CMO 1984 Problem 2

Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place: Alice: Are you going to cover your keys? Bob: I would like to, but there are only $7$ colours and I have $8$ keys. Alice: Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key. Bob: You must be careful what you mean by "next to" or "three keys over from" since you can turn the key ring over and the keys are arranged in a circle. Alice: Even so, you don't need $8$ colours.Problem: What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered.