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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

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
h1; each second thereafter, if it is at ht, it hops to a new integer ht+1 such that |ht+1-ht|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 htin{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 h1. For which smaller C can the hunter also devise
a strategy to guarantee shooting the rabbit eventually?
(b) Suppose instead of requiring htin{1,2,dots,6}, we allow ht to be any integer but demand
that no number appears in the sequence (h1,h2,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.
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