Office Applications and Entertainment, Latin Squares
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Construction of order 17 Self Orthogonal Composed Latin Diagonal Squares

Construct an order 16 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} and {13, 14, 15, 16}

with respectively the magic constants s4 = 6, 22, 42 and 58.

Step 1
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
14 13 16 15
15 16 13 14
13 14 15 16
16 15 14 13
10 9 12 11
11 12 9 10
9 10 11 12
12 11 10 9
10 9 12 11
11 12 9 10
9 10 11 12
12 11 10 9
14 13 16 15
15 16 13 14
13 14 15 16
16 15 14 13
1 0 3 2
2 3 0 1
0 1 2 3
3 2 1 0
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
5 4 7 6
6 7 4 5
4 5 6 7
7 6 5 4
10 9 12 11
11 12 9 10
9 10 11 12
12 11 10 9
14 13 16 15
15 16 13 14
13 14 15 16
16 15 14 13
14 13 16 15
15 16 13 14
13 14 15 16
16 15 14 13
10 9 12 11
11 12 9 10
9 10 11 12
12 11 10 9
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
5 1 14 10
10 14 1 5
1 5 10 14
14 10 5 1

The order 4 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 17 square by adding a Center Cross, filled with the number 8 (= center), to the order 16 Self Orthogonal Composed Latin Diagonal Square as shown below:

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

The Intermediate Square is magic but has to be transformed to a Self Orthogonal Latin Diagonal Square, which can be achieved by means of a set of four order 5 Auxiliary Latin Diagonal Squares:

A51
5 4 8 7 6
8 7 6 5 4
6 5 4 8 7
4 8 7 6 5
7 6 5 4 8
A52
14 13 8 16 15
8 16 15 14 13
15 14 13 8 16
13 8 16 15 14
16 15 14 13 8
A53
8 11 12 9 10
12 9 10 8 11
10 8 11 12 9
11 12 9 10 8
9 10 8 11 12
A54
8 2 3 0 1
3 0 1 8 2
1 8 2 3 0
2 3 0 1 8
0 1 8 2 3

The four Auxiliary Squares are based on the four sub series defined above and the number 8 (= 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
5 4 8 7 1 0 3 2 6 14 13 16 15 10 9 12 11
8 7 6 5 2 3 0 1 4 15 16 13 14 11 12 9 10
6 5 4 8 0 1 2 3 7 13 14 15 16 9 10 11 12
4 8 7 6 3 2 1 0 5 16 15 14 13 12 11 10 9
10 9 12 11 14 13 8 16 15 1 0 3 2 5 4 7 6
11 12 9 10 8 16 15 14 13 2 3 0 1 6 7 4 5
9 10 11 12 15 14 13 8 16 0 1 2 3 4 5 6 7
12 11 10 9 13 8 16 15 14 3 2 1 0 7 6 5 4
7 6 5 4 16 15 14 13 8 11 12 9 10 2 3 0 1
1 0 3 2 5 4 7 6 12 9 10 8 11 14 13 16 15
2 3 0 1 6 7 4 5 10 8 11 12 9 15 16 13 14
0 1 2 3 4 5 6 7 11 12 9 10 8 13 14 15 16
3 2 1 0 7 6 5 4 9 10 8 11 12 16 15 14 13
14 13 16 15 10 9 12 11 3 5 4 7 6 0 1 8 2
15 16 13 14 11 12 9 10 1 6 7 4 5 8 2 3 0
13 14 15 16 9 10 11 12 2 4 5 6 7 3 0 1 8
16 15 14 13 12 11 10 9 0 7 6 5 4 1 8 2 3

The order 17 Self Orthogonal Composed Latin Diagonal Square shown above is ready to be used for the construction of an order 17 Composed Simple Magic Square.

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