The problem is:
There is a mole and n holes named with numbers $1, 2, 3, \ldots, n-1, n$. The mole can start from any hole and each day it can move only from a hole to a consecutive hole. So if the mole is at hole $3$ the next day it can move to hole $2$ or hole $4$. To catch the mole you place a trap in a single hole everyday and each day you can change the position of the trap to anywhere you like.
There is a winning strategy for catching the mole and I proved it does catch the mole eventually. If there are $n$ holes and $n$ is odd place the trap in the sequence $$2, 3, 4, 5, \ldots, n-2, n-1, 2, 3, 4, 5, \ldots, n-2, n-1$$ and if $n$ is even place the trap in the sequence $$2, 3, 4, 5, \ldots, n-2, n-1, 1, 2, 3, 4, \ldots, n-2, n-1\text{.}$$ So for odd $n$ $2(n-2)$ days are required and for even $n$ $2(n-2)+1$ days are required with this method.
How can I show that this is the minimum number of days required to guarantee catching the mole?
Any help is appreciated.