Office Applications and Entertainment, Latin Squares

Construction of order 29 Self Orthogonal Composed Latin Diagonal Squares

Construct an order 28 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, 16 ... 21} and {22, 23 ... 28}

with respectively the magic constants s7 = 21, 70, 126 and 175

Sqrs7

24	17	2	9
2	9	24	17
9	2	17	24
17	24	9	2

The order 4 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.

Construct an intermediate order 29 square by adding a Center Cross, to the order 28 Self Orthogonal Composed Latin Diagonal Square as shown below:

Step 2

24	26	25	22	27	28	23	17	19	18	15	20	21	16	0	2	4	3	0	5	6	1	9	11	10	7	12	13	8
28	23	27	26	24	22	25	21	16	20	19	17	15	18	0	6	1	5	4	2	0	3	13	8	12	11	9	7	10
23	24	22	27	26	25	28	16	17	15	20	19	18	21	0	1	2	0	5	4	3	6	8	9	7	12	11	10	13
26	27	28	25	22	23	24	19	20	21	18	15	16	17	0	4	5	6	3	0	1	2	11	12	13	10	7	8	9
22	25	24	23	28	26	27	15	18	17	16	21	19	20	0	0	3	2	1	6	4	5	7	10	9	8	13	11	12
25	28	26	24	23	27	22	18	21	19	17	16	20	15	0	3	6	4	2	1	5	0	10	13	11	9	8	12	7
27	22	23	28	25	24	26	20	15	16	21	18	17	19	0	5	0	1	6	3	2	4	12	7	8	13	10	9	11
2	4	3	0	5	6	1	9	11	10	7	12	13	8	0	24	26	25	22	27	28	23	17	19	18	15	20	21	16
6	1	5	4	2	0	3	13	8	12	11	9	7	10	0	28	23	27	26	24	22	25	21	16	20	19	17	15	18
1	2	0	5	4	3	6	8	9	7	12	11	10	13	0	23	24	22	27	26	25	28	16	17	15	20	19	18	21
4	5	6	3	0	1	2	11	12	13	10	7	8	9	0	26	27	28	25	22	23	24	19	20	21	18	15	16	17
0	3	2	1	6	4	5	7	10	9	8	13	11	12	0	22	25	24	23	28	26	27	15	18	17	16	21	19	20
3	6	4	2	1	5	0	10	13	11	9	8	12	7	0	25	28	26	24	23	27	22	18	21	19	17	16	20	15
5	0	1	6	3	2	4	12	7	8	13	10	9	11	0	27	22	23	28	25	24	26	20	15	16	21	18	17	19
0	0	0	0	0	0	0	0	0	0	0	0	0	0	14	0	0	0	0	0	0	0	0	0	0	0	0	0	0
9	11	10	7	12	13	8	2	4	3	0	5	6	1	0	17	19	18	15	20	21	16	24	26	25	22	27	28	23
13	8	12	11	9	7	10	6	1	5	4	2	0	3	0	21	16	20	19	17	15	18	28	23	27	26	24	22	25
8	9	7	12	11	10	13	1	2	0	5	4	3	6	0	16	17	15	20	19	18	21	23	24	22	27	26	25	28
11	12	13	10	7	8	9	4	5	6	3	0	1	2	0	19	20	21	18	15	16	17	26	27	28	25	22	23	24
7	10	9	8	13	11	12	0	3	2	1	6	4	5	0	15	18	17	16	21	19	20	22	25	24	23	28	26	27
10	13	11	9	8	12	7	3	6	4	2	1	5	0	0	18	21	19	17	16	20	15	25	28	26	24	23	27	22
12	7	8	13	10	9	11	5	0	1	6	3	2	4	0	20	15	16	21	18	17	19	27	22	23	28	25	24	26
17	19	18	15	20	21	16	24	26	25	22	27	28	23	0	9	11	10	7	12	13	8	2	4	3	0	5	6	1
21	16	20	19	17	15	18	28	23	27	26	24	22	25	0	13	8	12	11	9	7	10	6	1	5	4	2	0	3
16	17	15	20	19	18	21	23	24	22	27	26	25	28	0	8	9	7	12	11	10	13	1	2	0	5	4	3	6
19	20	21	18	15	16	17	26	27	28	25	22	23	24	0	11	12	13	10	7	8	9	4	5	6	3	0	1	2
15	18	17	16	21	19	20	22	25	24	23	28	26	27	0	7	10	9	8	13	11	12	0	3	2	1	6	4	5
18	21	19	17	16	20	15	25	28	26	24	23	27	22	0	10	13	11	9	8	12	7	3	6	4	2	1	5	0
20	15	16	21	18	17	19	27	22	23	28	25	24	26	0	12	7	8	13	10	9	11	5	0	1	6	3	2	4

