Н. Макарова
ГРУППЫ MOLS 26-го и 38-го ПОРЯДКА
Порядок 26 – очень сложный порядок. Метод составных квадратов для него не работает. Этот порядок относится к серии порядков n = 2(mod 6), для которой у меня нет алгоритма построения пары ОЛК. Так что до сегодняшнего дня у меня не было даже пары ОЛК 26-го порядка. Наконец-то по книге “Handbook of Combinatorial Designs” мне удалось построить группу MOLS 26-го порядка, состоящую из четырёх квадратов (см. Главу 3, пункт 3.53).
На рис. 1 изображена квази-разностная матрица, построенная из матрицы, приведённой в книге.
|
x1 |
1 |
2 |
4 |
7 |
5 |
x2 |
1 |
7 |
12 |
11 |
20 |
x3 |
1 |
8 |
21 |
15 |
6 |
x4 |
1 |
9 |
19 |
2 |
13 |
x5 |
1 |
15 |
11 |
6 |
3 |
|
1 |
2 |
4 |
7 |
5 |
x1 |
1 |
7 |
12 |
11 |
20 |
x2 |
1 |
8 |
21 |
15 |
6 |
x3 |
1 |
9 |
19 |
2 |
13 |
x4 |
1 |
15 |
11 |
6 |
3 |
x5 |
|
2 |
4 |
7 |
5 |
x1 |
1 |
7 |
12 |
11 |
20 |
x2 |
1 |
8 |
21 |
15 |
6 |
x3 |
1 |
9 |
19 |
2 |
13 |
x4 |
1 |
15 |
11 |
6 |
3 |
x5 |
1 |
|
4 |
7 |
5 |
x1 |
1 |
2 |
12 |
11 |
20 |
x2 |
1 |
7 |
21 |
15 |
6 |
x3 |
1 |
8 |
19 |
2 |
13 |
x4 |
1 |
9 |
11 |
6 |
3 |
x5 |
1 |
15 |
|
7 |
5 |
x1 |
1 |
2 |
4 |
11 |
20 |
x2 |
1 |
7 |
12 |
15 |
6 |
x3 |
1 |
8 |
21 |
2 |
13 |
x4 |
1 |
9 |
19 |
6 |
3 |
x5 |
1 |
15 |
11 |
|
5 |
x1 |
1 |
2 |
4 |
7 |
20 |
x2 |
1 |
7 |
12 |
11 |
6 |
x3 |
1 |
8 |
21 |
15 |
13 |
x4 |
1 |
9 |
19 |
2 |
3 |
x5 |
1 |
15 |
11 |
6 |
Рис. 1
По этой матрице строю латинские квадраты. Первый латинский квадрат построен с переменными x1, x2, x3, x4, x5 (рис. 2).
Первый латинский квадрат (с переменными)
|
1 |
10 |
x1 |
x5 |
17 |
19 |
13 |
18 |
16 |
x3 |
x4 |
5 |
x2 |
11 |
14 |
6 |
21 |
12 |
20 |
4 |
3 |
2 |
7 |
8 |
9 |
15 |
|
4 |
2 |
11 |
x1 |
x5 |
18 |
20 |
14 |
19 |
17 |
x3 |
x4 |
6 |
x2 |
12 |
15 |
7 |
1 |
13 |
21 |
5 |
3 |
8 |
9 |
10 |
16 |
|
6 |
5 |
3 |
12 |
x1 |
x5 |
19 |
21 |
15 |
20 |
18 |
x3 |
x4 |
7 |
x2 |
13 |
16 |
8 |
2 |
14 |
1 |
4 |
9 |
10 |
11 |
17 |
|
2 |
7 |
6 |
4 |
13 |
x1 |
x5 |
20 |
1 |
16 |
21 |
19 |
x3 |
x4 |
8 |
x2 |
14 |
17 |
9 |
3 |
15 |
5 |
10 |
11 |
12 |
18 |
|
16 |
3 |
8 |
7 |
5 |
14 |
x1 |
x5 |
21 |
2 |
17 |
1 |
20 |
x3 |
x4 |
9 |
x2 |
15 |
18 |
10 |
4 |
6 |
11 |
12 |
13 |
19 |
|
5 |
17 |
4 |
9 |
8 |
6 |
15 |
x1 |
x5 |
1 |
3 |
18 |
2 |
21 |
x3 |
x4 |
10 |
x2 |
16 |
19 |
11 |
7 |
12 |
13 |
14 |
20 |
|
12 |
6 |
18 |
5 |
10 |
9 |
7 |
16 |
x1 |
x5 |
2 |
4 |
19 |
3 |
1 |
x3 |
x4 |
11 |
x2 |
17 |
20 |
8 |
13 |
14 |
15 |
21 |
|
21 |
13 |
7 |
19 |
6 |
11 |
10 |
8 |
17 |
x1 |
x5 |
3 |
5 |
20 |
4 |
2 |
