
15 9 12 16 13 10 14 17 11
14 13 11 15 9 16 10 12 17
10 16 14 9 11 12 17 15 13
12 17 15 11 10 13 16 14 9
16 10 13 17 12 14 9 11 15
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
21 25 27 22 24 19 23 26 20
27 22 20 23 26 25 21 19 24
23 21 19 24 25 27 22 20 26
19 24 26 20 27 21 25 23 22
25 19 22 26 23 20 24 27 21
24 23 21 25 19 26 20 22 27
20 26 24 19 21 22 27 25 23
22 27 25 21 20 23 26 24 19
26 20 23 27 22 24 19 21 25
30 34 36 31 33 28 32 35 29
36 31 29 32 35 34 30 28 33
32 30 28 33 34 36 31 29 35
28 33 35 29 36 30 34 32 31
34 28 31 35 32 29 33 36 30
33 32 30 34 28 35 29 31 36
29 35 33 28 30 31 36 34 32
31 36 34 30 29 32 35 33 28
35 29 32 36 31 33 28 30 34
21 25 27 22 24 19 23 26 20
27 22 20 23 26 25 21 19 24
23 21 19 24 25 27 22 20 26
19 24 26 20 27 21 25 23 22
25 19 22 26 23 20 24 27 21
24 23 21 25 19 26 20 22 27
20 26 24 19 21 22 27 25 23
22 27 25 21 20 23 26 24 19
26 20 23 27 22 24 19 21 25
30 34 36 31 33 28 32 35 29
36 31 29 32 35 34 30 28 33
32 30 28 33 34 36 31 29 35
28 33 35 29 36 30 34 32 31
34 28 31 35 32 29 33 36 30
33 32 30 34 28 35 29 31 36
29 35 33 28 30 31 36 34 32
31 36 34 30 29 32 35 33 28
35 29 32 36 31 33 28 30 34
11 15 17 12 14 9 13 16 10
17 12 10 13 16 15 11 9 14
13 11 9 14 15 17 12 10 16
9 14 16 10 17 11 15 13 12
15 9 12 16 13 10 14 17 11
14 13 11 15 9 16 10 12 17
10 16 14 9 11 12 17 15 13
12 17 15 11 10 13 16 14 9
16 10 13 17 12 14 9 11 15
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

Construct an intermediate order 37 square by adding a Center Cross, to the order 36 Self Orthogonal Composed Latin Diagonal Square as shown below:

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

The Intermediate Square has to be transformed to a Self Orthogonal Latin Diagonal Square, which can be achieved by means of a set of four order 10 Auxiliary Latin Diagonal Squares:

A101
28 33 29 32 31 36 18 30 34 35
36 29 28 35 30 34 33 31 18 32
35 36 30 33 34 18 31 32 28 29
30 34 18 31 35 29 36 33 32 28
34 35 36 28 32 31 29 18 30 33
32 18 35 36 29 33 30 28 31 34
31 32 33 30 18 28 34 29 35 36
18 30 34 29 36 32 28 35 33 31
29 28 31 18 33 35 32 34 36 30
33 31 32 34 28 30 35 36 29 18
A102
9 14 10 13 12 17 18 11 15 16
17 10 9 16 11 15 14 12 18 13
16 17 11 14 15 18 12 13 9 10
11 15 18 12 16 10 17 14 13 9
15 16 17 9 13 12 10 18 11 14
13 18 16 17 10 14 11 9 12 15
12 13 14 11 18 9 15 10 16 17
18 11 15 10 17 13 9 16 14 12
10 9 12 18 14 16 13 15 17 11
14 12 13 15 9 11 16 17 10 18
A103
18 23 19 22 21 26 27 20 24 25
26 19 18 25 20 24 23 21 27 22
25 26 20 23 24 27 21 22 18 19
20 24 27 21 25 19 26 23 22 18
24 25 26 18 22 21 19 27 20 23
22 27 25 26 19 23 20 18 21 24
21 22 23 20 27 18 24 19 25 26
27 20 24 19 26 22 18 25 23 21
19 18 21 27 23 25 22 24 26 20
23 21 22 24 18 20 25 26 19 27
A104
18 4 0 3 2 7 8 1 5 6
7 0 18 6 1 5 4 2 8 3
6 7 1 4 5 8 2 3 18 0
1 5 8 2 6 0 7 4 3 18
5 6 7 18 3 2 0 8 1 4
3 8 6 7 0 4 1 18 2 5
2 3 4 1 8 18 5 0 6 7
8 1 5 0 7 3 18 6 4 2
0 18 2 8 4 6 3 5 7 1
4 2 3 5 18 1 6 7 0 8

The four Auxiliary Squares are based on the four sub series defined above and the number 18 (= center).

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 below:

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

The order 37 Self Orthogonal Composed Latin Diagonal Square shown above is ready to be used for the construction of an order 37 Composed Simple Magic Square.

Vorige Pagina About the Author