Question 57
$k$ cards numbered $1$ to $k$ are arranged at random in a row. In a move, we may change any block of consecutive cards whose numbers are in ascending or descending order and switch the block around. For example, in the case $k = 4$, $4132$ may be changed to $1432$ and $3214$ may be changed to $1234$. Suppose that in at most $n_k$ moves, we can arrange the $k$ cards so that their numbers are in ascending or descending order.
a. Prove that $n_4 = 3$.
b. Determine or estimate $n_5$.
a. Prove that $n_4 = 3$.
b. Determine or estimate $n_5$.