How was Stanley Kubrick able to make a film that could be interpreted in so many ways? Is there a technique that allows you to make a film that can be interpreted in many different ways? The Shining has several interpretations and none of them are completely satisfactory.
One way to do this is to seed the question by having characters deliberately ask it or invoke the audience to ask it... but do not give the answer.
Consider the film Inception where the characters can enter someone's dreams and may even enter their dream persona's own dream... the further down they go, the more likely they are to be lost in the dream world... so they each keep totems... items that behave a certain way in the dream world, with Leo's character keeping a top that will spin indefinitely in the dream world. When he thinks he emerges, he spins the top and watches it fall over to conlcude he's back in the real world. The big question that is seeded in the audience's mind comes at the end, where Leo's character spins the top... but looks away and chooses not to observe if it falls. The film ends on the still spinning top after it starts to seemingly wobble like it will fall, but before it actually does.
The audience is then left wondering what the implications and if the ending is really a happy one (with Leo back in the waking world) or just another level of dream that he's trapped in.
Another example is to show that there is an unreliable narrator in your presence. Edgar Allen Poe's Cask of Amantilado is narrated by one such device, and he demonstrates in the opening of the story that you cannot trust the man telling you the story. Although he admits to entombing a living person in the cell with the titular cask in retribution for an undisclosed crime, the narrator's character and motivation for his cruel act of murder are called into doubt as even he is unwilling to justify his own actions and relies on the implicit trust of the narrative voice. Whether his victim was truly deserving of his fate or totally innocent and slandered by the narrator in his quest to justify his deeds is left up to the reader to decide. The unreliable narrator need not be malicious. In "How I Met Your Mother" the show is framed as a Future Ted telling his children the story that answers the titular question... but he's unreliable out of both sheltering his children by censoring his own actions, such as a sexist term he called a friend in a Christmas episode getting turned into him calling her "a Grinch" or anytime he brings up the fact that he and his friends were smoking pot, he quickly changes the activity to "eating sandwiches". The characters in the show proper go along with these changes as if that was what really happened, but the context clues imply the actual term used. The use of Grinch is pretty clearly the "c-word"... and the narrator has to stop the story to reveal that someone actually called another person "A Grinch" in the correct context it would be used, as an insult for someone who is ruining Christmas. In the "sandwiches" example, the characters have subs and hoagies in their hands but are clearly high and in one episode, successfully make "Sandwich Brownies".
In other instances, the narrator might not be unreliable but the exact nature of the events is perceived differently by all involved (normally in homage to Roshamone). While this is normally a done-to-death premise, the film they are parodying is actually the reverse... rather than characters remembering the same sequence of events differently to deflect blame, in Rashamone, the same sequence of events is told differently in acceptance of blame. The facts in evidence in the film are that a Samurai and his wife encounter a thief while traveling and this encounter leads to the Samurai being fatally stabbed with his wife's ornate dagger, which is not present when the body is found. The account as to what happened is told four times (the thief's tale, the wife's tale, the Samurai/Medium's tale, and a spoiler character's tale) and each is largely incompatible with each other. The thief says he raped the wife, then dueled the Samurai in an epic combat and stabbed the Samurai with the dagger after being goaded to by the wife, though she flees him after this happens. The thief also laments leaving the dagger, as it would have sold for a high price. The wife said she killed the Samurai after begging for forgiveness for her unfaithfulness in the sexual encounter with the Samurai, which he coldly refuses to acknowledge, causing her to kill him and leave the knife in his corpse as she fled. The Samurai (channeled from beyond by a medium) says that the wife ran off with the thief after his defeat, but left him alive. The wife had dropped the knife and in his own shame, he took his life. While he was still dying someone came and pulled the knife from his chest, but he was too far gone to identify who this person was. Finally, the spoiler character does is unique in that while he denies any hand in the death and confirms that portions of the story were true but exaggerates, he is the only person whose version of the events places the Samurai's murder on someone other than himself (he confirms the thief killed the Samurai) but where his culpability lies is that he stole the missing knife from the scene to sell the blade.
Through the production, the actors themselves were confused as to which of the stories was the real version of the events and demanded the director, Akira Kurosawa, tell them what really happened... but for Kurosawa, that was the point of the film... that people will lie when logic dictates they should tell the truth. Of the four characters, each one admits to being criminally responsible for something in their story (the first three are guilty of murder, while the last one is a thief, by their own admittance) and each one tells a story that conflicts directly with everyone else's (though the fourth's seems to confirm some elements of the tale are true but exaggerated... but is still in conflict with what took place).
Here the question is not only which story is true... but who can we trust... as the differences in each reflect the character's own lies and while the fourth character does justify his actions... we have to remember he is lying about even seeing the conflict from the beginning and could be lying in his sympathetic back story. And given the fact that this is fiction about people who are telling fictional stories... who's to say what really happened.
