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Problem of the Month (August 2017)

For a set S of U.S. states, we say its adjacency set A(S) is the set of numbers that count how many states in S are adjacent to each state in S. (You can find the neighboring states here except we do not consider Colorado to neighbor Arizona, or New Mexico to neighbor Utah so that the resulting graph is planar.) If T is a non-empty subset of {1,2,3,4,5,6,7,8}, does there exist a collection of states S with A(S)=T? If so, what is the largest such set? Which largest sets are unique? For example, one such set for T={2} is shown below.


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