Find Positive Integer Solution for a Given Equation
<p>Given a callable function <code>f(x, y)</code> <strong>with a hidden formula</strong> and a value <code>z</code>, reverse engineer the formula and return <em>all positive integer pairs </em><code>x</code><em> and </em><code>y</code><em> where </em><code>f(x,y) == z</code>. You may return the pairs in any order.</p>

<p>While the exact formula is hidden, the function is monotonically increasing, i.e.:</p>

<ul>
	<li><code>f(x, y) &lt; f(x + 1, y)</code></li>
	<li><code>f(x, y) &lt; f(x, y + 1)</code></li>
</ul>

<p>The function interface is defined like this:</p>

<pre>
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);
};
</pre>

<p>We will judge your solution as follows:</p>

<ul>
	<li>The judge has a list of <code>9</code> hidden implementations of <code>CustomFunction</code>, along with a way to generate an <strong>answer key</strong> of all valid pairs for a specific <code>z</code>.</li>
	<li>The judge will receive two inputs: a <code>function_id</code> (to determine which implementation to test your code with), and the target <code>z</code>.</li>
	<li>The judge will call your <code>findSolution</code> and compare your results with the <strong>answer key</strong>.</li>
	<li>If your results match the <strong>answer key</strong>, your solution will be <code>Accepted</code>.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> function_id = 1, z = 5
<strong>Output:</strong> [[1,4],[2,3],[3,2],[4,1]]
<strong>Explanation:</strong> 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 -&gt; f(1, 4) = 1 + 4 = 5.
x=2, y=3 -&gt; f(2, 3) = 2 + 3 = 5.
x=3, y=2 -&gt; f(3, 2) = 3 + 2 = 5.
x=4, y=1 -&gt; f(4, 1) = 4 + 1 = 5.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> function_id = 2, z = 5
<strong>Output:</strong> [[1,5],[5,1]]
<strong>Explanation:</strong> 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 -&gt; f(1, 5) = 1 * 5 = 5.
x=5, y=1 -&gt; f(5, 1) = 5 * 1 = 5.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= function_id &lt;= 9</code></li>
	<li><code>1 &lt;= z &lt;= 100</code></li>
	<li>It is guaranteed that the solutions of <code>f(x, y) == z</code> will be in the range <code>1 &lt;= x, y &lt;= 1000</code>.</li>
	<li>It is also guaranteed that <code>f(x, y)</code> will fit in 32 bit signed integer if <code>1 &lt;= x, y &lt;= 1000</code>.</li>
</ul>

[]
[]
[{"value": "cpp", "text": "C++", "defaultCode": "/*\n * // This is the custom function interface.\n * // You should not implement it, or speculate about its implementation\n * class CustomFunction {\n * public:\n *     // Returns f(x, y) for any given positive integers x and y.\n *     // Note that f(x, y) is increasing with respect to both x and y.\n *     // i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n *     int f(int x, int y);\n * };\n */\n\nclass Solution {\npublic:\n    vector<vector<int>> findSolution(CustomFunction& customfunction, int z) {\n        \n    }\n};"}, {"value": "java", "text": "Java", "defaultCode": "/*\n * // This is the custom function interface.\n * // You should not implement it, or speculate about its implementation\n * class CustomFunction {\n *     // Returns f(x, y) for any given positive integers x and y.\n *     // Note that f(x, y) is increasing with respect to both x and y.\n *     // i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n *     public int f(int x, int y);\n * };\n */\n\nclass Solution {\n    public List<List<Integer>> findSolution(CustomFunction customfunction, int z) {\n        \n    }\n}"}, {"value": "python", "text": "Python", "defaultCode": "\"\"\"\n   This is the custom function interface.\n   You should not implement it, or speculate about its implementation\n   class CustomFunction:\n       # Returns f(x, y) for any given positive integers x and y.