Star Number In C++

Introduction

A "star number" refers to a type of figurate number that represents a centered hexagram, a six-pointed star. These numbers are part of a broader category of numbers that visually form geometric patterns. The nth star number can be calculated using a specific formula, and this concept finds applications in mathematical visualizations and combinatorics.

This article provides an understanding of star numbers, a mathematical formula to compute them, and an implementation in C++ to calculate and display the star numbers.

Problem Statement:

Given an integer n, compute the nth star number using its formula:

Star number=6n(n−1) +1

Write a C++ program that:

✅ Accepts the value of n from the user.
✅ Calculates the nth star number using the formula.
✅ Displays the result to the user.

Understanding the Formula:

The formula 6n(n−1)+1 works as follows:

✅ The term 6n(n−1) calculates the number of points added to the star pattern as n increases.
✅ Adding 1 accounts for the central point of the star.

For example:

✅ The 1st star number: 6(1)(1−1)+1=1
✅ The 2nd star number: 6(2)(2−1)+1=13
✅ The 3rd star number: 6(3)(3−1)+1=37

Algorithm:

✅ Prompt the user to enter the value of n.
✅ Verify if n is positive, as star numbers are defined for n≥1.
✅ Apply the formula 6n(n−1)+1.
✅ Print the calculated star number.

Example 1:

Let us take an example to illustrate the Star Number in C++.

Example


#include <iostream>
using namespace std;

int calculateStarNumber(int n) {
    // Formula for calculating star number
    return 6 * n * (n - 1) + 1;
}

int main() {
    int n;

    // Prompt user for input
    cout << "Enter the value of n to calculate the nth star number: ";
    cin >> n;

    // Input validation
    if (n < 1) {
        cout << "Star numbers are defined for n >= 1. Please enter a valid value.\n";
        return 1;
    }

    // Calculate and display the star number
    int starNumber = calculateStarNumber(n);
    cout << "The " << n << "th star number is: " << starNumber << endl;

    return 0;
}

Output:

Output


Enter the value of n to calculate the nth star number: 4
The 4th star number is: 73

Example 2:

Let us take another example to illustrate the Star Number in C++.

Example


#include <iostream>
#include <fstream> // For file handling
#include <iomanip> // For formatted output
using namespace std;

// Function to calculate the nth star number
int calculateStarNumber(int n) {
    return 6 * n * (n - 1) + 1;
}

// Function to calculate multiple star numbers and return a dynamic array
int* calculateStarNumbers(int maxN, int& size) {
    size = maxN;
    int* starNumbers = new int[maxN];
    for (int i = 1; i <= maxN; ++i) {
        starNumbers[i - 1] = calculateStarNumber(i);
    }
    return starNumbers;
}

// Function to display a single star number
void displaySingleStarNumber() {
    int n;
    cout << "Enter the value of n to calculate the nth star number: ";
    cin >> n;

    if (n < 1) {
        cout << "Error: Star numbers are defined for n >= 1.\n";
        return;
    }

    int result = calculateStarNumber(n);
    cout << "The " << n << "th star number is: " << result << endl;
}

// Function to display multiple star numbers
void displayMultipleStarNumbers() {
    int maxN, size;
    cout << "Enter the maximum value of n to calculate star numbers up to nth: ";
    cin >> maxN;

    if (maxN < 1) {
        cout << "Error: Star numbers are defined for n >= 1.\n";
        return;
    }

    int* starNumbers = calculateStarNumbers(maxN, size);
    cout << "\nStar Numbers up to " << maxN << ":\n";
    cout << left << setw(5) << "n" << setw(15) << "Star Number\n";
    cout << "-------------------------\n";
    for (int i = 0; i < size; ++i) {
        cout << setw(5) << (i + 1) << setw(15) << starNumbers[i] << endl;
    }

    // Free dynamically allocated memory
    delete[] starNumbers;
}

// Function to save star numbers to a file
void saveStarNumbersToFile() {
    int maxN, size;
    cout << "Enter the maximum value of n to calculate star numbers up to nth: ";
    cin >> maxN;

    if (maxN < 1) {
        cout << "Error: Star numbers are defined for n >= 1.\n";
        return;
    }

    int* starNumbers = calculateStarNumbers(maxN, size);

    ofstream outFile("StarNumbers.txt");
    if (!outFile) {
        cout << "Error: Could not create file.\n";
        delete[] starNumbers;
        return;
    }

    outFile << "Star Numbers up to " << maxN << ":\n";
    outFile << left << setw(5) << "n" << setw(15) << "Star Number\n";
    outFile << "-------------------------\n";
    for (int i = 0; i < size; ++i) {
        outFile << setw(5) << (i + 1) << setw(15) << starNumbers[i] << endl;
    }

    outFile.close();
    cout << "Star numbers have been saved to 'StarNumbers.txt'.\n";

    // Free dynamically allocated memory
    delete[] starNumbers;
}

// Menu-driven interface
void menu() {
    int choice;
    do {
        cout << "\n==== Star Number Calculator ====\n";
        cout << "1. Calculate a single star number\n";
        cout << "2. Calculate and display multiple star numbers\n";
        cout << "3. Save star numbers to a file\n";
        cout << "4. Exit\n";
        cout << "Enter your choice: ";
        cin >> choice;

        switch (choice) {
            case 1:
                displaySingleStarNumber();
                break;
            case 2:
                displayMultipleStarNumbers();
                break;
            case 3:
                saveStarNumbersToFile();
                break;
            case 4:
                cout << "Exiting program. Goodbye!\n";
                break;
            default:
                cout << "Invalid choice! Please try again.\n";
        }
    } while (choice != 4);
}

int main() {
    menu();
    return 0;
}

Output:

