Question 10
There is a row of $n$ paddocks, labelled from left to right with the integers 1 to $n$, where $n \geq 3$. Skippy the Kangaroo grazes in the paddocks according to the following rule:
If Skippy is grazing in paddock $k$, then she makes a sequence of $k$ hops from a paddock to an adjacent paddock. The first of the $k$ hops is always to the right, unless Skippy is in paddock $n$, in which case it is to the left. Each of the following $k-1$ hops is in the same direction as the previous hop, unless Skippy is in paddock 1 or $n$. Skippy grazes in the paddock that she is in after the $k^{\mathrm{th}}$ hop.
For example, if n = 8 and Skippy is grazing in paddock 3, then the next four paddocks she grazes in are 6, 4, 8 and 2, in this order.
Skippy continues grazing in this way indefinitely. A paddock is overgrazed if Skippy eventually grazes in it, regardless of the paddock that she starts in. Determine all $ n \geq 3 $ for which there is exactly one overgrazed paddock.
If Skippy is grazing in paddock $k$, then she makes a sequence of $k$ hops from a paddock to an adjacent paddock. The first of the $k$ hops is always to the right, unless Skippy is in paddock $n$, in which case it is to the left. Each of the following $k-1$ hops is in the same direction as the previous hop, unless Skippy is in paddock 1 or $n$. Skippy grazes in the paddock that she is in after the $k^{\mathrm{th}}$ hop.
For example, if n = 8 and Skippy is grazing in paddock 3, then the next four paddocks she grazes in are 6, 4, 8 and 2, in this order.
Skippy continues grazing in this way indefinitely. A paddock is overgrazed if Skippy eventually grazes in it, regardless of the paddock that she starts in. Determine all $ n \geq 3 $ for which there is exactly one overgrazed paddock.