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Construction of order 24 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}, {16, 17, 18, 19} and {20, 21, 22, 23}

with respectively the magic constants s4 = 6, 22, 38, 70 and 86.

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
21 20 23 22
22 23 20 21
20 21 22 23
23 22 21 20
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
21 20 23 22
22 23 20 21
20 21 22 23
23 22 21 20
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
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
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
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
21 20 23 22
22 23 20 21
20 21 22 23
23 22 21 20
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
21 20 23 22
22 23 20 21
20 21 22 23
23 22 21 20
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
9 8 11 10
10 11 8 9
8 9 10 11
11 10 9 8
21 20 23 22
22 23 20 21
20 21 22 23
23 22 21 20
17 16 19 18
18 19 16 17
16 17 18 19
19 18 17 16
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 21 17
1 21 17 9 5
17 9 5 1 21
5 1 21 17 9
21 17 9 5 1
Aux4
12 14 15 13
15 13 12 14
13 15 14 12
14 12 13 15

The order 5 Self orthogonal Latin Diagonal Square right/top (Sqrs4) is based on the first elemnets of the Sub Squares, and has been used as a guideline for the construction.

The order 4 Self orthogonal Latin Diagonal Square right/bottom (Aux4) is based on the sub series {12, 13, 14, 15}.

Construct an intermediate order 24 square by adding the Auxilliary Square Aux4 and the related rows and columns, to the order 20 Self Orthogonal Composed Latin Diagonal Square as shown below:

Step 2
9 8 11 10 5 4 7 6 0 0 1 0 3 2 0 0 21 20 23 22 17 16 19 18
10 11 8 9 6 7 4 5 0 0 2 3 0 1 0 0 22 23 20 21 18 19 16 17
8 9 10 11 4 5 6 7 0 0 0 1 2 3 0 0 20 21 22 23 16 17 18 19
11 10 9 8 7 6 5 4 0 0 3 2 1 0 0 0 23 22 21 20 19 18 17 16
1 0 3 2 21 20 23 22 0 0 17 16 19 18 0 0 9 8 11 10 5 4 7 6
2 3 0 1 22 23 20 21 0 0 18 19 16 17 0 0 10 11 8 9 6 7 4 5
0 1 2 3 20 21 22 23 0 0 16 17 18 19 0 0 8 9 10 11 4 5 6 7
3 2 1 0 23 22 21 20 0 0 19 18 17 16 0 0 11 10 9 8 7 6 5 4
0 0 0 0 0 0 0 0 12 14 0 0 0 0 15 13 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 15 13 0 0 0 0 12 14 0 0 0 0 0 0 0 0
17 16 19 18 9 8 11 10 0 0 5 4 7 6 0 0 1 0 3 2 21 20 23 22
18 19 16 17 10 11 8 9 0 0 6 7 4 5 0 0 2 3 0 1 22 23 20 21
16 17 18 19 8 9 10 11 0 0 4 5 6 7 0 0 0 1 2 3 20 21 22 23
19 18 17 16 11 10 9 8 0 0 7 6 5 4 0 0 3 2 1 0 23 22 21 20
0 0 0 0 0 0 0 0 13 15 0 0 0 0 14 12 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 14 12 0 0 0 0 13 15 0 0 0 0 0 0 0 0
5 4 7 6 1 0 3 2 0 0 21 20 23 22 0 0 17 16 19 18 9 8 11 10
6 7 4 5 2 3 0 1 0 0 22 23 20 21 0 0 18 19 16 17 10 11 8 9
4 5 6 7 0 1 2 3 0 0 20 21 22 23 0 0 16 17 18 19 8 9 10 11
7 6 5 4 3 2 1 0 0 0 23 22 21 20 0 0 19 18 17 16 11 10 9 8
21 20 23 22 17 16 19 18 0 0 9 8 11 10 0 0 5 4 7 6 1 0 3 2
22 23 20 21 18 19 16 17 0 0 10 11 8 9 0 0 6 7 4 5 2 3 0 1
20 21 22 23 16 17 18 19 0 0 8 9 10 11 0 0 4 5 6 7 0 1 2 3
23 22 21 20 19 18 17 16 0 0 11 10 9 8 0 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 twenty order 5 Auxiliary Latin Diagonal Squares:

