Number of cycles formed by joining vertices of n sided polygon at the center
Given an N-sided regular polygon, we have connected all the vertices at the center of the polygon, thus dividing the polygon into N equal parts. Our task is to the Count of the total number of cycles in the polygon.
Note: A Cycle is a closed-loop starting and ending on same point.
Examples:
Input: N = 3
Output: 7
Explanation:
When a 3 sided polygon is connected by vertices at the center then we get 7 cycles possible for it as shown in the image.
Input: N = 5
Output: 21
Explanation:
When a 5 sided polygon is connected by vertices at the center then we get 21 cycles possible for it as shown in the image.
Approach: To the problem mentioned above we are supposed to count the total number of closed loops possible in the given polygon after division. The approach is based upon Mathematical Pattern. There will be N cycles already created due to the division of polygon. One out of N blocks will form a cycle with rest (N – 1) blocks. The remaining (N – 1) blocks will form cycle with other (N – 2) blocks. So the total cycles we have can be found out using the formula given below:
Total Cycles = N + 1 * (N – 1) + (N – 1) * (N – 2)
Total Cycles = 2 * N – 1) + (N – 1) * (N – 2)
Below is the implementation of the above approach:
C++
// C++ program for the above approach #include<bits/stdc++.h> using namespace std; // Function to calculate number of cycles int findCycles( int N) { int res = 0; int finalResult = 0; int val = 2 * N - 1; // BigInteger is used here // if N=10^9 then multiply // will result into value // greater than 10^18 int s = val; // BigInteger multiply function res = (N - 1) * (N - 2); finalResult = res + s; // Return the final result return finalResult; } // Driver code int main() { // Given N int N = 5; // Function Call cout << findCycles(N) << endl; return 0; } // This code is contributed by divyeshrabadiya07 |
Java
// Java program for the above approach import java.util.*; import java.math.*; class GFG { // Function to calculate number of cycles static BigInteger findCycles( int N) { BigInteger res, finalResult; long val = 2 * N - 1 ; String st = String.valueOf(val); // BigInteger is used here // if N=10^9 then multiply // will result into value // greater than 10^18 BigInteger str = new BigInteger(st); String n1 = String.valueOf((N - 1 )); String n2 = String.valueOf((N - 2 )); BigInteger a = new BigInteger(n1); BigInteger b = new BigInteger(n2); // BigInteger multiply function res = a.multiply(b); finalResult = res.add(str); // Return the final result return finalResult; } // Driver Code public static void main(String args[]) throws Exception { // Given N int N = 5 ; // Function Call System.out.println(findCycles(N)); } } |
Python3
# Python3 program for the above approach # Function to calculate number of cycles def findCycles(N): res = 0 finalResult = 0 val = 2 * N - 1 ; # BigInteger is used here # if N=10^9 then multiply # will result into value # greater than 10^18 s = val # BigInteger multiply function res = (N - 1 ) * (N - 2 ) finalResult = res + s; # Return the final result return finalResult; # Driver Code if __name__ = = '__main__' : # Given N N = 5 ; # Function Call print (findCycles(N)); # This code is contributed by pratham76 |
C#
// C# program for the above approach using System; class GFG { // Function to calculate number of cycles static int findCycles( int N) { int res = 0; int finalResult = 0; int val = 2 * N - 1; // BigInteger is used here // if N=10^9 then multiply // will result into value // greater than 10^18 int s = val; // BigInteger multiply function res = (N - 1) * (N - 2); finalResult = res + s; // Return the final result return finalResult; } // Driver code static void Main() { // Given N int N = 5; // Function Call Console.WriteLine(findCycles(N)); } } // This code is contributed by divyesh072019 |
Javascript
<script> // Javascript program for the above approach // Function to calculate number of cycles function findCycles(N) { let res = 0; let finalResult = 0; let val = 2 * N - 1; // BigInteger is used here // if N=10^9 then multiply // will result into value // greater than 10^18 let s = val; // BigInteger multiply function res = (N - 1) * (N - 2); finalResult = res + s; // Return the final result return finalResult; } // Given N let N = 5; // Function Call document.write(findCycles(N)); </script> |
21
Time Complexity: O(1)
Auxiliary Space: O(1)
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