x3 |
x4 |
12 |
x2 |
18 |
9 |
14 |
15 |
16 |
1 |
|
19 |
1 |
14 |
8 |
20 |
7 |
12 |
11 |
9 |
18 |
x1 |
x5 |
4 |
6 |
21 |
5 |
3 |
x3 |
x4 |
13 |
x2 |
10 |
15 |
16 |
17 |
2 |
|
x2 |
20 |
2 |
15 |
9 |
21 |
8 |
13 |
12 |
10 |
19 |
x1 |
x5 |
5 |
7 |
1 |
6 |
4 |
x3 |
x4 |
14 |
11 |
16 |
17 |
18 |
3 |
|
15 |
x2 |
21 |
3 |
16 |
10 |
1 |
9 |
14 |
13 |
11 |
20 |
x1 |
x5 |
6 |
8 |
2 |
7 |
5 |
x3 |
x4 |
12 |
17 |
18 |
19 |
4 |
|
x4 |
16 |
x2 |
1 |
4 |
17 |
11 |
2 |
10 |
15 |
14 |
12 |
21 |
x1 |
x5 |
7 |
9 |
3 |
8 |
6 |
x3 |
13 |
18 |
19 |
20 |
5 |
|
x3 |
x4 |
17 |
x2 |
2 |
5 |
18 |
12 |
3 |
11 |
16 |
15 |
13 |
1 |
x1 |
x5 |
8 |
10 |
4 |
9 |
7 |
14 |
19 |
20 |
21 |
6 |
|
8 |
x3 |
x4 |
18 |
x2 |
3 |
6 |
19 |
13 |
4 |
12 |
17 |
16 |
14 |
2 |
x1 |
x5 |
9 |
11 |
5 |
10 |
15 |
20 |
21 |
1 |
7 |
|
11 |
9 |
x3 |
x4 |
19 |
x2 |
4 |
7 |
20 |
14 |
5 |
13 |
18 |
17 |
15 |
3 |
x1 |
x5 |
10 |
12 |
6 |
16 |
21 |
1 |
2 |
8 |
|
7 |
12 |
10 |
x3 |
x4 |
20 |
x2 |
5 |
8 |
21 |
15 |
6 |
14 |
19 |
18 |
16 |
4 |
x1 |
x5 |
11 |
13 |
17 |
1 |
2 |
3 |
9 |
|
14 |
8 |
13 |
11 |
x3 |
x4 |
21 |
x2 |
6 |
9 |
1 |
16 |
7 |
15 |
20 |
19 |
17 |
5 |
x1 |
x5 |
12 |
18 |
2 |
3 |
4 |
10 |
|
13 |
15 |
9 |
14 |
12 |
x3 |
x4 |
1 |
x2 |
7 |
10 |
2 |
17 |
8 |
16 |
21 |
20 |
18 |
6 |
x1 |
x5 |
19 |
3 |
4 |
5 |
11 |
|
x5 |
14 |
16 |
10 |
15 |
13 |
x3 |
x4 |
2 |
x2 |
8 |
11 |
3 |
18 |
9 |
17 |
1 |
21 |
19 |
7 |
x1 |
20 |
4 |
5 |
6 |
12 |
|
x1 |
x5 |
15 |
17 |
11 |
16 |
14 |
x3 |
x4 |
3 |
x2 |
9 |
12 |
4 |
19 |
10 |
18 |
2 |
1 |
20 |
8 |
21 |
5 |
6 |
7 |
13 |
|
9 |
x1 |
x5 |
16 |
18 |
12 |
17 |
15 |
x3 |
x4 |
4 |
x2 |
10 |
13 |
5 |
20 |
11 |
19 |
3 |
2 |
21 |
1 |
6 |
7 |
8 |
14 |
|
18 |
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
x1 |
x2 |
x3 |
x4 |
x5 |
|
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
1 |
2 |
x2 |
x3 |
x4 |
x5 |
x1 |
|
17 |
18 |
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
x3 |
x4 |
x5 |
x1 |
x2 |
|
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
x4 |
x5 |
x1 |
x2 |
x3 |
|
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
x5 |
x1 |
x2 |
x3 |
x4 |
Рис. 2
Теперь присваиваю переменным конкретные значения: x1 = 22, x2 = 23, x3 = 24, x4 = 25, x5 = 26. На рис. 3 вы видите первый латинский квадрат с такими значениями переменных. Латинские квадраты данной группы содержат подквадрат 5-го порядка (на рисунках подквадрат выделен серым цветом). Понятно, что можно варьировать группы из 4 подквадратов, получая различные (неизоморфные) группы MOLS 26-го порядка.
