Number three doesn't play an important role just in fairy tales but in literature in general. It's also highly important in other arts, religion, sports, ... In short - number three is important for several reasons. After all, it's a divine number according to many beliefs (think about the Holy Trinity in Christianity, Tridevi in Hinduism, or even the Tripple Goddes of Wicca) and it deserves at least some explanation.
While number three pops up fifteen times among the titles (two hundred altogether) in the famous collection by the Brothers Grimm, none of these fifteen fairy tales belongs to the corpus of the most popular ones. The reason is simple - three is not the major player in any of these tales.
Way more important is the fact that number three is used in almost all fairy tales to establish at least one of two effects: repetition or escalation.
Repetition is part of storytelling from the beginning of time. We learn through repetition and storytelling is essentially just one way of transfer of information. Yes, we could say: Snow White was naive and didn't recognize an evil attempt by the witch, but it's way more effective to tell how she fell into the trap by the brought stuff which symbolizes the sins of the majority of teenagers.
Not once.
Not twice.
Three times!
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Well, some time is obviously needed to learn a thing or two about life. This, by the way, is one of the major lessons in fairy tales (very rarely mentioned even among scholars), which, essentially are stories about growing up.
Escalation is another important reason for the use of the number three which is obvious in the Wolf and Three Pigs (no, this one is not from the Grimm collection). The first pig is lazy, his house is weak, and the wolf eats him. The second pig is a bit less lazy, his house is a bit better, and the wolf eats him too. Only the third pig is hard-working, his house strong, and the wolf can't harm him. The message is obvious - you have to work hard to achieve something (in this case safety), superficiality and half-heartedness won't bring you any good. Or in other words: you need some time to grow up (grown-ups can typically take care of their own safety).
Yes, we could say this directly, but learning such life truth by example, through graphic scenery of the wolf puffing and huffing at pigs' houses makes a way more memorable experience.
Escalation can, of course, be achieved with other numbers, but three is the lowest where we can establish some kind of rhythm. Many nations of the world use the same pattern when something important is about to happen (for instance the race starts or something heavy has to be lifted): one, two, hop!
Three is simply the most economical number for this case.
We have already mentioned repetition and escalation but sometimes three just means 'a lot'. Somebody has to solve three puzzles, defeat three enemies, bring three gifts, wait for three years, ... In such cases, three could be substituted with a lesser number (but the feel of abundance would be lost) or a higher number (with risking to become boring and predictable). Again, three is often the best option thanks to its economics.
Number three is also the best number to quickly establish some kind of imbalance, which is necessary to make a story interesting. When two brothers fight, we really don't know what to expect. When two brothers attack the third one (like in Golden Bird, for instance), injustice is obvious and our sense of right and wrong (including the need for some kind of revenge) goes on trial. When one little tailor fights two giants at once, we have an epic scene with the possibility to enter the collective memory for centuries!
Three is in most fairy tales used to establish a pattern. Three sons in The Wishing Table failed to convince their father due to the capricious goat. After one year each of the brothers gets a magic object. When the first and the second lose their precious treasuries we expect the repeat of the pattern, but this pattern is broken with the third brother and his magic object.
Patterns are interesting but the possibility of breaking the pattern is even more appealing. Yes, Snow White couldn't break her pattern and each time made a mistake. But eventually, the solution came in the form of the prince.
Some people would claim that such a solution is just another outdated stereotype, yet I firmly believe real life can show us numerous similar examples. Not every problem is solved due to the heroism, smartness, or similar characteristic of the person in trouble. Often the solution comes from the outside - from friends, relatives, complete strangers, or even higher force. And, by the way, classic fairy tales offer numerous examples of men in trouble being saved or at least helped by women.
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Let's go back to the subject - the rule of three. A great example of the rule of three can be seen in Goldilocks and Three Bears. Older versions have three similar bears but later they became a family. This way the option of repetition is upgraded with the option of escalation. As we can see, the pattern can be used both ways. More versions of a specific story we can find, the higher the possibility to find the use of the rule of three and exercising the possibilities which it offers.
Another fine example is Jack and the Bean-Stalk. In theory, he could return to the giant's castle again and again to improve his financial status (repetition), but the third time is also the last one. It's time to confront a much stronger enemy (escalation) and a great way to bring the story to the great finale.
What are your thoughts on the importance of number three, especially in fairy tales?