If they can trust Zorro has done his work well, 

>! at the end, everyone knows who is the youngest.  
>!  
>! No matter the information given (to Bob), Bob can conclude he is not the youngest. This is because who is the youngest will become common knowledge, thus known by him, and then he will know who is the oldest. -- Bob can share this conclusion to everyone, no matter what the others are told, since it will become common knowledge anyway. (And thus, even if he is the oldest, sharing it will not tell anyone he is the oldest, if the logician did his job well).  
>!  
>! Charlie will not be told he is the middle one, since then at the end he will know the eldest. -- However, he cannot tell he is not the middle one as long as there is a chance he is the oldest. (Since the youngest then will find out he is) -- Thus, Charlie cannot say anything safely.    
>!  
>! If Alice is the oldest, she can safely share she is not the youngest. (since that will become common knowledge anyway) -- If Alice is not the oldest, she may still be the middle one, but only if the logician made an error (since she will learn who is the youngest (not herself) and then herself not being the oldest will tell her who the oldest is). So, in that case she can conclude (and tell) she is the youngest.

Thus, assuming Zorro did his work right, there are two possibilities:
 
>! <ol><li> Alice is the oldest, and Charlie the youngest. Then, after Bob and Alice both have told the others they are not the youngest, they can start the process of dividing the cake happy.  
>! <li> Alice is the youngest and says so, Charlie is the oldest.</ol>