A hunter is searching for an invisible rabbit on the number line with a tranquilizer dart gun and

a tracking device. Earlier, the hunter has painted every integer one of $C$ colors; their tracking

device constantly reports the color of the rabbit's location. The rabbit starts at any integer

$h_1$; each second thereafter, if it is at $h_t,$ it hops to a new integer $h_{t+1}$ such that $|h_{t+1}-h_t|1.$

Before each of the rabbit's hops, the hunter may shoot a dart at an integer $($or not$).$

Since invisible rabbits are hard to detect, the hunter receives no information other than

tracking device outputs and whether they have successfully shot the rabbit. The hunter has to

pre$-$commit to a coloring and shooting strategy $($i$.$e$.,$ where and when to shoot as a function of

the tracking device outputs$)$ before the hunt starts, and the rabbit sees both the coloring and

strategy. The hunter's strategy cannot involve making randomized decisions.

$($a$)$ Suppose we add the additional condition $h_{t}in\{1,2,$dots,$6\}$ for all $t.$ Then the hunter can

obviously guarantee to hit the rabbit before its first hop by using six different colors and

shooting the color corresponding to $h_1.$ For which smaller $C$ can the hunter also devise

a strategy to guarantee shooting the rabbit eventually?

$($b$)$ Suppose instead of requiring $h_{t}in\{1,2,$dots,$6\},$ we allow $h_t$ to be any integer but demand

that no number appears in the sequence $(h_1,h_2,$dots$)$ infinitely many times. For what $C$

can the hunter guarantee they can shoot the rabbit?

$($c$)$ Same question as $($b$),$ but the hunter can only shoot once.