Output


==== Star Number Calculator ====
1. Calculate a single star number
2. Calculate and display multiple star numbers
3. Save star numbers to a file
4. Exit
Enter your choice: 1
Enter the value of n to calculate the nth star number: 5
The 5th star number is: 121

==== Star Number Calculator ====
1. Calculate a single star number
2. Calculate and display multiple star numbers
3. Save star numbers to a file
4. Exit
Enter your choice: 2
Enter the maximum value of n to calculate star numbers up to nth: 8

Star Numbers up to 8:
n    Star Number
   -------------------------
1    1              
2    13             
3    37             
4    73             
5    121            
6    181            
7    253            
8    337            

==== Star Number Calculator ====
1. Calculate a single star number
2. Calculate and display multiple star numbers
3. Save star numbers to a file
4. Exit
Enter your choice: 3
Enter the maximum value of n to calculate star numbers up to nth: 2
Star numbers have been saved to 'StarNumbers.txt'.

==== Star Number Calculator ====
1. Calculate a single star number
2. Calculate and display multiple star numbers
3. Save star numbers to a file
4. Exit
Enter your choice: 4
Exiting program. Goodbye!

Explanation:

✅ Dynamic Memory Allocation: It is used in calculateStarNumbers to allocate memory for storing star numbers dynamically, allowing flexibility in handling large sequences.
✅ File Handling: Results can be saved to a file (StarNumbers.txt) for offline use, which makes the program practical for long-term storage of computed values.
✅ Formatted Output: The use of iomanip ensures neat alignment of results in the console and file, which improves the readability.
✅ Menu-Driven Design: The user can choose between different functionalities, ensuring the program is versatile and easy to navigate.
✅ It is used in calculateStarNumbers to allocate memory for storing star numbers dynamically, allowing flexibility in handling large sequences.
✅ Results can be saved to a file (StarNumbers.txt) for offline use, which makes the program practical for long-term storage of computed values.
✅ The use of iomanip ensures neat alignment of results in the console and file, which improves the readability.
✅ The user can choose between different functionalities, ensuring the program is versatile and easy to navigate.

Applications of Star Number:

Several applications of the Star Number in C++ are as follows:

✅ Mathematics and Number Theory Geometric Visualization: Star numbers represent a centered hexagram (six-pointed star) when visualized geometrically, making them useful in studying patterns in combinatorics and geometry. Sequence Analysis: Researchers study star numbers as part of figurate number sequences, which helps in the exploration of number theory and sequence generation. Formula Validation: Star numbers provide a simple example for validating general formulas related to figurate numbers.
✅ Education and Learning Teaching Tools: Star numbers can be used to introduce students to advanced topics like sequence generation, arithmetic progression, and geometric patterns. Problem Solving: Exercises based on star numbers help students develop problem-solving and critical-thinking skills.
✅ Game Design Visual Representation: Star patterns are often used in game design for puzzles, board games, and collectible items, adding aesthetic and symbolic appeal. Scoring Systems: Unique number sequences, like star numbers, can be integrated into scoring algorithms for games to introduce complexity.
✅ Computer Graphics and Design Pattern Generation: Star numbers can guide the creation of symmetric and aesthetically pleasing patterns in computer graphics. Fractal Geometry: Star patterns can be used in recursive designs for fractal generation and animations.
✅ Cryptography Unique Identifiers: Due to their uniqueness and rapid growth, star numbers can be employed in generating pseudo-random keys or unique identifiers. Sequence-Based Encryption: Star numbers can form the basis of sequence-based cryptographic systems, leveraging their mathematical properties for secure encoding.
✅ Astronomy and Symbolism Cultural Symbolism: The six-pointed star is a significant symbol in various cultures and religions. Star numbers may be used in studies of cultural geometry and historical symbolism. Stellar Configurations: Researchers exploring mathematical models for star shapes in astronomy can use star numbers as a theoretical framework.
✅ Geometric Visualization: Star numbers represent a centered hexagram (six-pointed star) when visualized geometrically, making them useful in studying patterns in combinatorics and geometry.
✅ Sequence Analysis: Researchers study star numbers as part of figurate number sequences, which helps in the exploration of number theory and sequence generation.
✅ Formula Validation: Star numbers provide a simple example for validating general formulas related to figurate numbers.
✅ Teaching Tools: Star numbers can be used to introduce students to advanced topics like sequence generation, arithmetic progression, and geometric patterns.
✅ Problem Solving: Exercises based on star numbers help students develop problem-solving and critical-thinking skills.
✅ Visual Representation: Star patterns are often used in game design for puzzles, board games, and collectible items, adding aesthetic and symbolic appeal.
✅ Scoring Systems: Unique number sequences, like star numbers, can be integrated into scoring algorithms for games to introduce complexity.
✅ Pattern Generation: Star numbers can guide the creation of symmetric and aesthetically pleasing patterns in computer graphics.
✅ Fractal Geometry: Star patterns can be used in recursive designs for fractal generation and animations.
✅ Unique Identifiers: Due to their uniqueness and rapid growth, star numbers can be employed in generating pseudo-random keys or unique identifiers.
✅ Sequence-Based Encryption: Star numbers can form the basis of sequence-based cryptographic systems, leveraging their mathematical properties for secure encoding.
✅ Cultural Symbolism: The six-pointed star is a significant symbol in various cultures and religions. Star numbers may be used in studies of cultural geometry and historical symbolism.
✅ Stellar Configurations: Researchers exploring mathematical models for star shapes in astronomy can use star numbers as a theoretical framework.

Conclusion:

In conclusion, Star numbers are an intriguing mathematical concept with simple computational requirements. Using C++, this article demonstrates how to compute and display the nth star number effectively. With minor modifications, this program can be extended for more complex applications involving geometric patterns or sequences.