Three is a Magic Number
There were four Marx Brothers, but only three of them were funny. Good things come in threes: Musketeers, Wise Men, and Stooges. The same goes for comedy. Think of the Rule of Three as a laundry list where the third item is the punch line. Why three? It's the minimum number needed to set up and break a pattern. You build to a climax.
I've always had a great memory. Probably my earliest memory is from when I was a baby. I can even remember my christening like it was yesterday: lots of people, the noise — Queen Elizabeth hitting me over the head with a champagne bottle.
This technique lengthens the set-up by adding a list. The first two items on the list reinforce or “prove” the statement of fact at the beginning, and the third item on the list is the twist.
One great facet of the Rule of Three is that the audience is preconditioned to expect something funny at the punch line. They've heard the format so many times, it's almost as if they're in on the joke. They expect the joke to be funny at the point you want it to be funny — they know it's coming.
Sometimes the audience is so used to the Rule of Three that the best way to get them to laugh is to violate it. Make them expect the joke on the third beat, but drag it out in a funny way.
In this classic example from Monty Python and the Holy Grail (1975), the instructions for the Holy Hand Grenade of Antioch are solemnly read:
And the Lord spake, saying, “First shalt thou take out the Holy Pin. Then shalt thou count to three, no more, no less. Three shall be the number thou shalt count, and the number of the counting shall be three. Four shalt thou not count, neither count thou two, excepting that thou then proceed to three. Five is right out. Once the number three, being the third number, be reached, then lobbest thou thy Holy Hand Grenade of Antioch toward thy foe, who, being naughty in my sight, shall snuff it.
The audience thinks the joke is coming at three, but three is just the beginning. So whether you choose to use — or misuse — the rule of three, it will probably be funny. It works because you're playing with the audiences' expectations either way. It's all about timing; performers have timing when they tell a joke, but audiences have their own timing based on what they expect from a joke.

