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

I think, you need to strictly define the problem first. What you are allowed to do and what you are not. You can start by making it simple and only allowing multiplication, division, substraction and addition.

Now you know your problem space- set of inputs, set of available operations and desired input. If you have only 4 operations and x inputs, the number of combinations is less than:

The number of order in which you can carry out operations (x!) times the possible choices of operations on every step: 4^x. As you can see for 6 numbers it gives reasonable 2949120 operations. This means that this may be your limit for brute force algorithm.

Once you have brute force and you know it works, you can start improving your algorithm with some sort of A* algorithm which would require you to define heuristic functions.

In my opinion the best way to think about it is as the search problem. The main difficulty will be finding good heuristics, or ways to reduce your problem space (if you have numbers that do not add up to the answer, you will need at least one multiplication etc.). Start small, build on that and ask follow up questions once you have some code.