Broadly describe what conclusion the king and the queen draw about the madness of Hamlet.

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In general terms, by the end of Hamlet neither the King nor the Queen really think Hamlet is mad (crazy).  As noted above, Gertrude does seem to think he's only upset by his father's death and her "o'er hasty" marriage to Claudius. In turn, Claudius criticizes Hamlet for grieving too much and too openly, attributing any odd behavior to this excessive grief. 

Before long, though, we discover Claudius's role in the former king's death and we understand he's determined to find the cause of Hamlet's odd behavior simply to discover whether or not Hamlet knows who killed his father.  He claims outwardly that Hamlet is mad; inside he's convinced Hamlet knows his secret.

Gertrude really does wonder about her son's sanity when they talk in her bedroom and when Hamlet has inadvertantly killed Polonius.  Her son soon convinces her, though, that he's angry and upset but not mad.  She leaves their meeting and tells Claudius Hamlet is, indeed, mad; however, she does so to help maintain her son's cover not because she believes it. 

In the end, then, they're both quite aware of Hamlet's sanity but continue to act as if the charade were real. 

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In Act 2, sc. 2, Polonius comes to Claudius and Gertrude with the announcement that he knows what is bothering Hamlet. Prior to his announcement though, Gertrude says that she thinks what is bothering Hamlet is "His father's death and our hasty marriage."  Polonius says, based on the letters he read from Hamlet to Ophelia and on what Ophelia told him about Hamlet's actions, that Hamlet is love-sick.  He says all of Hamlet's actions can be explained because of his unrequited love for Ophelia.  Polonius had told Ophelia, in Act 1, sc. 3, that she was to cut off her relationship and all contact with Hamlet.  Polonius felt that Hamlet was only using her and that when Hamlet inevitably ended the relationship, Ophelia would be tainted and the family would be embarrassed.  Gertrude doesn't seem entirely convinced, but she is relieved and in Act 3, sc. 1, tells Ophelia that she hopes Hamlet's problems truly do stem from his love for her.  Claudius is eager to embrace this idea, but he wants proof.  For this reason, he and Polonius hide, in Act 3, sc. 1, and plan to have Ophelia return to Hamlet the letters he wrote to her while they watch the interaction between the two.  This sets up the famous scene where Hamlet ponders aloud about life and death in the "To be or not to be" soliloquy.

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