Act IV, scene III is very hard to interpret because there could be so many ways that one person could take it. I think that Hamlet is "mad" to an extent but that more of it is a show that he puts on and is using some of these characters as his puppets. His mockery and wit seem to go over all of their heads and he is completely amused by that. Hamlet's character is much more wise than one is lead to believe and in this scene I think he shows part of his mastery.

We start off with the King sending people to find both Hamlet and the body of Polonius. He cannot prosecute Hamlet under the law because "He's loved of the distracted multitude" (4.3, 4) He also must figure out a way for the trip to England to appear useful, trusted and must take careful consideration...