{
    "data": {
        "question": {
            "questionId": "1358",
            "questionFrontendId": "1237",
            "boundTopicId": null,
            "title": "Find Positive Integer Solution for a Given Equation",
            "titleSlug": "find-positive-integer-solution-for-a-given-equation",
            "content": "<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>\n\n<p>While the exact formula is hidden, the function is monotonically increasing, i.e.:</p>\n\n<ul>\n\t<li><code>f(x, y) &lt; f(x + 1, y)</code></li>\n\t<li><code>f(x, y) &lt; f(x, y + 1)</code></li>\n</ul>\n\n<p>The function interface is defined like this:</p>\n\n<pre>\ninterface CustomFunction {\npublic:\n  // Returns some positive integer f(x, y) for two positive integers x and y based on a formula.\n  int f(int x, int y);\n};\n</pre>\n\n<p>We will judge your solution as follows:</p>\n\n<ul>\n\t<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>\n\t<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>\n\t<li>The judge will call your <code>findSolution</code> and compare your results with the <strong>answer key</strong>.</li>\n\t<li>If your results match the <strong>answer key</strong>, your solution will be <code>Accepted</code>.</li>\n</ul>\n\n<p>&nbsp;</p>\n<p><strong>Example 1:</strong></p>\n\n<pre>\n<strong>Input:</strong> function_id = 1, z = 5\n<strong>Output:</strong> [[1,4],[2,3],[3,2],[4,1]]\n<strong>Explanation:</strong> The hidden formula for function_id = 1 is f(x, y) = x + y.\nThe following positive integer values of x and y make f(x, y) equal to 5:\nx=1, y=4 -&gt; f(1, 4) = 1 + 4 = 5.\nx=2, y=3 -&gt; f(2, 3) = 2 + 3 = 5.\nx=3, y=2 -&gt; f(3, 2) = 3 + 2 = 5.\nx=4, y=1 -&gt; f(4, 1) = 4 + 1 = 5.\n</pre>\n\n<p><strong>Example 2:</strong></p>\n\n<pre>\n<strong>Input:</strong> function_id = 2, z = 5\n<strong>Output:</strong> [[1,5],[5,1]]\n<strong>Explanation:</strong> The hidden formula for function_id = 2 is f(x, y) = x * y.\nThe following positive integer values of x and y make f(x, y) equal to 5:\nx=1, y=5 -&gt; f(1, 5) = 1 * 5 = 5.\nx=5, y=1 -&gt; f(5, 1) = 5 * 1 = 5.\n</pre>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 &lt;= function_id &lt;= 9</code></li>\n\t<li><code>1 &lt;= z &lt;= 100</code></li>\n\t<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>\n\t<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>\n</ul>\n",
            "translatedTitle": null,
            "translatedContent": null,
            "isPaidOnly": false,
            "difficulty": "Medium",
            "likes": 337,
            "dislikes": 1129,
            "isLiked": null,
            "similarQuestions": "[]",
            "exampleTestcases": "1\n5\n2\n5",
            "categoryTitle": "Algorithms",
            "contributors": [],
            "topicTags": [
                {
                    "name": "Math",
                    "slug": "math",
                    "translatedName": null,
                    "__typename": "TopicTagNode"
                },
                {
                    "name": "Two Pointers",
                    "slug": "two-pointers",
                    "translatedName": null,
                    "__typename": "TopicTagNode"
                },
                {
                    "name": "Binary Search",
                    "slug": "binary-search",
                    "translatedName": null,
                    "__typename": "TopicTagNode"
                },
                {
                    "name": "Interactive",
                    "slug": "interactive",
                    "translatedName": null,
                    "__typename": "TopicTagNode"
                }
            ],
            "companyTagStats": null,
            "codeSnippets": [
                {
                    "lang": "C++",
                    "langSlug": "cpp",
                    "code": "/*\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};",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "Java",
                    "langSlug": "java",
                    "code": "/*\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}",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "Python",
                    "langSlug": "python",
                    "code": "\"\"\"\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        ",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "Python3",
                    "langSlug": "python3",
                    "code": "\"\"\"\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        ",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "C",
                    "langSlug": "c",
                    "code": "/*\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}",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "C#",
                    "langSlug": "csharp",
                    "code": "/*\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}",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "JavaScript",
                    "langSlug": "javascript",
                    "code": "/**\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};",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "Ruby",
                    "langSlug": "ruby",
                    "code": "# \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",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "Swift",
                    "langSlug": "swift",
                    "code": "/*\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}",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "Go",
                    "langSlug": "golang",
                    "code": "/** \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}",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "Scala",
                    "langSlug": "scala",
                    "code": "/*\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}",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "Kotlin",
                    "langSlug": "kotlin",
                    "code": "/*\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}",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "Rust",
                    "langSlug": "rust",
                    "code": "/*\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}",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "PHP",
                    "langSlug": "php",
                    "code": "/*\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}",
                    "__typename": "CodeSnippetNode"
                },
                {
                    "lang": "TypeScript",
                    "langSlug": "typescript",
                    "code": "/**\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};",
                    "__typename": "CodeSnippetNode"
                }
            ],
            "stats": "{\"totalAccepted\": \"47.7K\", \"totalSubmission\": \"68.7K\", \"totalAcceptedRaw\": 47736, \"totalSubmissionRaw\": 68727, \"acRate\": \"69.5%\"}",
            "hints": [
                "Loop over 1 ≤ x,y ≤ 1000 and check if f(x,y) == z."
