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common knowledge

The unstated assumptions are not that everyone in the room is good at logic and that the person who makes the announcement is telling the truth, but that these two facts are common knowledge (see http://en.wikipedia.org/wiki/Common_knowledge_%28logic%29). The mere facts themselves don't lead to the solution, since all assumptions of the form "(everyone thinks that)*k everyone in the room is good at logic" for k from 0 to n-2 and "(everyone believes that)*k the person who makes the announcement is telling the truth" for k from 0 to n-1 are used at some point in the deduction.

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