Find Positive Integer Solution for a Given Equation

The problem provides a hidden monotonically increasing function f(x, y), and the task is to find all pairs (x, y) such that f(x, y) == z.

Here’s a simple approach:

  1. Initialize the Result: Create an empty result list to store the pairs (x, y) that satisfy the condition.
  2. Iterate Through Possible x Values: Since the constraints provide 1 <= x, y <= 1000, we can iterate through all possible values of x from 1 to 1000. a. Find the Corresponding y Value: For each x, start with y as 1 and keep incrementing y until f(x, y) <= z. If f(x, y) == z, add the pair (x, y) to the result. b. Take Advantage of Monotonicity: Since f(x, y) is monotonically increasing, once we find f(x, y) > z, we can break the loop for y as the further values will only increase.
  3. Return the Result: Return the result list containing the pairs.

Here’s the code:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11

class Solution:
    def findSolution(self, customfunction: 'CustomFunction', z: int) -> List[List[int]]:
        result = []
        for x in range(1, 1001): # Iterate through x values
            for y in range(1, 1001): # Iterate through y values
                value = customfunction.f(x, y)
                if value == z:
                    result.append([x, y]) # Add the pair if it matches z
                elif value > z:
                    break # Break if the value is greater than z
        return result

This code efficiently leverages the monotonically increasing property of the function and the constraints to find the required pairs (x, y). By breaking the inner loop when f(x, y) > z, we avoid unnecessary calculations.