How to design an algorithm to calculate countdown style maths number puzzle

Question

I have always wanted to do this but every time I start thinking about the problem it blows my mind because of its exponential nature.

The problem solver I want to be

Answer 1

Input is obviously a set of digits and operators: D={1,3,3,6,7,8,3} and Op={+,-,*,/}. The most straight forward algorithm would be a brute force solver, which enumerates all possible combinations of these sets. Where the elements of set Op can be used as often as wanted, but elements from set D are used exactly once. Pseudo code:

D={1,3,3,6,7,8,3}
Op={+,-,*,/}
Solution=348
for each permutation D_ of D:
   for each binary tree T with D_ as its leafs:
       for each sequence of operators Op_ from Op with length |D_|-1:
           label each inner tree node with operators from Op_
           result = compute T using infix traversal
           if result==Solution
              return T
return nil

Other than that: read jedrus07's and HPM's answers.