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 wrote a terminal application to do this: https://github.com/pg328/CountdownNumbersGame/tree/main

Inside, I've included an illustration of the calculation of the size of the solution space (it's n*((n-1)!^2)*(2^n-1), so: n=6 -> 2,764,800. I know, gross), and more importantly why that is. My implementation is there if you care to check it out, but in case you don't I'll explain here.

Essentially, at worst it is brute force because as far as I know it's impossible to determine whether any specific branch will result in a valid answer without explicitly checking. Having said that, the average case is some fraction of that; it's {that number} divided by the number of valid solutions (I tend to see around 1000 on my program, where 10 or so are unique and the rest are permutations fo those 10). If I handwaved a number, I'd say roughly 2,765 branches to check which takes like no time. (Yes, even in Python.)

TL;DR: Even though the solution space is huge and it takes a couple million operations to find all solutions, only one answer is needed. Best route is brute force til you find one and spit it out.