What is Claudius and Laertes's three-part plan to kill Hamlet?

Claudius and Laertes's plan to kill Hamlet is to arrange a duel between Hamlet and Laertes. Hamlet could either be wounded by the (secretly) sharpened sword of Laertes, poisoned by the venom with which Laertes will anoint the tip, or killed by more poison which Claudius will place in his chalice when he calls for a drink.

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When Claudius and Laertes conspire to kill Hamlet, they plan three separate methods of attack in order to ensure his death.

The first part of the plan, which Claudius suggests, is to goad Hamlet into a fencing match in which Laertes may kill him. This, Claudius says, will be just revenge for Hamlet's murder of Laertes's father, Polonius, which was also carried out with a sword. Fencing matches of this sort were not meant to be dangerous, and both parties would normally be dueling with dulled blades to prevent injury. However, Laertes plans to secretly use a sharpened sword so that he may wound Hamlet.

Laertes devises the second part of the plan, proposing that he will anoint the end of his sword with a strong poison. Thus, Laertes doesn't need to win the duel to kill Hamlet—he simply needs to strike him once. As he says,

I'll touch my point

With this contagion, that, if I gall him slightly,
It may be death.

To "gall" Hamlet is to make contact with him. So, if Laertes even scratches Hamlet with his sword, he will still be killed by the poison.

The third part of the plan is a contingency thought up by Claudius. He wants to make sure that this plan is entirely failsafe and not leave their success entirely up to Laertes's skill with his sword. He tells Laertes,

When in your motion you are hot and dry—
As make your bouts more violent to that end—
And that he calls for drink, I'll have prepared him
A chalice for the nonce, whereon but sipping,
If he by chance escape your venom'd stuck,
Our purpose may hold there.

He intends to poison the chalice that Hamlet would drink from when he becomes thirsty during the duel. Thus, even if Hamlet proves such a skilled swordsman that Laertes is unable to touch him with his poisoned weapon, Hamlet will still be poisoned when he pauses to drink from his cup (something he will undoubtably do during an exhausting duel).

In summary, provided Claudius and Laertes can successfully goad Hamlet into a duel, they have set up three possibilities for his death: he may be killed by a fatal blow from Laertes's sword, a scratch from the poisoned sword tip, or a drink from the poisoned chalice. However, in placing so many hidden weapons and threats around the room, Claudius and Laertes also allow for the possibility of far more deaths than Hamlet's—including their own.

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The first part of the plan is for the rapiers that Hamlet and Laertes use for the fight to be left unblunted. Usually, for exhibits of this kind, the idea is that the fighters will show off their skills without anybody getting hurt, but in case, the two will be able to wound or kill each other with their weapons. Laertes is glad of this, as he wants revenge on Hamlet for killing his father, and Hamlet goes along because he has decided his fate is in God's hand.

Laertes has poison for which there is no antidote, so he suggests putting this on the end of his rapier as well. That way, he doesn't even have to stab Hamlet to kill him; even a scratch that barely breaks the skin will expose him to the powerful poison.

Claudius, who has been thwarted by Hamlet too many times—we can only imagine how surprised he was when the pesky nephew he thought would be killed in England popped back up in Denmark—has yet another fallback. He assumes that Hamlet will get thirsty from the fight, so he arranges for a poisoned chalice of wine to be offered to him at that point.

Laertes eventually repents of his revenge scheme and confesses it to Hamlet, but it is too late, and both end up killed by the poisoned rapier.

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When Claudius persuades Laertes to fight a duel against Hamlet, he tells him to choose a sword "unbated." By this, he means a sword that has not been blunted for the purposes of fencing and practice but a sharp, fully-functioning weapon, which will enable him to kill Hamlet in the duel.

Laertes, however, has an even better idea. He says that he will anoint the tip of the sword with a poison he has bought from a mountebank. This poison is so deadly that Hamlet will be killed even if his skin is only scratched by the point. Even this is not enough for Claudius, who decides that there should be a third stratagem. When the two of them are hot and tired from the fight, he will bring refreshments, and there will be more poison in Hamlet's chalice of wine.

Hamlet is therefore threatened by the sharpened sword, the venom on its point, and the poison in his drink. This means that there are many ways to die available on stage in the final scene, making it easier for Shakespeare to pile up the bodies. These include Hamlet's, but the same means of death they prepare for him will first kill Laertes and Claudius, as well as his queen.

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OK. It's the fencing match. And I think it's actually a two-point plan. First of all, Laertes once to cut Hamlet's throat in the church, but Claudius persuades him there's a better way. To somehow rig the fencing match so that Hamlet ends up dead. Laertes agrees:

I will do't!
And for that purpose I'll anoint my sword.
I bought an unction of a mountebank,
So mortal that but dip a knife in it,
Where it draws blood no cataplasm so rare,
Collected from all simples that have virtue
Under the moon, can save the thing from death
This is but scratch'd withal. I'll touch my point
With this contagion, that, if I gall him slightly,
It may be death.

So, the poison on the rapier will kill Hamlet. But Claudius organises a back up plan:

Therefore this project
Should have a back or second, that might hold
If this did blast in proof. Soft! let me see.
We'll make a solemn wager on your cunnings—
I ha't!
When in your motion you are hot and dry—
As make your bouts more violent to that end—
And that he calls for drink, I'll have prepared him
A chalice for the nonce; whereon but sipping,
If he by chance escape your venom'd stuck,
Our purpose may hold there.

Claudius will put poison into a chalice, and offer it to Hamlet as a toast, if the rapier plan fails to work. So it's a two-point plan for killing him off.

Hope it helps!

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