So, I'm playing around with a puzzle in a game where I have to figure out the correct quantities of ingredients to go into a recipe, and I'm trying to figure out the most efficient way to search the space of possible quantities given a few conditions:

• Recipes can have any number of ingredients from 1 to 12, they always contain at least 1 of each ingredient, and the quantities are all natural numbers (1, 2, 3, etc.)

• There's a condition for "closeness" - if you're within X total units away, you get some feedback that you're close. This range is about 4 total units, I think.

• "Closeness" depends on the sum of the absolute values of differences between the recipe and the guess - one being 1 unit too high and one being 2 units too low counts as being 3 units away total.

So for example, if I have a recipe that is (2, 3, 4, 5) and I guess (2, 2, 5, 5), then I'm 2 units away and therefore I get told I'm "close".

I'm trying to figure out a way to basically sample the space of possible recipes so if I have one with N ingredients, I can just systematically test as few recipes as possible until I find something that's "close". I'm not sure how to formalize this and come up with a solution though - it feels like something that could be mapped out with N-dimensional taxicab distance, treating each ingredient as a dimension, and basically covering the space with N-dimensional "balls" with a radius equal to the "closeness" condition to generate guesses, but I don't know how to actually work with that. I'd love some ideas on how else I could visualize this puzzle and how I could build up an algorithm for generating guesses efficiently.