 The problem that John Nash writes on the blackboard in his lecture  is a real one (unlike in other movies, where math on boards is  usually either too simple or fake). There is an important theorem in  mathematical physics that directly says the answer to this is 1.  Later, when he discusses the problem with Alicia, he makes  additional restrictions for the solution, without which the problem  is much harder, so he is pretty confident she didnt solve it.