Первый латинский квадрат (с конкретными значениями переменных)
|
1 |
10 |
22 |
26 |
17 |
19 |
13 |
18 |
16 |
24 |
25 |
5 |
23 |
11 |
14 |
6 |
21 |
12 |
20 |
4 |
3 |
2 |
7 |
8 |
9 |
15 |
|
4 |
2 |
11 |
22 |
26 |
18 |
20 |
14 |
19 |
17 |
24 |
25 |
6 |
23 |
12 |
15 |
7 |
1 |
13 |
21 |
5 |
3 |
8 |
9 |
10 |
16 |
|
6 |
5 |
3 |
12 |
22 |
26 |
19 |
21 |
15 |
20 |
18 |
24 |
25 |
7 |
23 |
13 |
16 |
8 |
2 |
14 |
1 |
4 |
9 |
10 |
11 |
17 |
|
2 |
7 |
6 |
4 |
13 |
22 |
26 |
20 |
1 |
16 |
21 |
19 |
24 |
25 |
8 |
23 |
14 |
17 |
9 |
3 |
15 |
5 |
10 |
11 |
12 |
18 |
|
16 |
3 |
8 |
7 |
5 |
14 |
22 |
26 |
21 |
2 |
17 |
1 |
20 |
24 |
25 |
9 |
23 |
15 |
18 |
10 |
4 |
6 |
11 |
12 |
13 |
19 |
|
5 |
17 |
4 |
9 |
8 |
6 |
15 |
22 |
26 |
1 |
3 |
18 |
2 |
21 |
24 |
25 |
10 |
23 |
16 |
19 |
11 |
7 |
12 |
13 |
14 |
20 |
|
12 |
6 |
18 |
5 |
10 |
9 |
7 |
16 |
22 |
26 |
2 |
4 |
19 |
3 |
1 |
24 |
25 |
11 |
23 |
17 |
20 |
8 |
13 |
14 |
15 |
21 |
|
21 |
13 |
7 |
19 |
6 |
11 |
10 |
8 |
17 |
22 |
26 |
3 |
5 |
20 |
4 |
2 |
24 |
25 |
12 |
23 |
18 |
9 |
14 |
15 |
16 |
1 |
|
19 |
1 |
14 |
8 |
20 |
7 |
12 |
11 |
9 |
18 |
22 |
26 |
4 |
6 |
21 |
5 |
3 |
24 |
25 |
13 |
23 |
10 |
15 |
16 |
17 |
2 |
|
23 |
20 |
2 |
15 |
9 |
21 |
8 |
13 |
12 |
10 |
19 |
22 |
26 |
5 |
7 |
1 |
6 |
4 |
24 |
25 |
14 |
11 |
16 |
17 |
18 |
3 |
|
15 |
23 |
21 |
3 |
16 |
10 |
1 |
9 |
14 |
13 |
11 |
20 |
22 |
26 |
6 |
8 |
2 |
7 |
5 |
24 |
25 |
12 |
17 |
18 |
19 |
4 |
|
25 |
16 |
23 |
1 |
4 |
17 |
11 |
2 |
10 |
15 |
14 |
12 |
21 |
22 |
26 |
7 |
9 |
3 |
8 |
6 |
24 |
13 |
18 |
19 |
20 |
5 |
|
24 |
25 |
17 |
23 |
2 |
5 |
18 |
12 |
3 |
11 |
16 |
15 |
13 |
1 |
22 |
26 |
8 |
10 |
4 |
9 |
7 |
14 |
19 |
20 |
21 |
6 |
|
8 |
24 |
25 |
18 |
23 |
3 |
6 |
19 |
13 |
4 |
12 |
17 |
16 |
14 |
2 |
22 |
26 |
9 |
11 |
5 |
10 |
15 |
20 |
21 |
1 |
7 |
|
11 |
9 |
24 |
25 |
19 |
23 |
4 |
7 |
20 |
14 |
5 |
13 |
18 |
17 |
15 |
3 |
22 |
26 |
10 |
12 |
6 |
16 |
21 |
1 |
2 |
8 |
|
7 |
12 |
10 |
24 |
25 |
20 |
23 |
5 |
8 |
21 |
15 |
6 |
14 |
19 |
18 |
16 |
4 |
22 |
26 |
11 |
13 |
17 |
1 |
2 |
3 |
9 |
|
14 |
8 |
13 |
11 |
24 |
25 |
21 |
23 |
6 |
9 |
1 |
16 |
7 |
15 |
20 |
19 |
17 |
5 |
22 |
26 |
12 |
18 |
2 |
3 |
4 |
10 |
|
13 |
15 |
9 |
14 |
12 |
24 |
25 |
1 |
23 |
7 |
10 |
2 |
17 |
8 |
16 |
21 |
20 |
18 |
6 |
22 |
26 |
19 |
3 |
4 |
5 |
11 |
|
26 |
14 |
16 |
10 |
15 |
13 |
24 |
25 |
2 |
23 |
8 |
11 |
3 |
18 |
9 |
17 |
1 |
21 |
19 |
7 |
22 |
20 |
4 |
5 |
6 |
12 |
|
22 |
26 |
15 |
17 |
11 |
16 |
14 |
24 |
25 |
3 |
23 |
9 |
12 |
4 |
19 |
10 |
18 |
2 |
1 |
20 |
8 |
21 |
5 |
6 |
7 |
13 |
|
9 |
22 |
26 |
16 |
18 |
12 |
17 |
15 |
24 |
25 |
4 |
23 |
10 |
13 |
5 |
20 |
11 |
19 |
3 |
2 |
21 |
1 |
6 |
7 |
8 |
14 |
|
18 |
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
22 |
23 |
24 |
25 |
26 |
|
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
1 |
2 |
23 |
24 |
25 |
26 |
22 |
|
17 |
18 |
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
24 |
25 |
26 |
22 |
23 |
|
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
25 |
26 |
22 |
23 |
24 |
|
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
26 |
22 |
23 |
24 |
25 |
Рис. 3
Второй латинский квадрат сразу построила с заданными выше значениями переменных (рис. 4).
