In "Hamlet" refer to Polonius, Claudius, Hamlet, and Laertes by explaining the purpose each had and how it fell on his own head.
At the end of the play, Horatio refers to "Purposes mistook/Fall'n on th' inventor's heads." Refer to Polonius, Claudius, Hamlet, and Laertes by explaining the purpose each had and how it fell on his own head.
Please Answer the Question
The way that all of these characters' plans "fall'n on th' inventor's heads" is death. Death is a common part of Shakespeare's tragedies, and "Hamlet" is no exception to that trend.
Polonius had the purpose of scouting for information for the king. He decides, with the king's promptings and concerns, to snoop into Hamlet's behavior. Hamlet has been behaving strangely towards Ophelia, his daughter, and towards everyone in the castle. And advisor to the king, Polonius takes it upon himself to meddle, spy, and figure out the puzzle of Hamlet's odd behavior. It is for this reason that he hides himself in the queen's bedroom, and that is how he ends up dead. Hamlet rashly kills the unknown intruder, and Polonius is brought down by his own meddling "inventions."
Claudius schemes to murder his brother and become king--he is successful for a while, but unfortunately for him, Hamlet finds out. This leads him to plot Hamlet's death also; he fails with Rosencrantz and Guildenstern, so then manipulates Laertes into the duel situation. To ensure death, he tosses in the poisoned wine. All this scheming to kill his brother and nephew lead to his own death at the end.
Hamlet's purpose is revenge; his snooping, strange behavior, and play-casting alert Claudius to his possible knowledge of the matter, which leads, eventually, to his death. Hamlet's design was to enact revenge, but he did it so poorly and slowly that it gave Claudius time to plot his murder. So, it was his plot to avenge his father that led to his death, to the plot coming down on his own head.
Laertes schemed to kill Hamlet to avenge his father's death. If he hadn't done that, he wouldn't have died. In the process of enacting that revenge, he is poisoned and killed.
Shakespeare, when he does something, does it with drama and flair, and all of the schemes, inventions and plotting by characters in this play end up in death for all. I hope that helped; good luck!