\n       # Note that f(x, y) is increasing with respect to both x and y.\n       # i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n       def f(self, x, y):\n\"\"\"\n\nclass Solution(object):\n    def findSolution(self, customfunction, z):\n        \"\"\"\n        :type num: int\n        :type z: int\n        :rtype: List[List[int]]\n        \"\"\"\n        "}, {"value": "python3", "text": "Python3", "defaultCode": "\"\"\"\n   This is the custom function interface.\n   You should not implement it, or speculate about its implementation\n   class CustomFunction:\n       # Returns f(x, y) for any given positive integers x and y.\n       # Note that f(x, y) is increasing with respect to both x and y.\n       # i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n       def f(self, x, y):\n  \n\"\"\"\n\nclass Solution:\n    def findSolution(self, customfunction: 'CustomFunction', z: int) -> List[List[int]]:\n        "}, {"value": "c", "text": "C", "defaultCode": "/*\n * // This is the definition for customFunction API.\n * // You should not implement it, or speculate about its implementation\n *\n * // Returns f(x, y) for any given positive integers x and y.\n * // Note that f(x, y) is increasing with respect to both x and y.\n * // i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n */\n\n/**\n * Return an array of arrays of size *returnSize.\n * The sizes of the arrays are returned as *returnColumnSizes array.\n * Note: Both returned array and *columnSizes array must be malloced, assume caller calls free().\n */\nint** findSolution(int (*customFunction)(int, int), int z, int* returnSize, int** returnColumnSizes) {\n\t\n}"}, {"value": "csharp", "text": "C#", "defaultCode": "/*\n * // This is the custom function interface.\n * // You should not implement it, or speculate about its implementation\n * public class CustomFunction {\n *     // Returns f(x, y) for any given positive integers x and y.\n *     // Note that f(x, y) is increasing with respect to both x and y.\n *     // i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n *     public int f(int x, int y);\n * };\n */\n\npublic class Solution {\n    public IList<IList<int>> FindSolution(CustomFunction customfunction, int z) {\n        \n    }\n}"}, {"value": "javascript", "text": "JavaScript", "defaultCode": "/**\n * // This is the CustomFunction's API interface.\n * // You should not implement it, or speculate about its implementation\n * function CustomFunction() {\n *     @param {integer, integer} x, y\n *     @return {integer}\n *     this.f = function(x, y) {\n *         ...\n *     };\n * };\n */\n\n/**\n * @param {CustomFunction} customfunction\n * @param {integer} z\n * @return {integer[][]}\n */\nvar findSolution = function(customfunction, z) {\n    \n};"}, {"value": "typescript", "text": "TypeScript", "defaultCode": "/**\n * // This is the CustomFunction's API interface.\n * // You should not implement it, or speculate about its implementation\n * class CustomFunction {\n *      f(x: number, y: number): number {}\n * }\n */\n\nfunction findSolution(customfunction: CustomFunction, z: number): number[][] {\n\t\n};"}, {"value": "php", "text": "PHP", "defaultCode": "/*\n * // This is the custom function interface.\n * // You should not implement it, or speculate about its implementation\n * class CustomFunction {\n *     // Returns f(x, y) for any given positive integers x and y.\n *     // Note that f(x, y) is increasing with respect to both x and y.\n *     // i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n *     public function f($x, $y){}\n * };\n */\n\nclass Solution {\n    /**\n     * @param  CustomFunction  $customfunction\n     * @param  Integer  $z\n     * @return Integer[][]\n     */\n    function findSolution($customfunction, $n) {\n        \n    }\n}"}, {"value": "swift", "text": "Swift", "defaultCode": "/*\n * // This is the custom function interface.