|
1 |
23 |
18 |
20 |
14 |
26 |
24 |
13 |
7 |
17 |
10 |
16 |
3 |
15 |
8 |
5 |
9 |
25 |
22 |
2 |
6 |
4 |
12 |
21 |
19 |
11 |
|
7 |
2 |
23 |
19 |
21 |
15 |
26 |
24 |
14 |
8 |
18 |
11 |
17 |
4 |
16 |
9 |
6 |
10 |
25 |
22 |
3 |
5 |
13 |
1 |
20 |
12 |
|
4 |
8 |
3 |
23 |
20 |
1 |
16 |
26 |
24 |
15 |
9 |
19 |
12 |
18 |
5 |
17 |
10 |
7 |
11 |
25 |
22 |
6 |
14 |
2 |
21 |
13 |
|
22 |
5 |
9 |
4 |
23 |
21 |
2 |
17 |
26 |
24 |
16 |
10 |
20 |
13 |
19 |
6 |
18 |
11 |
8 |
12 |
25 |
7 |
15 |
3 |
1 |
14 |
|
25 |
22 |
6 |
10 |
5 |
23 |
1 |
3 |
18 |
26 |
24 |
17 |
11 |
21 |
14 |
20 |
7 |
19 |
12 |
9 |
13 |
8 |
16 |
4 |
2 |
15 |
|
14 |
25 |
22 |
7 |
11 |
6 |
23 |
2 |
4 |
19 |
26 |
24 |
18 |
12 |
1 |
15 |
21 |
8 |
20 |
13 |
10 |
9 |
17 |
5 |
3 |
16 |
|
11 |
15 |
25 |
22 |
8 |
12 |
7 |
23 |
3 |
5 |
20 |
26 |
24 |
19 |
13 |
2 |
16 |
1 |
9 |
21 |
14 |
10 |
18 |
6 |
4 |
17 |
|
15 |
12 |
16 |
25 |
22 |
9 |
13 |
8 |
23 |
4 |
6 |
21 |
26 |
24 |
20 |
14 |
3 |
17 |
2 |
10 |
1 |
11 |
19 |
7 |
5 |
18 |
|
2 |
16 |
13 |
17 |
25 |
22 |
10 |
14 |
9 |
23 |
5 |
7 |
1 |
26 |
24 |
21 |
15 |
4 |
18 |
3 |
11 |
12 |
20 |
8 |
6 |
19 |
|
12 |
3 |
17 |
14 |
18 |
25 |
22 |
11 |
15 |
10 |
23 |
6 |
8 |
2 |
26 |
24 |
1 |
16 |
5 |
19 |
4 |
13 |
21 |
9 |
7 |
20 |
|
5 |
13 |
4 |
18 |
15 |
19 |
25 |
22 |
12 |
16 |
11 |
23 |
7 |
9 |
3 |
26 |
24 |
2 |
17 |
6 |
20 |
14 |
1 |
10 |
8 |
21 |
|
21 |
6 |
14 |
5 |
19 |
16 |
20 |
25 |
22 |
13 |
17 |
12 |
23 |
8 |
10 |
4 |
26 |
24 |
3 |
18 |
7 |
15 |
2 |
11 |
9 |
1 |
|
8 |
1 |
7 |
15 |
6 |
20 |
17 |
21 |
25 |
22 |
14 |
18 |
13 |
23 |
9 |
11 |
5 |
26 |
24 |
4 |
19 |
16 |
3 |
12 |
10 |
2 |
|
20 |
9 |
2 |
8 |
