Given a number n, find two pairs that can represent the number as sum of two cubes. In other words, find two pairs (a, b) and (c, d) such that given number n can be expressed asĀ
n = a^3 + b^3 = c^3 + d^3
where a, b, c and d are four distinct numbers.
Examples:Ā
Input: N = 1729 Output: (1, 12) and (9, 10) Explanation: 1729 = 1^3 + 12^3 = 9^3 + 10^3 Input: N = 4104 Output: (2, 16) and (9, 15) Explanation: 4104 = 2^3 + 16^3 = 9^3 + 15^3 Input: N = 13832 Output: (2, 24) and (18, 20) Explanation: 13832 = 2^3 + 24^3 = 18^3 + 20^3
Any number n that satisfies the constraint will have two distinct pairs (a, b) and (c, d) such that a, b, c and d are all less than n1/3. The idea is very simple. For every distinct pair (x, y) formed by numbers less than the n1/3, if their sum (x3 + y3) is equal to given number, we store them in a hash table using sum as a key. If pairs with sum equal to given number appears again, we simply print both pairs.
1) Create an empty hash map, say s.
2) cubeRoot = n1/3
3) for (int x = 1; x < cubeRoot; x++)
for (int y = x + 1; y <= cubeRoot; y++)
int sum = x3 + y3;
if (sum != n) continue;
if sum exists in s,
we found two pairs with sum, print the pairs
else
insert pair(x, y) in s using sum as key
Below is the implementation of above idea āĀ
C++
// C++ program to find pairs that can represent// the given number as sum of two cubes#include <bits/stdc++.h>using namespace std;Ā
// Function to find pairs that can represent// the given number as sum of two cubesvoid findPairs(int n){Ā Ā Ā Ā // find cube root of nĀ Ā Ā Ā int cubeRoot = pow(n, 1.0/3.0);Ā
Ā Ā Ā Ā // create an empty mapĀ Ā Ā Ā unordered_map<int, pair<int, int> > s;Ā
Ā Ā Ā Ā // Consider all pairs such with values lessĀ Ā Ā Ā // than cuberootĀ Ā Ā Ā for (int x = 1; x < cubeRoot; x++)Ā Ā Ā Ā {Ā Ā Ā Ā Ā Ā Ā Ā for (int y = x + 1; y <= cubeRoot; y++)Ā Ā Ā Ā Ā Ā Ā Ā {Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // find sum of current pair (x, y)Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā int sum = x*x*x + y*y*y;Ā
Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // do nothing if sum is not equal toĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // given numberĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if (sum != n)Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā continue;Ā
Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // if sum is seen before, we found two pairsĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if (s.find(sum) != s.end())Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā {Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā cout << "(" << s[sum].first << ", "Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā << s[sum].second << ") and ("Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā << x << ", " << y << ")" << endl;Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā elseĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // if sum is seen for the first timeĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā s[sum] = make_pair(x, y);Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā }}Ā
// Driver functionint main(){Ā Ā Ā Ā int n = 13832;Ā Ā Ā Ā findPairs(n);Ā Ā Ā Ā return 0;} |
Java
// Java program to find pairs that can represent// the given number as sum of two cubesimport java.util.*;Ā
class GFG{Ā Ā Ā Ā static class pairĀ Ā Ā Ā { Ā Ā Ā Ā Ā Ā Ā Ā int first, second; Ā Ā Ā Ā Ā Ā Ā Ā public pair(int first, int second) Ā Ā Ā Ā Ā Ā Ā Ā { Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā this.first = first; Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā this.second = second; Ā Ā Ā Ā Ā Ā Ā Ā } Ā Ā Ā Ā }Ā Ā Ā Ā Ā // Function to find pairs that can represent// the given number as sum of two cubesstatic void findPairs(int n){Ā Ā Ā Ā // find cube root of nĀ Ā Ā Ā int cubeRoot = (int) Math.pow(n, 1.0/3.0);Ā
Ā Ā Ā Ā // create an empty mapĀ Ā Ā Ā HashMap<Integer, pair> s = new HashMap<Integer, pair>();Ā
Ā Ā Ā Ā // Consider all pairs such with values lessĀ Ā Ā Ā // than cuberootĀ Ā Ā Ā for (int x = 1; x < cubeRoot; x++)Ā Ā Ā Ā {Ā Ā Ā Ā Ā Ā Ā Ā for (int y = x + 1; y <= cubeRoot; y++)Ā Ā Ā Ā Ā Ā Ā Ā {Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // find sum of current pair (x, y)Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā int sum = x*x*x + y*y*y;Ā
Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // do nothing if sum is not equal toĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // given numberĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if (sum != n)Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā continue;Ā
Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // if sum is seen before, we found two pairsĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if (s.containsKey(sum))Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā {Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā System.out.print("(" + s.get(sum).first+ ", "Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā + s.get(sum).second+ ") and ("Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā + x+ ", " + y+ ")" +"\n");Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā elseĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // if sum is seen for the first timeĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā s.put(sum, new pair(x, y));Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā }}Ā
// Driver codepublic static void main(String[] args){Ā Ā Ā Ā int n = 13832;Ā Ā Ā Ā findPairs(n);}}Ā
// This code is contributed by PrinciRaj1992 |
Python3
