I know that once I have a graph or decision tree representation of a problem, there are many different ways I can get a computer to solve the problem. They're called search algorithms, and go by names such as Depth-First Search, or Breadth-First Search or A* Search, etc.
However, I have the following game that I would like to input to my computer. The game is simple, you pick ten numbers. If all ten of the numbers are the number 1, then you win.
However, there are an infinite amount of decisions that the computer could take even with picking the first number. It could choose 8, or pi, or -3, etc.
Thus, even if I use something like A* and program in a heuristic such that it's obvious to the computer that choosing 1 is the best option, the computer still wont have enough memory to evaluate the heuristic on other the other options and make an informed decision.
How can computers handle cases like this? What sort of data structures are used when dealing with situations where there is not enough memory to hold all of the data, but there is enough to hold the relevant data at any given time?
(Compressed sparse data structures come to mind, but I'm not exactly sure how they can be used here.)