## Algorithm

Problem Name: 1237. Find Positive Integer Solution for a Given Equation

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 = 5
Output: [[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 = 5
Output: [[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`.

## Code Examples

### #1 Code Example with Python Programming

```Code - Python Programming```

``````
from itertools import product as pr

class Solution(object):
def findSolution(self, customfunction, z):
return [
[i, j]
for i, j in pr(range(1, z + 1), repeat=2)
if customfunction.f(i, j) == z
]

``````
Copy The Code &

Input

cmd
function_id = 1, z = 5

Output

cmd
[[1,4],[2,3],[3,2],[4,1]]

### #2 Code Example with C# Programming

```Code - C# Programming```

``````
using System;
using System.Collections.Generic;

namespace LeetCode
{
public class _1237_FindPositiveIntegerSolutionForAGivenEquation
{
public IList> FindSolution(CustomFunction customfunction, int z)
{
var solutions = new List>();

int minX = MinX(customfunction, z);
int maxX = MaxX(customfunction, z);

for (int x = minX; x <= maxX; x++)
{
var solution = BinarySearchY(customfunction, x, z);
if (solution != null)
}

return solutions;
}

private int MinX(CustomFunction customfunction, int z)
{
int min = 1;
int max = 1000;

while (min <= max)
{
int mid = (min + max) / 2;
int fMid = customfunction.f(mid, 1000);
if (fMid == z)
return mid;

if (fMid < z)
min = mid + 1;
else
max = mid - 1;
}
return min;
}

private int MaxX(CustomFunction customfunction, int z)
{
int min = 1;
int max = 1000;

while (min <= max)
{
int mid = (min + max) / 2;
int fMid = customfunction.f(mid, 1);
if (fMid == z)
return mid;

if (fMid < z)
min = mid + 1;
else
max = mid - 1;
}
return max;
}

private IList BinarySearchY(CustomFunction customfunction, int x, int z)
{
int min = 1;
int max = 1000;

while (min <= max)
{
int mid = (min + max) / 2;
int fMid = customfunction.f(x, mid);
if (fMid == z)
return new int[] { x, mid };

if (fMid < z)
min = mid + 1;
else
max = mid - 1;
}

return null;
}

public class CustomFunction
{
private Func func;

public CustomFunction(Func func)
{
this.func = func;
}

// 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)
public int f(int x, int y)
{
return func(x, y);
}
};
}
}
``````
Copy The Code &

Input

cmd
function_id = 1, z = 5

Output

cmd
[[1,4],[2,3],[3,2],[4,1]]