# Python3 program to find pairs# that can represent the given # number as sum of two cubesĀ
# Function to find pairs that # can represent the given number# as sum of two cubes def findPairs(n):Ā
Ā Ā Ā Ā # Find cube root of n Ā Ā Ā Ā cubeRoot = pow(n, 1.0 / 3.0); Ā Ā Ā Ā Ā Ā Ā # Create an empty map Ā Ā Ā Ā s = {} Ā Ā Ā Ā Ā Ā Ā # Consider all pairs such with Ā Ā Ā Ā # values less than cuberoot Ā Ā Ā Ā for x in range(int(cubeRoot)):Ā Ā Ā Ā Ā Ā Ā Ā for y in range(x + 1, Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā int(cubeRoot) + 1):Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā # Find sum of current pair (x, y) Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā sum = x * x * x + y * y * y; Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā # Do nothing if sum is not equal to Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā # given number Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if (sum != n):Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā continue; Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā # If sum is seen before, weĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā # found two pairs Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if sum in s.keys():Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā print("(" + str(s[sum][0]) +Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā ", " + str(s[sum][1]) +Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā ") and (" + str(x) +Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā ", " + str(y) +Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā ")" + "\n")Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā else:Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā # If sum is seen for the first time Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā s[sum] = [x, y] Ā
# Driver codeif __name__=="__main__":Ā Ā Ā Ā Ā Ā Ā Ā Ā n = 13832Ā Ā Ā Ā Ā Ā Ā Ā Ā findPairs(n)Ā Ā Ā Ā Ā # This code is contributed by rutvik_56 |
C#
// C# program to find pairs that can represent// the given number as sum of two cubesusing System;using System.Collections.Generic;Ā
class GFG{Ā Ā Ā Ā class pairĀ Ā Ā Ā { Ā Ā Ā Ā Ā Ā Ā Ā public int first, second; Ā Ā Ā Ā Ā Ā Ā Ā public pair(int first, int second) Ā Ā Ā Ā Ā Ā Ā Ā { Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā this.first = first; Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā this.second = second; Ā Ā Ā Ā Ā Ā Ā Ā } Ā Ā Ā Ā }Ā Ā Ā Ā Ā Ā // Function to find pairs that can represent// the given number as sum of two cubesstatic void findPairs(int n){Ā Ā Ā Ā // find cube root of nĀ Ā Ā Ā int cubeRoot = (int) Math.Pow(n, 1.0/3.0);Ā Ā Ā Ā Ā Ā // create an empty mapĀ Ā Ā Ā Dictionary<int, pair> s = new Dictionary<int, pair>();Ā Ā Ā Ā Ā Ā // Consider all pairs such with values lessĀ Ā Ā Ā // than cuberootĀ Ā Ā Ā for (int x = 1; x < cubeRoot; x++)Ā Ā Ā Ā {Ā Ā Ā Ā Ā Ā Ā Ā for (int y = x + 1; y <= cubeRoot; y++)Ā Ā Ā Ā Ā Ā Ā Ā {Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // find sum of current pair (x, y)Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā int sum = x*x*x + y*y*y;Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // do nothing if sum is not equal toĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // given numberĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if (sum != n)Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā continue;Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // if sum is seen before, we found two pairsĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if (s.ContainsKey(sum))Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā {Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Console.Write("(" + s[sum].first+ ", "Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā + s[sum].second+ ") and ("Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā + x+ ", " + y+ ")" +"\n");Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā elseĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // if sum is seen for the first timeĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā s.Add(sum, new pair(x, y));Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā }}Ā Ā // Driver codepublic static void Main(String[] args){Ā Ā Ā Ā int n = 13832;Ā Ā Ā Ā findPairs(n);}}Ā
// This code is contributed by PrinciRaj1992 |
Javascript
// JavaScript program to find pairs that can represent// the given number as sum of two cubesĀ
// Function to find pairs that can represent// the given number as sum of two cubesfunction findPairs(n){Ā Ā Ā Ā // find cube root of nĀ Ā Ā Ā let cubeRoot = Math.floor(Math.pow(n, 1/3));Ā
Ā Ā Ā Ā // create an empty mapĀ Ā Ā Ā let s = new Map();Ā
Ā Ā Ā Ā // Consider all pairs such with values lessĀ Ā Ā Ā // than cuberootĀ Ā Ā Ā for (let x = 1; x < cubeRoot; x++){Ā Ā Ā Ā Ā Ā Ā Ā for (let y = x + 1; y <= cubeRoot; y++){Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // find sum of current pair (x, y)Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā let sum = x*x*x + y*y*y;Ā
Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // do nothing if sum is not equal toĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // given numberĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if (sum != n){Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā continue;Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // if sum is seen before, we found two pairsĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā if (s.has(sum)){Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā console.log("(", s.get(sum)[0], ",", s.get(sum)[1], ") and (",x,"," ,y,")");Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā else{Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā // if sum is seen for the first timeĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā s.set(sum, [x, y]);Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā Ā Ā Ā Ā }Ā Ā Ā Ā }}Ā
// Driver function{Ā Ā Ā Ā let n = 13832;Ā Ā Ā Ā findPairs(n);}Ā
// The code is contributed by Gautam goel (gautamgoel962) |
Output:Ā
(2, 24) and (18, 20)
Time Complexity of above solution is O(n2/3) which is much less than O(n).
Can we solve the above problem in O(n1/3) time? We will be discussing that in next post.
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