16 |
7 |
21 |
18 |
1 |
25 |
22 |
15 |
19 |
14 |
23 |
10 |
12 |
6 |
26 |
24 |
5 |
17 |
4 |
13 |
11 |
3 |
|
6 |
21 |
10 |
3 |
9 |
17 |
8 |
1 |
19 |
2 |
25 |
22 |
16 |
20 |
15 |
23 |
11 |
13 |
7 |
26 |
24 |
18 |
5 |
14 |
12 |
4 |
|
24 |
7 |
1 |
11 |
4 |
10 |
18 |
9 |
2 |
20 |
3 |
25 |
22 |
17 |
21 |
16 |
23 |
12 |
14 |
8 |
26 |
19 |
6 |
15 |
13 |
5 |
|
26 |
24 |
8 |
2 |
12 |
5 |
11 |
19 |
10 |
3 |
21 |
4 |
25 |
22 |
18 |
1 |
17 |
23 |
13 |
15 |
9 |
20 |
7 |
16 |
14 |
6 |
|
10 |
26 |
24 |
9 |
3 |
13 |
6 |
12 |
20 |
11 |
4 |
1 |
5 |
25 |
22 |
19 |
2 |
18 |
23 |
14 |
16 |
21 |
8 |
17 |
15 |
7 |
|
17 |
11 |
26 |
24 |
10 |
4 |
14 |
7 |
13 |
21 |
12 |
5 |
2 |
6 |
25 |
22 |
20 |
3 |
19 |
23 |
15 |
1 |
9 |
18 |
16 |
8 |
|
16 |
18 |
12 |
26 |
24 |
11 |
5 |
15 |
8 |
14 |
1 |
13 |
6 |
3 |
7 |
25 |
22 |
21 |
4 |
20 |
23 |
2 |
10 |
19 |
17 |
9 |
|
23 |
17 |
19 |
13 |
26 |
24 |
12 |
6 |
16 |
9 |
15 |
2 |
14 |
7 |
4 |
8 |
25 |
22 |
1 |
5 |
21 |
3 |
11 |
20 |
18 |
10 |
|
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
22 |
23 |
24 |
25 |
26 |
|
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
24 |
25 |
26 |
22 |
23 |
|
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
1 |
2 |
26 |
22 |
23 |
24 |
25 |
|
18 |
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
23 |
24 |
25 |
26 |
22 |
|
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
25 |
26 |
22 |
23 |
24 |
Рис. 4
Третий (зелёная строка матрицы) и четвёртый (жёлтая строка матрицы) латинские квадраты предлагаю построить читателям. У меня готова пара ОЛК, которую я сейчас преобразую, чтобы она была пригодна для построения магических квадратов.
