This rule has been used for centuries in all sorts of creative writing, particularly kids' stories:
The three blind mice, three little pigs, Goldilocks and the three bears etc...
It works because the audience / reader are compelled to wait for pay-off, in our case the punch-line... and compelled to wait for just the right amount of time.
Please bear in mind that generally you shouldn't over-use this trick. The majority of your gags are best as singles - but variety is the spice of life.
Jokes delivered in 2 parts don't always work. Somehow the delay can be too short. And a 4 part joke tends to fizzle out - you've overplayed your hand and perhaps the audience have momentarily become a bit bored.
There is something about a joke set up as a three which is just right, and it is often more effective than the same joke told as a single.
The most legendary type of 'Rule of Three' joke is that on the lines of the legendary: "An Englishman, an Irishman and a Scotsman walked into a pub..."
Here's an example:
An Englishman, an Irishman and a Scotsman were in a field. The Englishman said: "Look at that English cow over there."
The Irishman says: "No, it's Irish," whereupon the Scotsman says: "Actually it's Scottish. Look, it's got bagpipes under it."
Would the following work as well?
A Scotsman was in a field and said: Look, there's a Scottish cow. You can tell because it's got bagpipes under it."
I don't think so. In this case the tripling up certainly does improve the joke... and it's probably just as well.
I'm not suggesting you pepper a speech or a stand-up gig with Rule of Three jokes. One is probably enough, maybe two if you're going to be delivering for a while.
We'll continue our look at the Rule of Three next time.
Look after yourselves,