\n * // You should not implement it, or speculate about its implementation\n * class CustomFunction {\n *     // Returns f(x, y) for any given positive integers x and y.\n *     // Note that f(x, y) is increasing with respect to both x and y.\n *     // i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n *     func f(_ x: Int, _ y: Int) -> Int {}\n * }\n */\n\nclass Solution {\n    func findSolution(_ customfunction: CustomFunction, _ z: Int) -> [[Int]] {\n        \n    }\n}"}, {"value": "kotlin", "text": "Kotlin", "defaultCode": "/*\n * // This is the custom function interface.\n * // You should not implement it, or speculate about its implementation\n * class CustomFunction {\n *     // Returns f(x, y) for any given positive integers x and y.\n *     // Note that f(x, y) is increasing with respect to both x and y.\n *     // i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n *     fun f(x:Int, y:Int):Int {}\n * };\n */\n\nclass Solution {\n\tfun findSolution(customfunction:CustomFunction, z:Int):List<List<Int>> {\n        \n    }\n}"}, {"value": "golang", "text": "Go", "defaultCode": "/** \n * This is the declaration of customFunction API.\n * @param  x    int\n * @param  x    int\n * @return \t    Returns f(x, y) for any given positive integers x and y.\n *\t\t\t    Note that f(x, y) is increasing with respect to both x and y.\n *              i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n */\n\nfunc findSolution(customFunction func(int, int) int, z int) [][]int {\n    \n}"}, {"value": "ruby", "text": "Ruby", "defaultCode": "# \tThis is the custom function interface.\n#\tYou should not implement it, or speculate about its implementation\n#\tclass CustomFunction:\n#\t\tdef f(self, x, y):\n# \t\t\tReturns f(x, y) for any given positive integers x and y.\n# \t\t\tNote that f(x, y) is increasing with respect to both x and y.\n# \t\t\ti.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n# \t\tend\n# \tend\n# \n\n# @param {CustomFunction} customfunction\n# @param {Integer} z\n# @return {List[List[Integer]]}\ndef findSolution(customfunction, z)\n    \nend"}, {"value": "scala", "text": "Scala", "defaultCode": "/*\n * // This is the custom function interface.\n * // You should not implement it, or speculate about its implementation\n * class CustomFunction {\n *     // Returns f(x, y) for any given positive integers x and y.\n *     // Note that f(x, y) is increasing with respect to both x and y.\n *     // i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)\n *     def f(x: Int, y: Int): Int = {}\n * };\n */\n\nobject Solution {\n    def findSolution(customfunction: CustomFunction, z: Int): List[List[Int]] = {\n        \n    }\n}"}, {"value": "rust", "text": "Rust", "defaultCode": "/*\n * // This is the custom function interface.\n * // You should not implement it, or speculate about its implementation\n * struct CustomFunction;\n * impl CustomFunction {\n *    pub fn f(x:i32,y:i32)->i32{}\n * }\n */\n\nimpl Solution {\n    pub fn find_solution(customfunction: &CustomFunction, z: i32) -> Vec<Vec<i32>> {\n\t\n    }\n}"}]
{
  "name": "findSolution",
  "params": [
    {
      "name": "num",
      "type": "integer"
    },
    {
      "type": "integer",
      "name": "z"
    }
  ],
  "return": {
    "type": "list<list<integer>>"
  },
  "manual": true,
