Question 43
Let $n \ge 3$ be a positive integer. Positive integers are written in two rows on the whiteboard $a_1\ a_2\ \dots\ a_n\ b_1\ b_2\ \dots\ b_n$, where the sets $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ are some permutations of the numbers $1, 2, \dots, n$. The teacher allows the student Dima to use the following two operations as many times as he likes and in any order:
(A) replace each number $a$ of the first line with the number that is in the $a$'th place (from left to right) of the second line;
(B) replace each number $b$ of the second line with the number that is in the $b$'th place (from left to right) of the first line.
Thus, for example, the operation $A$ transforms the table
$$ \begin{array}{ccc} 1 & 2 & 3 \\ 2 & 3 & 1 \end{array} $$
into the table
$$ \begin{array}{ccc} 2 & 3 & 1 \\ 2 & 3 & 1 \end{array} $$
, which in turn is transformed by the operation $B$ into the table
$$ \begin{array}{ccc} 2 & 3 & 1 \\ 3 & 1 & 2 \end{array} $$
Can Dima obtain the table
$$ \begin{array}{ccccccc} 2 & 3 & 1 & 4 & \dots & n-1 & n \\ 2 & 3 & 4 & 5 & \dots & n & 1 \end{array} $$
from the table
$$ \begin{array}{ccccccc} 2 & 1 & 3 & 4 & \dots & n-1 & n \\ 2 & 3 & 4 & 5 & \dots & n & 1 \end{array} $$
?
(A) replace each number $a$ of the first line with the number that is in the $a$'th place (from left to right) of the second line;
(B) replace each number $b$ of the second line with the number that is in the $b$'th place (from left to right) of the first line.
Thus, for example, the operation $A$ transforms the table
$$ \begin{array}{ccc} 1 & 2 & 3 \\ 2 & 3 & 1 \end{array} $$
into the table
$$ \begin{array}{ccc} 2 & 3 & 1 \\ 2 & 3 & 1 \end{array} $$
, which in turn is transformed by the operation $B$ into the table
$$ \begin{array}{ccc} 2 & 3 & 1 \\ 3 & 1 & 2 \end{array} $$
Can Dima obtain the table
$$ \begin{array}{ccccccc} 2 & 3 & 1 & 4 & \dots & n-1 & n \\ 2 & 3 & 4 & 5 & \dots & n & 1 \end{array} $$
from the table
$$ \begin{array}{ccccccc} 2 & 1 & 3 & 4 & \dots & n-1 & n \\ 2 & 3 & 4 & 5 & \dots & n & 1 \end{array} $$
?