Find Pythagorean Triplet with given sum
Last Updated :
15 Oct, 2024
Given a positive integer target, the task is to find all Pythagorean Triplets whose sum of the elements is equal to the given target. A triplet {a, b, c} is considered a Pythagorean triplet if it satisfies the condition a2 + b2 = c2.
Examples :
Input: target = 60
Output: {{10, 24, 26}, {15, 20, 25}}
Explanation: There are two Pythagorean triplets: {10, 24, 26} and {15, 20, 25}, both having a sum equal to 60.
Input : target = 4
Output : {}
Explanation: No Pythagorean triplet exists with a sum of 4.
[Naive Approach] - Using three nested loops - O(n^3) Time and O(1) Space
A simple solution is to generate all possible triplets using three nested loops and for every triplet, check if it is a Pythagorean Triplet and the sum of its elements is equal to given target.
C++
// C++ program to find Pythagorean triplets
// having given sum using three nested loops
#include <iostream>
#include <vector>
using namespace std;
vector<vector<int>> pythagoreanTriplet(int target) {
vector<vector<int>> res;
// Generating all possible triplets
for(int a = 0; a < target - 2; a++) {
for(int b = a + 1; b < target - 1; b++) {
for(int c = b + 1; c < target; c++) {
// If this triplet satisfies the condition
// then, add it to the result
if(a + b + c == target && a*a + b*b == c*c)
res.push_back({a, b, c});
}
}
}
return res;
}
int main() {
int target = 60;
vector<vector<int>> ans = pythagoreanTriplet(target);
for(vector<int> triplet : ans)
cout << triplet[0] << " " << triplet[1] << " " <<
triplet[2] << endl;
return 0;
}
// C program to find Pythagorean triplets
// having given sum using three nested loops
#include <stdio.h>
#define MAX_SIZE 100
void pythagoreanTriplet(int triplets[][3], int *count, int target) {
*count = 0;
// Generating all possible triplets
for (int a = 0; a < target - 2; a++) {
for (int b = a + 1; b < target - 1; b++) {
for (int c = b + 1; c < target; c++) {
// If this triplet satisfies the condition
// then, add it to the result
if (a + b + c == target && a * a + b * b == c * c) {
triplets[*count][0] = a;
triplets[*count][1] = b;
triplets[*count][2] = c;
(*count)++;
}
}
}
}
}
int main() {
int triplets[MAX_SIZE][3];
int count;
int target = 60;
pythagoreanTriplet(triplets, &count, target);
for (int i = 0; i < count; i++)
printf("%d %d %d\n", triplets[i][0], triplets[i][1],
triplets[i][2]);
return 0;
}
// Java program to find Pythagorean triplets
// having given sum using three nested loops
import java.util.ArrayList;
import java.util.List;
class GfG {
static List<List<Integer>> pythagoreanTriplet(int target) {
List<List<Integer>> res = new ArrayList<>();
// Generating all possible triplets
for (int a = 0; a < target - 2; a++) {
for (int b = a + 1; b < target - 1; b++) {
for (int c = b + 1; c < target; c++) {
// If this triplet satisfies the condition
// then, add it to the result
if (a + b + c == target && a * a + b * b == c * c) {
res.add(List.of(a, b, c));
}
}
}
}
return res;
}
public static void main(String[] args) {
int target = 60;
List<List<Integer>> ans = pythagoreanTriplet(target);
for (List<Integer> triplet : ans) {
System.out.println(triplet.get(0) + " " + triplet.get(1) + " "
+ triplet.get(2));
}
}
}
# Python program to find Pythagorean triplets
# having given sum using three nested loops
def pythagoreanTriplet(target):
res = []
# Generating all possible triplets
for a in range(target - 2):
for b in range(a + 1, target - 1):
for c in range(b + 1, target):
# If this triplet satisfies the condition
# then, add it to the result
if a + b + c == target and a * a + b * b == c * c:
res.append([a, b, c])
return res
target = 60
ans = pythagoreanTriplet(target)
for triplet in ans:
print(*triplet)
// C# program to find Pythagorean triplets
// having given sum using three nested loops
using System;
using System.Collections.Generic;
class GfG {
static List<List<int>> PythagoreanTriplet(int target) {
List<List<int>> res = new List<List<int>>();
// Generating all possible triplets
for (int a = 0; a < target - 2; a++) {
for (int b = a + 1; b < target - 1; b++) {
for (int c = b + 1; c < target; c++) {
// If this triplet satisfies the condition
// then, add it to the result
if (a + b + c == target && a * a + b * b == c * c) {
res.Add(new List<int> { a, b, c });
}
}
}
}
return res;
}
static void Main() {
int target = 60;
List<List<int>> ans = PythagoreanTriplet(target);
foreach (var triplet in ans) {
Console.WriteLine($"{triplet[0]} {triplet[1]} {triplet[2]}");
}
}
}
// JavaScript program to find Pythagorean triplets
// having given sum using three nested loops
function pythagoreanTriplet(target) {
const res = [];
// Generating all possible triplets
for (let a = 0; a < target - 2; a++) {
for (let b = a + 1; b < target - 1; b++) {
for (let c = b + 1; c < target; c++) {
// If this triplet satisfies the condition
// then, add it to the result
if (a + b + c === target && a * a + b * b === c * c) {
res.push([a, b, c]);
}
}
}
}
return res;
}
const target = 60;
const ans = pythagoreanTriplet(target);
ans.forEach(triplet => console.log(triplet.join(' ')));
[Better Approach] - Using two nested loops - O(n^2) Time and O(1) Space
This approach is an optimization to the previous one. In this, we run two nested loops to generate all possible values of first two elements of the triplet. The third element can be found by subtracting these values from the given target. We then check whether these elements form a Pythagorean triplet.