"typescriptCustomType" : "class CustomFunction {\n    num: number;\n    constructor(x: number) {\n        this.num = x;\n    }\n\n    f(x: number, y: number): number {\n        switch (this.num) {\n            case 1 : return x + y;\n            case 2 : return x * y;\n            case 3 : return x * x + y;\n            case 4 : return x + y * y;\n            case 5 : return x * x + y * y;\n            case 6 : return (x + y) * (x + y);\n            case 7 : return x * x * x + y * y * y;\n            case 8 : return x * x * y;\n            case 9 : return x * y * y;\n        }\n        return 0;\n    }\n};"
}
{"cpp": ["C++", "<p>Compiled with <code> clang 19 </code> using the latest C++ 23 standard, and <code>libstdc++</code> provided by GCC 14.</p>\r\n\r\n<p>Your code is compiled with level two optimization (<code>-O2</code>). <a href=\"https://github.com/google/sanitizers/wiki/AddressSanitizer\" target=\"_blank\">AddressSanitizer</a> is also enabled to help detect out-of-bounds and use-after-free bugs.</p>\r\n\r\n<p>Most standard library headers are already included automatically for your convenience.</p>"], "java": ["Java", "<p><code>OpenJDK 21</code>. Using compile arguments: <code>--enable-preview --release 21</code></p>\r\n\r\n<p>Most standard library headers are already included automatically for your convenience.</p>\r\n<p>Includes <code>Pair</code> class from https://docs.oracle.com/javase/8/javafx/api/javafx/util/Pair.html.</p>"], "python": ["Python", "<p><code>Python 2.7.18</code>.</p>\r\n\r\n<p>Most libraries are already imported automatically for your convenience, such as <a href=\"https://docs.python.org/2/library/array.html\" target=\"_blank\">array</a>, <a href=\"https://docs.python.org/2/library/bisect.html\" target=\"_blank\">bisect</a>, <a href=\"https://docs.python.org/2/library/collections.html\" target=\"_blank\">collections</a>. If you need more libraries, you can import it yourself.</p>\r\n\r\n<p>For Map/TreeMap data structure, you may use <a href=\"http://www.grantjenks.com/docs/sortedcontainers/\" target=\"_blank\">sortedcontainers</a> library.</p>\r\n\r\n<p>Note that Python 2.7 <a href=\"https://www.python.org/dev/peps/pep-0373/\" target=\"_blank\">is not maintained anymore</a>. For the latest Python, please choose Python3 instead.</p>"], "c": ["C", "<p>Compiled with <code>gcc 14</code> using the gnu11 standard.</p>\r\n\r\n<p>Your code is compiled with level one optimization (<code>-O2</code>). <a href=\"https://github.com/google/sanitizers/wiki/AddressSanitizer\" target=\"_blank\">AddressSanitizer</a> is also enabled to help detect out-of-bounds and use-after-free bugs.</p>\r\n\r\n<p>Most standard library headers are already included automatically for your convenience.</p>\r\n\r\n<p>For hash table operations, you may use <a href=\"https://troydhanson.github.io/uthash/\" target=\"_blank\">uthash</a>. \"uthash.h\" is included by default. Below are some examples:</p>\r\n\r\n<p><b>1. Adding an item to a hash.</b>\r\n<pre>\r\nstruct hash_entry {\r\n    int id;            /* we'll use this field as the key */\r\n    char name[10];\r\n    UT_hash_handle hh; /* makes this structure hashable */\r\n};\r\n\r\nstruct hash_entry *users = NULL;\r\n\r\nvoid add_user(struct hash_entry *s) {\r\n    HASH_ADD_INT(users, id, s);\r\n}\r\n</pre>\r\n</p>\r\n\r\n<p><b>2. Looking up an item in a hash:</b>\r\n<pre>\r\nstruct hash_entry *find_user(int user_id) {\r\n    struct hash_entry *s;\r\n    HASH_FIND_INT(users, &user_id, s);\r\n    return s;\r\n}\r\n</pre>\r\n</p>\r\n\r\n<p><b>3. Deleting an item in a hash:</b>\r\n<pre>\r\nvoid delete_user(struct hash_entry *user) {\r\n    HASH_DEL(users, user);  \r\n}\r\n</pre>\r\n</p>"], "csharp": ["C#", "<p><a href=\"https://learn.microsoft.com/en-us/dotnet/csharp/whats-new/csharp-13\" target=\"_blank\">C# 13</a> with .NET 9 runtime</p>"], "javascript": ["JavaScript", "<p><code>Node.js 22.14.0</code>.</p>\r\n\r\n<p>Your code is run with <code>--harmony</code> flag, enabling <a href=\"http://node.green/\" target=\"_blank\">new ES6 features</a>.