I created a very simple mix task to run the program. It has a recurisve solve/1 function so I can solve each pair of inputs given independently and handle cases where no inputs are given, or an odd number of inputs are given.

mix warehouse_id

This was a very interesting problem to work on. Having recently graduated from school, I recognized the triangular sequence of numbers that follows the base of the warehouse triangle. My solution works based off of that by getting the x-th triangular number, then figuring out what we need to do to go up y levels to come up with the ID we needed.

Because I've worked with triangular numbers before, the recursive and more efficient triangular formla solutions were easy to come up with. You'll find them as triangular_number/2 and fast_triangular_number/1 in the code. From there, I just needed to figure out a recursive function, which I called upward_sum/2 that is very similar to a recursive triangular sum, but starts at a particular base and starts the incrementer at its own base as well. From there, it's easy to find the offset that I need, add it to the triangular number, and out comes the solution to the problem. I'm sure that a more efficient solution to upward_sum/2 exists because it's a very simple summation, but I couldn't come up with its closed-form solution on my own. Also, the instructions say to prefer recursion, so I didn't dig too far into finding a non-recursive way of solving that part of it.

Once I solved the problem, I was able to identify that the bad test case was [100000,100000] = 20,000,000,001. The correct answer should have been 19_999_800_001.

I solved the stretch goal by simply guessing numbers for x, checking if they're correct, and continuing to guess. Of course, the naive way of doing this (guessing that x is 1, then 2, then 3, and so on) is incredibly slow. I took advantage of the fact that the maximum x coordinate is 100,000 and implemented a binary search for the number. So, first we'll guess 50,000. If our number is less than that, we'll guess 25,000, then 12,5000 if we're still lower, and so on until we reach the number. That way we only have to do ln(n) guesses, rather than n guesses. The answer to the specific stretch goal question came out to 13,457.