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Suppose n is a positive integer. The imaginary sea of Babab has islands each of which has n-letter name that uses only the letters "a" and "b", and such that for each n-letter name that uses only the letter "a" and "b", there is an island. For example, if n=3, then Aaa, Aab, Aba, Baa, Abb, Bab, Bba, and Bbb are the islands in the sea of Babab. The transportation system for Babab consists of ferries travelling back and forth between each pair of islands that differ in exactly one letter. For example, there is a ferry connecting Bab and Bbb since they differ only in the second letter.

a) How many islands and how many ferry routes are there in terms of n? Count the ferry route for both directions as a single ferry route, so for example, the ferry from Bab to Bbb is the same ferry route.

Babab does not have much in the way of natural resources or farm land so nearly all food and supplies are provided by the Babab All Bulk Company (BABCO). The people of Babab (Bababian) desire easy access to a BABCO store, where "easy access" means there is a BABCO store on their own island or on one that they can get to with a single ferry ride. However, BABCO finds it uneconomical to give the people on one island easy access to two different BABCO stores, and BABCO is willing to deny some Bababians easy access to a BABCO store in order to meet this restriction.

b) In the case n=3, n=4, and n=5, what is the maximum number of stores that BABCO can build while satisfying the restriction that no one has easy access to more than one BABCO store?

c) Now suppose BABCO changes it's strategy and decides it wants to be sure every Bababian has access to a BABCO store even if it means some Bababian have access to two stores. What is the minimum number of stores needed to satisfying this condition in the cases n=3, n=4, and n=5.

I can do question a) For question b) and c) I can do it by trying all the possible ways. But I am wondering is there a systematically way to solve b and c. And is there a general solution for n? Thanks.

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    As an aside which might help your search for more materials, the graph formed with islands as the vertices and edges as the ferry routes is what is referred to as a [hypercube graph](https://en.wikipedia.org/wiki/Hypercube_graph). I have not personally encountered the type of vertex/neighborhood coverings described in parts b and c, so maybe someone else can chime in for that. – 2017-02-21
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    in case b), do you want the minimum number of such arrangements? – 2017-02-21
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    @Lelouch I mean I can list all the possibilities. But is there any theory or systematic way behind it? Let's say n =10, obviously it is not possibility to try by listing possibilities. Is there a general solution for any n? – 2017-02-21
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    @Lelouch question b is the maximum number. I forgot to type. Question c is the minimum stores. – 2017-02-21

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