Suppose that there exists a neighborhood of k > 5 houses, arranged in a circle. The landlords want to connect the houses with roads, according to one of the two following plans: (a) Connect the houses in a circular manner, such that the roads form a single cycle. (b) Construct the complement of plan (a). That is, if two houses had a road between them in plan (a), there is no road between them now. If two houses did not have a road between them in plan (a), there is a road between them now. Prove that there will always be more roads used in plan (b) than in (a).

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