Neighbours in a circle with similar interests I've tried this problem (from British FST $2011$) for a long time, with an induction and swapping approach, but nothing has worked. Any ideas please?
Problem:
There is a conference attended by $512$ mathematicians, who have been assigned to share $256$ twin rooms. Each is interested in some subset of the following nine subjects: algebraic topology, Banach spaces, combinatorics, differential manifolds, Euclidean geometry, fluid dynamics, group theory, harmonic analysis and inequalities. Every mathematician has a distinct set of interests (so, in particular, one is interested in nothing, and one in everything).
Show that, at the conference dinner, they can be sat in one big circle such that everyone is sat next to his roommate, and such that, if two people are sat next to one another who are not roommates, then they have sets of interests which are identical except for one subject.
 A: Edit: I didn't realize this was such an old post, but someone may find this interesting. I've edited this to include a proper citation.
This is a whimsical application of a theorem from the paper 
Fink, Jiří, Perfect matchings extend to Hamilton cycles in hypercubes, J. Comb. Theory, Ser. B 97, No. 6, 1074-1076 (2007). ZBL1126.05080.
Consider $2^n$ mathematicians as nodes in a complete graph, each labeled by a unique binary sequence of length $n$ describing a set of interests. Considering only the edges that connect nodes whose corresponding sequences differ only by one bit gives an $n$-dimensional hypercube subgraph. The assignment of roommates corresponds to a perfect matching on the complete graph. A matching is a set of edges such that every vertex is incident with at most one edge. The matching is perfect if every vertex is incident with one of its edges. 
The theorem tells us that for every perfect matching $P$ on the complete graph described above, there is another perfect matching $R$ on the $n$-dimensional hypercube subgraph such that $P\cup R$ is a Hamiltonian circuit. This Hamiltonian circuit gives an assignment of mathematicians around the round table as sought. 