There are simple techniques that already make a story more open for interpretation - I'm not sure if they'll generate countless interpretations, but they make for a good start :)
1 An unreliable narrator
An unreliable narrator tells the story from a first person singular view ("I"). That also means that the narrator tells the story the way he/she interprets it. This might not be how the other characters interpreted it, or might even be a total lie.
Example: Wuthering Heights is told by a man named Mr. Lockwood, who heard the story from a lady's maid, Nelly. Nelly might have painted the story in a very personal way --> there is a lot of room for interpretation on what Cathy really was like, what Heathcliff really was like.
Example: stories told from the point of view of someone who's in love with the main character, someone who despises the main character, someone who has a very vivid imagination...
2 Several unreliable narrators
You can tell the story throught different narrators, and let them contradict each other. That way, there are several interpretations about what actually happened. Bret Easton Ellis does this in The Laws of Attraction.: the character Paul gives long descriptions of his love affair with a boy names Sean. But Sean barely even mentions Paul. As a reader, you don't know if Paul is imagining things or if Sean is denying his feelings. Or perhaps there's a third, forth, fifth... explanation.
3 Show don't tell
Don't explain people's behaviour. Only show their actions. Let the reader decide what they mean. In The Great Gatsby, we don't know if Gatsby ever truly loved Daisy, if Daisy ever truly loved Tom, and so on. In the Harry Potter series, we don't know why the Dursleys were so mean to Harry. We only know they were. Is it because Petunia was jealous of Lily's magic, is it because Harry is a Horcrux, is it because they're just awful people...? These "blanks" automatically lift the story to a higher level, and leave room for interpretations.
I hope this helps :) Good luck!
Leaving plenty of room for interpretation:
This sounds much easier than it is. In my answer, I will share just some of the ways to accomplish this. Some of my ideas are based on other answers to this question, but I feel that mine elaborate in greater detail, which is why I decided to add my answer to this discussion.
Before you get to writing your ideas, you must brainstorm them. Often, you have a set plan of what you will write before you start writing. However, if your goal is to write a piece that could be interpreted in many ways, a clever approach would be to not exactly know how you are interpreting your own story ideas while you write.
To elaborate on this, before setting your ideas in stone depending on your judgment and opinion, let the writing decide where to go. Even if that means your scrawling down half-written ideas. This way, your writing will remain very open to reader interpretation (of course, you should go back later to fill in any gaping holes). This way, your reader will follow along and interpret as they would like to.
This was pointed out by Charlotte in her answer. If your story is first-person, then you have it easy. As people, our first assumptions, judgments, and interpretations are often wrong. Your character can be wrong as well. This will make your reader second guess their first assumptions, and then triple guess their second guess (and so on). If your story is in third-person, this is a bit trickier, because you would have to flat out lie if you were hoping to provide misleading information. A blog post on penultimateword.com describes this perfectly and provides more information. It says:
An unreliable narrator is almost always written in the first person, making your protagonist the unreliable party. If you try to write an unreliable protagonist in the third person, that’s going to make you, the author, the unreliable party—and trust me, that likely won’t work. Readers may accept one of your characters as a liar, but they won’t accept being misled by you, the author.
So, I would side to not provide false information as the third-person narrator. In the next section, I will provide an alternative to this.
If you choose vagueness, you need to be careful to not be too vague and skip over major details. But if you can perfectly balance just the right amount of information portrayed to your reader, then you've hit the sweet spot.
Without knowing everything, your reader will infer what they do know and suddenly, your story is being "interpreted in countless number of ways."
The other option is to show not tell. I won't talk about this one very much, the other answers cover it pretty well.
There probably is a better term for this, but who cares ;) - I like mystery style better.
This style deliberately withholds information from the reader, usually to create suspense and a better mystery. If the narrator just told us who did it at the beginning, what's the point of reading it?
In your case, this will still work. Actually, your desired style often goes hand-in-hand with this one. This way, if the narrator purposely withholds certain information, your reader will have to come up with their own interpretations. It's important to note that this style is different from straight out lying to your reader because you (as narrator) can be subtle enough that the reader doesn't even realize that they are missing info or that you are responsible.
You can create something with two (or more) intended, but mutually exclusive interpretations (like the famous duck/rabbit or vase/faces optical illusions). In the book Freshwater, is the heroine a victim of mischievous spirits, or is she mentally ill and unwilling to take responsibility for her actions?
You can create something where there is deliberately no resolvable answer, forcing the viewer/reader to invent interpretations. In the book Dhalgren, there are several mysteries which seem on the verge of being solved, only to have the author introduce a new fact which invalidates the previous theory.
You can create something where you yourself don't have a correct interpretation in mind. Author Haruki Murakami admits that the dream-logic mysteries of his writing come straight from his subconscious --they have no one single intended conclusion.