</p>\r\n\r\n<p><a href=\"https://lodash.com\" target=\"_blank\">lodash.js</a> library is included by default.</p>\r\n\r\n<p>For Priority Queue / Queue data structures, you may use 6.3.2 version of <a href=\"https://github.com/datastructures-js/priority-queue/tree/v6.3.2\" target=\"_blank\">datastructures-js/priority-queue</a> and 4.2.3 version of <a href=\"https://github.com/datastructures-js/queue/tree/v4.2.3\" target=\"_blank\">datastructures-js/queue</a> and 1.0.4 version of <a href=\"https://github.com/datastructures-js/deque/tree/v1.0.4\" target=\"_blank\">datastructures-js/deque</a>.</p>"], "ruby": ["Ruby", "<p><code>Ruby 3.2</code></p>\r\n\r\n<p>Some common data structure implementations are provided in the Algorithms module: https://www.rubydoc.info/github/kanwei/algorithms/Algorithms</p>"], "swift": ["Swift", "<p><code>Swift 6.0</code>.</p>\r\n\r\n<p>You may use the following packages:<br/>\r\n<a href=\"https://github.com/apple/swift-algorithms/tree/1.2.0\" target=\"_blank\">swift-algorithms 1.2.0</a><br/>\r\n<a href=\"https://github.com/apple/swift-collections/tree/1.1.4\" target=\"_blank\">swift-collections 1.1.4</a><br/>\r\n<a href=\"https://github.com/apple/swift-numerics/tree/1.0.2\" target=\"_blank\">swift-numerics 1.0.2</a></p>"], "golang": ["Go", "<p><code>Go 1.23</code></p>\r\n<p>Support <a href=\"https://pkg.go.dev/github.com/emirpasic/gods@v1.18.1\" target=\"_blank\">https://pkg.go.dev/github.com/emirpasic/gods@v1.18.1</a> and <a href=\"https://pkg.go.dev/github.com/emirpasic/gods/v2@v2.0.0-alpha\" target=\"_blank\">https://pkg.go.dev/github.com/emirpasic/gods/v2@v2.0.0-alpha</a> library.</p>"], "python3": ["Python3", "<p><code>Python 3.11</code>.</p>\r\n\r\n<p>Most libraries are already imported automatically for your convenience, such as <a href=\"https://docs.python.org/3/library/array.html\" target=\"_blank\">array</a>, <a href=\"https://docs.python.org/3/library/bisect.html\" target=\"_blank\">bisect</a>, <a href=\"https://docs.python.org/3/library/collections.html\" target=\"_blank\">collections</a>. If you need more libraries, you can import it yourself.</p>\r\n\r\n<p>For Map/TreeMap data structure, you may use <a href=\"http://www.grantjenks.com/docs/sortedcontainers/\" target=\"_blank\">sortedcontainers</a> library.</p>"], "scala": ["Scala", "<p><code>Scala 3.3.1</code>.</p>"], "kotlin": ["Kotlin", "<p><code>Kotlin 2.1.10</code>.</p>"], "rust": ["Rust", "<p><code>Rust 1.85.0</code>. Your code will be compiled with <code>opt-level</code> 2.</p>\r\n\r\n<p>Supports <a href=\"https://crates.io/crates/rand\" target=\"_blank\">rand v0.8</a> and <a href=\"https://crates.io/crates/regex\" target=\"_blank\">regex\u00a0v1</a> and <a href=\"https://crates.io/crates/itertools\" target=\"_blank\">itertools\u00a0v0.14</a> from crates.io</p>"], "php": ["PHP", "<p><code>PHP 8.2</code>.</p>\r\n<p>With bcmath module</p>"], "typescript": ["Typescript", "<p><code>TypeScript 5.7.3, Node.js 22.14.0</code>.</p>\r\n\r\n<p>Compile Options: <code>--alwaysStrict --strictBindCallApply --strictFunctionTypes --target ES2024</code></p>\r\n\r\n<p><a href=\"https://lodash.com\" target=\"_blank\">lodash.js</a> library is included by default.</p>\r\n\r\n<p>For Priority Queue / Queue data structures, you may use 6.3.2 version of <a href=\"https://github.com/datastructures-js/priority-queue/tree/v6.3.2\" target=\"_blank\">datastructures-js/priority-queue</a> and 4.2.3 version of <a href=\"https://github.com/datastructures-js/queue/tree/v4.2.3\" target=\"_blank\">datastructures-js/queue</a> and 1.0.4 version of <a href=\"https://github.com/datastructures-js/deque/tree/v1.0.4\" target=\"_blank\">datastructures-js/deque</a>.</p>"]}
['Math', 'Two Pointers', 'Binary Search', 'Interactive']