Behind which door is the princess? Because of Alladin's forbidden love for the king's daughter, he has been sentenced to the usual punishment: death by tiger.
As the king is a fair man, he has a fighting chance to escape his doom. He is kept in a room with four doors. 
Behind two doors are tigers. Behind the third a dragon. Behind the fourth is the princess; if he manages to open her door, the king will forgive him and marry his daughter to Alladin.
On each door is a number and a plaque. Alladin has been assured that the door leading to the princess has a plaque that tells the truth, and that the doors leading to a tiger have plaques that lie. Alladin does not know whether the plaque on the door leading to the dragon is a lie or the truth.
Door 1: If a tiger waits behind Door 3, then the princess waits behind Door 4.
Door 2: If the princess waits behind Door 3, then the dragon waits behind Door 1.
Door 3: If a dragon waits behind Door 4, then a tiger waits behind Door 2.
Door 4: If a tiger waits behind Door 2, then the princess waits behind Door 3.
I have attached an image of my attempt. I am not sure how to go further or how to approach in a different manner.My approach
 A: Remember that a statement of the form "if $p$, then $q$" is true if $p$ is false.  So your first case is checking whether the princess is in door 1.  If so, then the door 1 statement is true, which means that a tiger cannot be behind the third door.  Therefore we must have the tigers in doors 2 and 4.  But this is impossible: since the princess is not behind door 3, the door 2 statement is automatically true, and there cannot be a tiger there.  You should be able to evaluate your other cases similarly.
A: Building on Michael Harrisons reasoning:  
If door 1 is the princess (meaning door 1 is true): 

Door 1: If a tiger waits behind Door 3, then the princess waits behind Door 4.  

The princess is not door 4, so door 3 is the dragon.
Door 2 and 4 are tigers and lies.  

Door 2: If the princess waits behind Door 3, then the dragon waits behind Door 1.  

The princess is not door 3, so this is true, but it should be a lie.
Wrong solution.  
If door 2 is the princess (=true): 

Door 2: If the princess waits behind Door 3, then the dragon waits behind Door 1.  

No princess in door 3, so it's true. Ok.  

Door 4: If a tiger waits behind Door 2, then the princess waits behind Door 3.  

Door 2 is the princess, so door 4 says the truth too. Dragon (honest variant).
Door 1 and 3 are tigers / lies.

Door 1: If a tiger waits behind Door 3, then the princess waits behind Door 4.  

While door 3 is a tiger, it's a lie, so it's ok.  

Door 3: If a dragon waits behind Door 4, then a tiger waits behind Door 2.  

Door 4 is the dragon, but door 2 no tiger, but it's a lie again, so it's ok.  
The princess in door 2 is a solution 
If door 3 is the princess (=true): 

Door 1: If a tiger waits behind Door 3, then the princess waits behind Door 4.  

There's no tiger behind door 3, so door 1 is true too. Dragon.
Door 2 and 4 have to be tigers.  

Door 2: If the princess waits behind Door 3, then the dragon waits behind Door 1.  

This is true, but it should be a lie.
Wrong solution.  
If door 4 is the princess (=true): 

Door 3: If a dragon waits behind Door 4, then a tiger waits behind Door 2.  

No dragon at 4, so it's true. Door 3 is the dragon.
Door 1 and 2 should be tigers.  

Door 1: If a tiger waits behind Door 3, then the princess waits behind Door 4.  

No tiger at 3, so it's true, but it should be a lie.
Wrong solution.  
A: The princess can be behind any of four doors, and then there are three doors left for the dragon, so there are 12 possibilities in all. 
In four of the 12 cases, the first statement is false, in two cases the second statement is false, in one case the third statement is false, and in one case the fourth statement is false. 
For example, the only way that the third statement can be false is if the dragon is behind door 4 and the princess behind door 2. 
At most two of the four statements can be true, and the only way this can happen is if the dragon is behind door 2 and the princess behind door 3.