Преобразовываю первый латинский квадрат следующими заменами: 8 à 18, 18 à 8; 20 à 21, 21 à 20. На рис. 5 изображён преобразованный первый латинский квадрат.
Преобразованный первый латинский квадрат
|
1 |
10 |
22 |
26 |
17 |
19 |
13 |
8 |
16 |
24 |
25 |
5 |
23 |
11 |
14 |
6 |
20 |
12 |
21 |
4 |
3 |
2 |
7 |
18 |
9 |
15 |
|
4 |
2 |
11 |
22 |
26 |
8 |
21 |
14 |
19 |
17 |
24 |
25 |
6 |
23 |
12 |
15 |
7 |
1 |
13 |
20 |
5 |
3 |
18 |
9 |
10 |
16 |
|
6 |
5 |
3 |
12 |
22 |
26 |
19 |
20 |
15 |
21 |
8 |
24 |
25 |
7 |
23 |
13 |
16 |
18 |
2 |
14 |
1 |
4 |
9 |
10 |
11 |
17 |
|
2 |
7 |
6 |
4 |
13 |
22 |
26 |
21 |
1 |
16 |
20 |
19 |
24 |
25 |
18 |
23 |
14 |
17 |
9 |
3 |
15 |
5 |
10 |
11 |
12 |
8 |
|
16 |
3 |
18 |
7 |
5 |
14 |
22 |
26 |
20 |
2 |
17 |
1 |
21 |
24 |
25 |
9 |
23 |
15 |
8 |
10 |
4 |
6 |
11 |
12 |
13 |
19 |
|
5 |
17 |
4 |
9 |
18 |
6 |
15 |
22 |
26 |
1 |
3 |
8 |
2 |
20 |
24 |
25 |
10 |
23 |
16 |
19 |
11 |
7 |
12 |
13 |
14 |
21 |
|
12 |
6 |
8 |
5 |
10 |
9 |
7 |
16 |
22 |
26 |
2 |
4 |
19 |
3 |
1 |
24 |
25 |
11 |
23 |
17 |
21 |
18 |
13 |
14 |
15 |
20 |
|
20 |
13 |
7 |
19 |
6 |
11 |
10 |
18 |
17 |
22 |
26 |
3 |
5 |
21 |
4 |
2 |
24 |
25 |
12 |
23 |
8 |
9 |
14 |
15 |
16 |
1 |
|
19 |
1 |
14 |
18 |
21 |
7 |
12 |
11 |
9 |
8 |
22 |
26 |
4 |
6 |
20 |
5 |
3 |
24 |
25 |
13 |
23 |
10 |
15 |
16 |
17 |
2 |
|
23 |
21 |
2 |
15 |
9 |
20 |
18 |
13 |
12 |
10 |
19 |
22 |
26 |
5 |
7 |
1 |
6 |
4 |
24 |
25 |
14 |
11 |
16 |
17 |
8 |
3 |
|
15 |
23 |
20 |
3 |
16 |
10 |
1 |
9 |
14 |
13 |
11 |
21 |
22 |
26 |
6 |
18 |
2 |
7 |
5 |
24 |
25 |
12 |
17 |
8 |
19 |
4 |
|
25 |
16 |
23 |
1 |
4 |
17 |
11 |
2 |
10 |
15 |
14 |
12 |
20 |
22 |
26 |
7 |
9 |
3 |
18 |
6 |
24 |
13 |
8 |
19 |
21 |
5 |
|
24 |
25 |
17 |
23 |
2 |
5 |
8 |
12 |
3 |
11 |
16 |
15 |
13 |
1 |
22 |
26 |
18 |
10 |
4 |
9 |
7 |
14 |
19 |
21 |
20 |
6 |
|
18 |
24 |
25 |
8 |
23 |
3 |
6 |
19 |
13 |
4 |
12 |
17 |
16 |
14 |
2 |
22 |
26 |
9 |
11 |
5 |
10 |
15 |
21 |
20 |
1 |
7 |
|
11 |
9 |
24 |
25 |
19 |
23 |
4 |
7 |
21 |
14 |
5 |
13 |
8 |
17 |
15 |
3 |
22 |
26 |
10 |
12 |
6 |
16 |
20 |
1 |
2 |
18 |
|
7 |
12 |
10 |
24 |
25 |
21 |
23 |
5 |
18 |
20 |
