Office Applications and Entertainment, Latin Squares

Construction of order 33 Self Orthogonal Composed Latin Diagonal Squares

Construct an order 32 Self Orthogonal Composed Latin Diagonal Square.

The required order 8 Self orthogonal Latin Diagonal Sub Squares can be constructed based on the sub series:

    {0, 1 ... 7}, {8, 9 ... 15}, {16}, {17, 18 ... 24} and {25, 26 ... 32}

with respectively the magic constants s8 = 28, 92, 164 and 228

Sqrs8

25	17	0	8
0	8	25	17
8	0	17	25
17	25	8	0

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.

Construct an intermediate order 33 square by adding a Center Cross, to the order 32 Self Orthogonal Composed Latin Diagonal Square as shown below:

Step 2

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

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 9 Auxiliary Latin Diagonal Squares:

The four Auxiliary Squares are based on the four sub series defined above and the number 16 (= 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

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

The order 33 Self Orthogonal Composed Latin Diagonal Square shown above is ready to be used for the construction of an order 33 Composed Simple Magic Square.