12
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
13
5 4 13 7 6
13 7 6 5 4
6 5 4 13 7
4 13 7 6 5
7 6 5 4 13
14
1 0 14 3 2
14 3 2 1 0
2 1 0 14 3
0 14 3 2 1
3 2 1 0 14
15
22 21 20 15 23
20 15 23 22 21
23 22 21 20 15
21 20 15 23 22
15 23 22 21 20
13
1 0 13 3 2
13 3 2 1 0
2 1 0 13 3
0 13 3 2 1
3 2 1 0 13
14
21 20 14 23 22
14 23 22 21 20
22 21 20 14 23
20 14 23 22 21
23 22 21 20 14
15
17 16 15 19 18
15 19 18 17 16
18 17 16 15 19
16 15 19 18 17
19 18 17 16 15
12
6 5 4 12 7
4 12 7 6 5
7 6 5 4 12
5 4 12 7 6
12 7 6 5 4
14
17 16 14 19 18
14 19 18 17 16
18 17 16 14 19
16 14 19 18 17
19 18 17 16 14
15
9 8 15 11 10
15 11 10 9 8
10 9 8 15 11
8 15 11 10 9
11 10 9 8 15
12
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
13
13 21 20 23 22
20 23 22 13 21
22 13 21 20 23
21 20 23 22 13
23 22 13 21 20
15
6 5 4 7 15
4 7 15 6 5
15 6 5 4 7
5 4 7 15 6
7 15 6 5 4
12
12 21 20 23 22
20 23 22 12 21
22 12 21 20 23
21 20 23 22 12
23 22 12 21 20
13
13 17 16 19 18
16 19 18 13 17
18 13 17 16 19
17 16 19 18 13
19 18 13 17 16
14
14 9 8 11 10
8 11 10 14 9
10 14 9 8 11
9 8 11 10 14
11 10 14 9 8
12
18 17 16 19 12
16 19 12 18 17
12 18 17 16 19
17 16 19 12 18
19 12 18 17 16
13
13 9 8 11 10
8 11 10 13 9
10 13 9 8 11
9 8 11 10 13
11 10 13 9 8
14
14 5 4 7 6
4 7 6 14 5
6 14 5 4 7
5 4 7 6 14
7 6 14 5 4
15
15 1 0 3 2
0 3 2 15 1
2 15 1 0 3
1 0 3 2 15
3 2 15 1 0

The twenty Auxiliary Squares are based on the five sub series defined above and the series {12, 13, 14, 15}.

Replace the applicable Sub Squares (of the Intermediate Square) together with the corresponding sections of the 'Crosses' by the contents of these Auxiliary Squares as shown below:

Step 3
9 8 12 11 5 4 13 7 10 6 1 0 14 3 2 22 21 20 15 23 17 16 19 18
12 11 10 9 13 7 6 5 8 4 14 3 2 1 0 20 15 23 22 21 18 19 16 17
10 9 8 12 6 5 4 13 11 7 2 1 0 14 3 23 22 21 20 15 16 17 18 19
8 12 11 10 4 13 7 6 9 5 0 14 3 2 1 21 20 15 23 22 19 18 17 16
1 0 13 3 21 20 14 23 6 2 17 16 15 19 22 18 9 8 11 10 5 4 12 7
13 3 2 1 14 23 22 21 4 0 15 19 18 17 20 16 10 11 8 9 12 7 6 5
2 1 0 13 22 21 20 14 7 3 18 17 16 15 23 19 8 9 10 11 6 5 4 12
0 13 3 2 20 14 23 22 5 1 16 15 19 18 21 17 11 10 9 8 4 12 7 6
11 10 9 8 18 17 16 19 12 14 21 20 23 22 15 13 1 0 3 2 7 6 5 4
3 2 1 0 7 6 5 4 15 13 9 8 11 10 12 14 17 16 19 18 21 20 23 22
17 16 14 19 9 8 15 11 0 20 5 4 7 6 18 10 3 2 12 1 23 22 13 21
14 19 18 17 15 11 10 9 2 22 6 7 4 5 16 8 12 1 0 3 13 21 20 23
18 17 16 14 10 9 8 15 1 21 4 5 6 7 19 11 0 3 2 12 20 23 22 13
16 14 19 18 8 15 11 10 3 23 7 6 5 4 17 9 2 12 1 0 22 13 21 20
19 18 17 16 23 22 21 20 13 15 3 2 1 0 14 12 5 4 7 6 9 8 11 10
6 5 4 7 11 10 9 8 14 12 19 18 17 16 13 15 23 22 21 20 1 0 3 2
4 7 15 6 1 0 3 2 20 16 23 22 12 21 8 5 19 18 13 17 11 10 14 9
15 6 5 4 2 3 0 1 22 18 12 21 20 23 10 7 13 17 16 19 14 9 8 11
5 4 7 15 0 1 2 3 21 17 20 23 22 12 9 6 16 19 18 13 8 11 10 14
7 15 6 5 3 2 1 0 23 19 22 12 21 20 11 4 18 13 17 16 10 14 9 8
21 20 23 22 16 19 12 18 17 8 11 10 13 9 4 0 7 6 14 5 3 2 15 1
22 23 20 21 12 18 17 16 19 10 13 9 8 11 6 2 14 5 4 7 15 1 0 3
20 21 22 23 17 16 19 12 18 9 8 11 10 13 5 1 4 7 6 14 0 3 2 15
23 22 21 20 19 12 18 17 16 11 10 13 9 8 7 3 6 14 5 4 2 15 1 0

The order 24 Self Orthogonal Composed Latin Diagonal Square shown above is ready to be used for the construction of an order 24 Composed Simple Magic Square.

Vorige Pagina About the Author