15 |
6 |
14 |
19 |
8 |
16 |
4 |
22 |
26 |
11 |
13 |
17 |
1 |
2 |
3 |
9 |
|
14 |
18 |
13 |
11 |
24 |
25 |
20 |
23 |
6 |
9 |
1 |
16 |
7 |
15 |
21 |
19 |
17 |
5 |
22 |
26 |
12 |
8 |
2 |
3 |
4 |
10 |
|
13 |
15 |
9 |
14 |
12 |
24 |
25 |
1 |
23 |
7 |
10 |
2 |
17 |
18 |
16 |
20 |
21 |
8 |
6 |
22 |
26 |
19 |
3 |
4 |
5 |
11 |
|
26 |
14 |
16 |
10 |
15 |
13 |
24 |
25 |
2 |
23 |
18 |
11 |
3 |
8 |
9 |
17 |
1 |
20 |
19 |
7 |
22 |
21 |
4 |
5 |
6 |
12 |
|
22 |
26 |
15 |
17 |
11 |
16 |
14 |
24 |
25 |
3 |
23 |
9 |
12 |
4 |
19 |
10 |
8 |
2 |
1 |
21 |
18 |
20 |
5 |
6 |
7 |
13 |
|
9 |
22 |
26 |
16 |
8 |
12 |
17 |
15 |
24 |
25 |
4 |
23 |
10 |
13 |
5 |
21 |
11 |
19 |
3 |
2 |
20 |
1 |
6 |
7 |
18 |
14 |
|
8 |
19 |
21 |
20 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
18 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
22 |
23 |
24 |
25 |
26 |
|
3 |
4 |
5 |
6 |
7 |
18 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
8 |
19 |
21 |
20 |
1 |
2 |
23 |
24 |
25 |
26 |
22 |
|
17 |
8 |
19 |
21 |
20 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
18 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
24 |
25 |
26 |
22 |
23 |
|
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
8 |
19 |
21 |
20 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
18 |
9 |
25 |
26 |
22 |
23 |
24 |
|
21 |
20 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
18 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
8 |
19 |
26 |
22 |
23 |
24 |
25 |
Рис. 5
Преобразовываю второй латинский квадрат такой взаимозаменой: 3 à 25, 25 à 3. Полученный квадрат показан на рис. 6.
Преобразованный второй латинский квадрат
|
1 |
23 |
18 |
20 |
14 |
26 |
24 |
13 |
7 |
17 |
10 |
16 |
25 |
15 |
8 |
5 |
9 |
3 |
22 |
2 |
6 |
4 |
12 |
21 |
19 |
11 |
|
7 |
2 |
23 |
19 |
21 |
15 |
26 |
24 |
14 |
8 |
18 |
11 |
17 |
4 |
16 |
9 |
6 |
10 |
3 |
22 |
25 |
5 |
13 |
1 |
20 |
12 |
|
4 |
8 |
25 |
23 |
20 |
1 |
16 |
26 |
24 |
15 |
9 |
19 |
12 |
18 |
5 |
17 |
10 |
7 |
11 |
3 |
22 |
6 |
14 |
2 |
21 |
13 |
|
22 |
5 |
9 |
4 |
23 |
21 |
2 |
17 |
26 |
24 |
16 |
10 |
20 |
13 |
19 |
6 |
18 |
11 |
8 |
12 |
3 |
7 |
15 |