How to implement two stacks in one array? I want to begin by saying that this is NOT a homework question. I am reading Introduction to Algorithms - the famous CLRS text to become a better programmer. I am trying to solve the problems and exercises given in the book by myself. 

I am trying to solve Excercise 10.1-2 from Chapter 10 Elementary Data Structures from CLRS Second Edition. Here is what its states:


  Explain how to implement two stacks in one array A[1..n] in such a way that neither stack overflows unless the total number of elements in both stacks together is n. The PUSH and POP operations should run in O(1) time.


The solution that I have come up with so far is:


  Let array A[1..n] implement two stacks: S1[1..i] and S2[i..n]. 
  
  For the PUSH-S1 and PUSH-S2 operations, if the stack is 'full' then start pushing elements into the other stack (eg. if stack S1 is full when a new element is trying to be pushed in, then push that element into stack S2 and vice versa). 


The problem with this approach is I will not be able to POP-S1 or POP-S2 reliably as there is no way of 'remembering' which element belongs to which stack. If the elements of the stack are (key,value) pairs, the key being the stack number, then to pop an element I would have to search, in the worst case, i or (n-i) times - which will be O(n) (feel free to correct me if I am wrong here), which would not be O(1). 

I have been banging my head on the question for quite a while now. Am I on the right track? Can someone give my possible pointers for solving this problem?

In general, how should I 'think' about these problems? Or can only really intelligent people solve these types of problems? Will tackling/solving problems like these (i.e. gaining experience) help me become better at this?

I await enlightenment.
Another hint in addition to what Yuval said: it helps to place the stacks in a specific way in the arrays and fix their direction of growth accordingly. They don't have to grow in the same direction.