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Construction of order 21 Self Orthogonal Composed Latin Diagonal Squares

Construct an order 20 Self Orthogonal Composed Latin Diagonal Square.

The required order 4 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:

    {0, 1, 2, 3}, {4, 5, 6, 7}, {8, 9, 10, 11}, {12}, {13, 14, 15, 16} and {17, 18, 19, 20}

with respectively the magic constants s4 = 6, 22, 38, 58 and 74.

Step 1
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
13 12 15 14
14 15 12 13
12 13 14 15
15 14 13 12
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
Sqrs4
9 5 1 18 14
1 18 14 9 5
14 9 5 1 18
5 1 18 14 9
18 14 9 5 1

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

Construct an intermediate order 21 square by adding a row and a column, to the order 20 Self Orthogonal Composed Latin Diagonal Square as shown below:

Step 2
9 8 11 10 5 4 7 6 1 0 3 2 0 18 17 20 19 14 13 16 15
10 11 8 9 6 7 4 5 2 3 0 1 0 19 20 17 18 15 16 13 14
8 9 10 11 4 5 6 7 0 1 2 3 0 17 18 19 20 13 14 15 16
11 10 9 8 7 6 5 4 3 2 1 0 0 20 19 18 17 16 15 14 13
1 0 3 2 18 17 20 19 14 13 16 15 0 9 8 11 10 5 4 7 6
2 3 0 1 19 20 17 18 15 16 13 14 0 10 11 8 9 6 7 4 5
0 1 2 3 17 18 19 20 13 14 15 16 0 8 9 10 11 4 5 6 7
3 2 1 0 20 19 18 17 16 15 14 13 0 11 10 9 8 7 6 5 4
14 13 16 15 9 8 11 10 5 4 7 6 0 1 0 3 2 18 17 20 19
15 16 13 14 10 11 8 9 6 7 4 5 0 2 3 0 1 19 20 17 18
13 14 15 16 8 9 10 11 4 5 6 7 0 0 1 2 3 17 18 19 20
16 15 14 13 11 10 9 8 7 6 5 4 0 3 2 1 0 20 19 18 17
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 4 7 6 1 0 3 2 18 17 20 19 0 14 13 16 15 9 8 11 10
6 7 4 5 2 3 0 1 19 20 17 18 0 15 16 13 14 10 11 8 9
4 5 6 7 0 1 2 3 17 18 19 20 0 13 14 15 16 8 9 10 11
7 6 5 4 3 2 1 0 20 19 18 17 0 16 15 14 13 11 10 9 8
18 17 20 19 14 13 16 15 9 8 11 10 0 5 4 7 6 1 0 3 2
19 20 17 18 15 16 13 14 10 11 8 9 0 6 7 4 5 2 3 0 1
17 18 19 20 13 14 15 16 8 9 10 11 0 4 5 6 7 0 1 2 3
20 19 18 17 16 15 14 13 11 10 9 8 0 7 6 5 4 3 2 1 0

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 five order 5 Auxiliary Latin Diagonal Squares:

A51
9 8 12 11 10
12 11 10 9 8
10 9 8 12 11
8 12 11 10 9
11 10 9 8 12
A52
18 17 12 20 19
12 20 19 18 17
19 18 17 12 20
17 12 20 19 18
20 19 18 17 12
A53 (Center)
5 4 12 7 6
12 7 6 5 4
6 5 4 12 7
4 12 7 6 5
7 6 5 4 12
A54
12 14 13 16 15
13 16 15 12 14
15 12 14 13 16
14 13 16 15 12
16 15 12 14 13
A55
12 1 0 3 2
0 3 2 12 1
2 12 1 0 3
1 0 3 2 12
3 2 12 1 0

The five Auxiliary Squares are based on the five sub series defined above and the number 12.

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
9 8 12 11 5 4 7 6 1 0 3 2 10 18 17 20 19 14 13 16 15
12 11 10 9 6 7 4 5 2 3 0 1 8 19 20 17 18 15 16 13 14
10 9 8 12 4 5 6 7 0 1 2 3 11 17 18 19 20 13 14 15 16
8 12 11 10 7 6 5 4 3 2 1 0 9 20 19 18 17 16 15 14 13
1 0 3 2 18 17 12 20 14 13 16 15 19 9 8 11 10 5 4 7 6
2 3 0 1 12 20 19 18 15 16 13 14 17 10 11 8 9 6 7 4 5
0 1 2 3 19 18 17 12 13 14 15 16 20 8 9 10 11 4 5 6 7
3 2 1 0 17 12 20 19 16 15 14 13 18 11 10 9 8 7 6 5 4
14 13 16 15 9 8 11 10 5 4 12 7 6 1 0 3 2 18 17 20 19
15 16 13 14 10 11 8 9 12 7 6 5 4 2 3 0 1 19 20 17 18
13 14 15 16 8 9 10 11 6 5 4 12 7 0 1 2 3 17 18 19 20
16 15 14 13 11 10 9 8 4 12 7 6 5 3 2 1 0 20 19 18 17
11 10 9 8 20 19 18 17 7 6 5 4 12 14 13 16 15 1 0 3 2
5 4 7 6 1 0 3 2 18 17 20 19 13 16 15 12 14 9 8 11 10
6 7 4 5 2 3 0 1 19 20 17 18 15 12 14 13 16 10 11 8 9
4 5 6 7 0 1 2 3 17 18 19 20 14 13 16 15 12 8 9 10 11
7 6 5 4 3 2 1 0 20 19 18 17 16 15 12 14 13 11 10 9 8
18 17 20 19 14 13 16 15 9 8 11 10 0 5 4 7 6 3 2 12 1
19 20 17 18 15 16 13 14 10 11 8 9 2 6 7 4 5 12 1 0 3
17 18 19 20 13 14 15 16 8 9 10 11 1 4 5 6 7 0 3 2 12
20 19 18 17 16 15 14 13 11 10 9 8 3 7 6 5 4 2 12 1 0

The order 21 Self Orthogonal Composed Latin Diagonal Square shown above is ready to be used for the construction of an order 21 Composed Simple Magic Square.

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