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 8 Auxiliary Latin Diagonal Squares:

The four Auxiliary Squares are based on the four sub series defined above and the number 14 (= 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

22	24	27	28	14	26	25	17	19	18	15	20	21	16	23	2	4	3	0	5	6	1	9	11	10	7	12	13	8
25	23	14	26	27	28	22	21	16	20	19	17	15	18	24	6	1	5	4	2	0	3	13	8	12	11	9	7	10
23	27	24	14	28	25	26	16	17	15	20	19	18	21	22	1	2	0	5	4	3	6	8	9	7	12	11	10	13
26	22	28	25	24	14	23	19	20	21	18	15	16	17	27	4	5	6	3	0	1	2	11	12	13	10	7	8	9
24	28	22	27	26	23	14	15	18	17	16	21	19	20	25	0	3	2	1	6	4	5	7	10	9	8	13	11	12
14	25	26	23	22	27	24	18	21	19	17	16	20	15	28	3	6	4	2	1	5	0	10	13	11	9	8	12	7
27	14	23	24	25	22	28	20	15	16	21	18	17	19	26	5	0	1	6	3	2	4	12	7	8	13	10	9	11
2	4	3	0	5	6	1	7	9	12	13	14	11	10	8	24	26	25	22	27	28	23	17	19	18	15	20	21	16
6	1	5	4	2	0	3	10	8	14	11	12	13	7	9	28	23	27	26	24	22	25	21	16	20	19	17	15	18
1	2	0	5	4	3	6	8	12	9	14	13	10	11	7	23	24	22	27	26	25	28	16	17	15	20	19	18	21
4	5	6	3	0	1	2	11	7	13	10	9	14	8	12	26	27	28	25	22	23	24	19	20	21	18	15	16	17
0	3	2	1	6	4	5	9	13	7	12	11	8	14	10	22	25	24	23	28	26	27	15	18	17	16	21	19	20
3	6	4	2	1	5	0	14	10	11	8	7	12	9	13	25	28	26	24	23	27	22	18	21	19	17	16	20	15
5	0	1	6	3	2	4	12	14	8	9	10	7	13	11	27	22	23	28	25	24	26	20	15	16	21	18	17	19
28	26	25	22	23	24	27	13	11	10	7	8	9	12	14	16	19	20	21	18	17	15	1	4	5	6	3	2	0
9	11	10	7	12	13	8	2	4	3	0	5	6	1	17	15	21	18	19	20	14	16	24	26	25	22	27	28	23
13	8	12	11	9	7	10	6	1	5	4	2	0	3	15	19	16	21	20	17	18	14	28	23	27	26	24	22	25
8	9	7	12	11	10	13	1	2	0	5	4	3	6	18	14	20	17	16	21	15	19	23	24	22	27	26	25	28
11	12	13	10	7	8	9	4	5	6	3	0	1	2	16	20	14	19	18	15	21	17	26	27	28	25	22	23	24
7	10	9	8	13	11	12	0	3	2	1	6	4	5	21	17	18	15	14	19	16	20	22	25	24	23	28	26	27
10	13	11	9	8	12	7	3	6	4	2	1	5	0	19	21	15	16	17	14	20	18	25	28	26	24	23	27	22
12	7	8	13	10	9	11	5	0	1	6	3	2	4	20	18	17	14	15	16	19	21	27	22	23	28	25	24	26
17	19	18	15	20	21	16	24	26	25	22	27	28	23	2	9	11	10	7	12	13	8	0	6	3	4	5	14	1
21	16	20	19	17	15	18	28	23	27	26	24	22	25	0	13	8	12	11	9	7	10	4	1	6	5	2	3	14
16	17	15	20	19	18	21	23	24	22	27	26	25	28	3	8	9	7	12	11	10	13	14	5	2	1	6	0	4
19	20	21	18	15	16	17	26	27	28	25	22	23	24	1	11	12	13	10	7	8	9	5	14	4	3	0	6	2
15	18	17	16	21	19	20	22	25	24	23	28	26	27	6	7	10	9	8	13	11	12	2	3	0	14	4	1	5
18	21	19	17	16	20	15	25	28	26	24	23	27	22	4	10	13	11	9	8	12	7	6	0	1	2	14	5	3
20	15	16	21	18	17	19	27	22	23	28	25	24	26	5	12	7	8	13	10	9	11	3	2	14	0	1	4	6

The order 29 Self Orthogonal Composed Latin Diagonal Square shown above is ready to be used for the construction of an order 29 Composed Simple Magic Square.