C++
// C++ program to find Pythagorean triplets having
// given sum using two nested loops
#include <iostream>
#include <vector>
using namespace std;
vector<vector<int>> pythagoreanTriplet(int target) {
vector<vector<int>> res;
// Generating all possible pairs
for(int a = 1; a < target - 2; a++) {
for(int b = a + 1; b < target - 1; b++) {
// Third value of triplet
int c = target - a - b;
// c should be the maximum value of triplet
if(c <= b)
continue;
// if {a, b, c} forms a pythagorean Triplet
// then add it to the reuslt.
if(a*a + b*b == c*c)
res.push_back({a, b, c});
}
}
return res;
}
int main() {
int target = 60;
vector<vector<int>> ans = pythagoreanTriplet(target);
for(vector<int> triplet : ans)
cout << triplet[0] << " " << triplet[1] << " " <<
triplet[2] << endl;
return 0;
}
// C program to find Pythagorean triplets
// having given sum using two nested loops
#include <stdio.h>
#define MAX_SIZE 100
void pythagoreanTriplet(int triplets[][3], int *count, int target) {
*count = 0;
// Generating all possible pairs
for (int a = 1; a < target - 2; a++) {
for (int b = a + 1; b < target - 1; b++) {
// Third value of triplet
int c = target - a - b;
// c should be the maximum value of triplet
if (c <= b)
continue;
// If {a, b, c} forms a Pythagorean triplet
// then add it to the result
if (a * a + b * b == c * c) {
triplets[*count][0] = a;
triplets[*count][1] = b;
triplets[*count][2] = c;
(*count)++;
}
}
}
}
int main() {
int triplets[MAX_SIZE][3];
int count;
int target = 60;
pythagoreanTriplet(triplets, &count, target);
for (int i = 0; i < count; i++)
printf("%d %d %d\n", triplets[i][0], triplets[i][1], triplets[i][2]);
return 0;
}
// Java program to find Pythagorean triplets having given sum
// using two nested loops
import java.util.ArrayList;
import java.util.List;
class GfG {
static List<List<Integer>> pythagoreanTriplet(int target) {
List<List<Integer>> res = new ArrayList<>();
// Generating all possible pairs
for (int a = 1; a < target - 2; a++) {
for (int b = a + 1; b < target - 1; b++) {
// Third value of the triplet
int c = target - a - b;
// c should be the maximum value of the triplet
if (c <= b)
continue;
// If {a, b, c} forms a Pythagorean triplet
// then add it to the result
if (a * a + b * b == c * c)
res.add(List.of(a, b, c));
}
}
return res;
}
public static void main(String[] args) {
int target = 60;
List<List<Integer>> ans = pythagoreanTriplet(target);
for (List<Integer> triplet : ans) {
System.out.println(triplet.get(0) + " " + triplet.get(1)
+ " " + triplet.get(2));
}
}
}
# Python program to find Pythagorean triplets having
# given sum using two nested loops
def pythagoreanTriplet(target):
res = []
# Generating all possible pairs
for a in range(1, target - 2):
for b in range(a + 1, target - 1):
# Third value of the triplet
c = target - a - b
# c should be the maximum value of the triplet
if c <= b:
continue
# If {a, b, c} forms a Pythagorean triplet
# then, add it to the result
if a * a + b * b == c * c:
res.append([a, b, c])
return res
target = 60
ans = pythagoreanTriplet(target)
for triplet in ans:
print(*triplet)
// C# program to find Pythagorean triplets having given
// sum using two nested loops
using System;
using System.Collections.Generic;
class GfG {
static List<List<int>> PythagoreanTriplet(int target) {
List<List<int>> res = new List<List<int>>();
// Generating all possible pairs
for (int a = 1; a < target - 2; a++) {
for (int b = a + 1; b < target - 1; b++) {
// Third value of triplet
int c = target - a - b;
// c should be the maximum value of triplet
if (c <= b)
continue;
// If {a, b, c} forms a Pythagorean triplet
// then, add it to the result
if (a * a + b * b == c * c)
res.Add(new List<int> { a, b, c });
}
}
return res;
}
static void Main() {
int target = 60;
List<List<int>> ans = PythagoreanTriplet(target);
foreach (var triplet in ans) {
Console.WriteLine($"{triplet[0]} {triplet[1]} {triplet[2]}");
}
}
}
// JavaScript program to find Pythagorean triplets
// having given sum using two nested loops
function pythagoreanTriplet(target) {
const res = [];
// Generating all possible pairs
for (let a = 1; a < target - 2; a++) {
for (let b = a + 1; b < target - 1; b++) {
// Third value of triplet
let c = target - a - b;
// It should be the maximum value of triplet
if (c <= b) continue;
// If {a, b, c} forms a Pythagorean triplet
// then, add it to the result
if (a * a + b * b === c * c) {
res.push([a, b, c]);
}
}
}
return res;
}
const target = 60;
const ans = pythagoreanTriplet(target);
ans.forEach(triplet => console.log(triplet.join(' ')));
[Expected Approach] - Using Mathematical Formula - O(n) Time and O(1) Space
We know that the sum of all the elements in a valid triplet is equal to the given target, i.e., a + b + c = target, and it is a Pythagorean Triplet, i.e., a2 + b2 = c2. Now, we can derive the value of c in terms of variable a using these two equations. Then, for all possible values of a, we calculate the corresponding valid c values. Finally, values of b can be found by subtracting a and c from the target.
You can refer to this article for complete implementation of this approach - Find Pythagorean Triplet with given sum using Mathematical Formula.
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