## Question

Given a callable function `f(x, y)`

**with a hidden formula** and a value `z`

, reverse engineer the formula and return *all positive integer pairs *`x`

* and *`y`

* where *`f(x,y) == z`

. You may return the pairs in any order.

While the exact formula is hidden, the function is monotonically increasing, i.e.:

`f(x, y) < f(x + 1, y)`

`f(x, y) < f(x, y + 1)`

The function interface is defined like this:

interface CustomFunction { public: // Returns some positive integer f(x, y) for two positive integers x and y based on a formula. int f(int x, int y); };

We will judge your solution as follows:

- The judge has a list of
`9`

hidden implementations of`CustomFunction`

, along with a way to generate an**answer key**of all valid pairs for a specific`z`

. - The judge will receive two inputs: a
`function_id`

(to determine which implementation to test your code with), and the target`z`

. - The judge will call your
`findSolution`

and compare your results with the**answer key**. - If your results match the
**answer key**, your solution will be`Accepted`

.

**Example 1:**

Input:function_id = 1, z = 5Output:[[1,4],[2,3],[3,2],[4,1]]Explanation:The hidden formula for function_id = 1 is f(x, y) = x + y. The following positive integer values of x and y make f(x, y) equal to 5: x=1, y=4 -> f(1, 4) = 1 + 4 = 5. x=2, y=3 -> f(2, 3) = 2 + 3 = 5. x=3, y=2 -> f(3, 2) = 3 + 2 = 5. x=4, y=1 -> f(4, 1) = 4 + 1 = 5.

**Example 2:**

Input:function_id = 2, z = 5Output:[[1,5],[5,1]]Explanation:The hidden formula for function_id = 2 is f(x, y) = x * y. The following positive integer values of x and y make f(x, y) equal to 5: x=1, y=5 -> f(1, 5) = 1 * 5 = 5. x=5, y=1 -> f(5, 1) = 5 * 1 = 5.

**Constraints:**

`1 <= function_id <= 9`

`1 <= z <= 100`

- It is guaranteed that the solutions of
`f(x, y) == z`

will be in the range`1 <= x, y <= 1000`

. - It is also guaranteed that
`f(x, y)`

will fit in 32 bit signed integer if`1 <= x, y <= 1000`

.

## Python Solution

""" This is the custom function interface. You should not implement it, or speculate about its implementation class CustomFunction: # Returns f(x, y) for any given positive integers x and y. # Note that f(x, y) is increasing with respect to both x and y. # i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1) def f(self, x, y): """ class Solution: def findSolution(self, customfunction: 'CustomFunction', z: int) -> List[List[int]]: res = [] y = 1000 for x in range(1, 1001): while y > 1 and customfunction.f(x, y) > z: y -= 1 if customfunction.f(x, y) == z: res.append([x, y]) return res