            ],
            "solution": null,
            "status": null,
            "sampleTestCase": "1\n5",
            "metaData": "{\n  \"name\": \"findSolution\",\n  \"params\": [\n    {\n      \"name\": \"num\",\n      \"type\": \"integer\"\n    },\n    {\n      \"type\": \"integer\",\n      \"name\": \"z\"\n    }\n  ],\n  \"return\": {\n    \"type\": \"list<list<integer>>\"\n  },\n  \"manual\": true\n}",
            "judgerAvailable": true,
            "judgeType": "large",
            "mysqlSchemas": [],
            "enableRunCode": true,
            "enableTestMode": false,
            "enableDebugger": true,
            "envInfo": "{\"cpp\": [\"C++\", \"<p>Compiled with <code> clang 11 </code> using the latest C++ 17 standard.</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 17 </code>. Java 8 features such as lambda expressions and stream API can be used. </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 2.7.12</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\\\">will not be maintained past 2020</a>. For the latest Python, please choose Python3 instead.</p>\"], \"c\": [\"C\", \"<p>Compiled with <code>gcc 8.2</code> using the gnu99 standard.</p>\\r\\n\\r\\n<p>Your code is compiled with level one optimization (<code>-O1</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://docs.microsoft.com/en-us/dotnet/csharp/whats-new/csharp-9\\\" target=\\\"_blank\\\">C# 10 with .NET 6 runtime</a></p>\\r\\n\\r\\n<p>Your code is compiled with debug flag enabled (<code>/debug</code>).</p>\"], \"javascript\": [\"JavaScript\", \"<p><code>Node.js 16.13.2</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 <a href=\\\"https://github.com/datastructures-js/priority-queue\\\" target=\\\"_blank\\\">datastructures-js/priority-queue</a> and <a href=\\\"https://github.com/datastructures-js/queue\\\" target=\\\"_blank\\\">datastructures-js/queue</a>.</p>\"], \"ruby\": [\"Ruby\", \"<p><code>Ruby 3.1</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 5.5.2</code>.</p>\"], \"golang\": [\"Go\", \"<p><code>Go 1.17.6</code>.</p>\\r\\n\\r\\n<p>Support <a href=\\\"https://godoc.org/github.com/emirpasic/gods\\\" target=\\\"_blank\\\">https://godoc.org/github.com/emirpasic/gods</a> library.</p>\"], \"python3\": [\"Python3\", \"<p><code>Python 3.10</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 2.13.7</code>.</p>\"], \"kotlin\": [\"Kotlin\", \"<p><code>Kotlin 1.3.10</code>.</p>\"], \"rust\": [\"Rust\", \"<p><code>Rust 1.58.1</code></p>\\r\\n\\r\\n<p>Supports <a href=\\\"https://crates.io/crates/rand\\\" target=\\\"_blank\\\">rand </a> v0.6\\u00a0from crates.io</p>\"], \"php\": [\"PHP\", \"<p><code>PHP 8.1</code>.</p>\\r\\n<p>With bcmath module</p>\"], \"typescript\": [\"Typescript\", \"<p><code>TypeScript 4.5.4, Node.js 16.13.2</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 ES2020 features</a>.</p>\\r\\n\\r\\n<p><a href=\\\"https://lodash.com\\\" target=\\\"_blank\\\">lodash.js</a> library is included by default.</p>\"]}",
            "libraryUrl": null,
            "adminUrl": null,
            "challengeQuestion": null,
            "__typename": "QuestionNode"
        }
    }
}