Suppose there are 6 holes in the ground in a line. Each day the gopher moves to an adjacent hole to the one he's in of his choice (but he never stays put). The farmer checks one hole per day for the gopher anywhere in the line, after he's moved. Does she have a strategy that is guaranteed to find him? What's the smallest number of checks she has to make to be sure she finds him eventually?
I drew a line with 6 circles for the holes.
If the farmer starts at hole 1, and for example the gopher is in hole 2, then the first day the gopher will not be caught, then the second day, the gopher must move to hole 1 or 3, so the farmer can check 3. If the gopher isn't there, then the farmer knows that on the next day, the gopher must be a hole 2, and the farmer captures the gopher. This is an example, but I can't find a pattern.
Hint 1: If on day 1 gopher was at hole $i$ and on day $n$ it was at hole $j$, what can be said about difference $i-j$?
Hint 2: If farmer checks the holes in sequence 1,2,3..., gopher can still slip away: $$\texttt{_F__G_} \to \texttt{__FG__} \to \texttt{__GF__} \to \texttt{_G__F_}$$ But what if farmer knows 3 holes where the gopher can't be every time?