Fascinating Number In C++

Introduction to Fascinating Numbers

Fascinating numbers are quite interesting concepts in number theory. Such numbers have some interesting properties when multiplied by certain numbers, as in producing sequences that have each of the digits from 1 to 9 exactly one time and not in the same order as before.

A number N is said to be fascinating if:

• ✅ N×2 and N×3 is multiplied by the value of N and these products are combined with N and they form a single figure.
• ✅ And this number contains all the information needed to count from the number 1 to 9, which has all the values and figures that are required, once.

In Mathematics,

If N, N×2 and N×3 are three values or figures. If N is set to equal N, N×2 and N×3, and N is rated for all nine figures being included, then N is defined as having the fascinating features ascribed to it.

Characteristics of Fascinating Numbers

• ✅ The first feature is that usually, there are at least three digits for anything fascinating. Please note that there are no fewer than three digits, 1 or 2 digits, because they fail to conform to the criteria specified.
• ✅ The second feature surrounds the combining of the three figures N, N×2 and N×3 in a pattern that places a great deal of excitement on the combining of the three figures.
• ✅ The placement of all the digits 1 through 9 must be in the figure comprising a circle once if the number has no placement for a zero integer zero.

Learning Fascinating Numbers Through Examples

Every fascinating number has something special about it that exists somewhere within it. Let's take it slow and explain the concept of fascinating numbers with the help of examples.

A detailed outline of what needs to be done in order to determine how to check if number

Step-by-step explanation of how to check if a number is fascinating

• ✅ Start with the Number (N) Take the given number N. Ensure it has at least three digits, as numbers with fewer digits cannot form a fascinating number.
• ✅ Multiply N by 2 and 3 Calculate N×2 and N×3.
• ✅ Concatenate the Results Combine N, N×2, and N×3 into a single string.
• ✅ Check for Digits 1 to 9 Verify if the concatenated string contains all digits from 1 to 9 exactly once. Ensure no digit is repeated, and the digit 0 is not included.
• ✅ Conclusion If the concatenated string meets the criteria, N is a fascinating number. Otherwise, it is not.
• ✅ Take the given number N. Ensure it has at least three digits, as numbers with fewer digits cannot form a fascinating number.
• ✅ Calculate N×2 and N×3.
• ✅ Combine N, N×2, and N×3 into a single string.
• ✅ Verify if the concatenated string contains all digits from 1 to 9 exactly once.
• ✅ Ensure no digit is repeated, and the digit 0 is not included.
• ✅ If the concatenated string meets the criteria, N is a fascinating number. Otherwise, it is not.

Examples illustrating fascinating numbers and non-fascinating numbers

Example 1: Fascinating Number (N=192)

• ✅ Given N=192
• ✅ Calculate: N×2=384 N×3=576
• ✅ Concatenate: 192384576
• ✅ Verify: The string 192384576 contains all digits from 1 to 9 exactly once.
• ✅ Conclusion: 192 is a fascinating number.
• ✅ N×2=384
• ✅ N×3=576
• ✅ 192384576
• ✅ The string 192384576 contains all digits from 1 to 9 exactly once.
• ✅ 192 is a fascinating number.

Example 2: Fascinating Number (N=273)

• ✅ Given N=273
• ✅ Calculate: N×2=546 N×3=819
• ✅ Concatenate: 273546819
• ✅ Verify: The string 273546819 includes all digits from 1 to 9 exactly once.
• ✅ Conclusion: 273 is a fascinating number.
• ✅ N×2=546
• ✅ N×3=819
• ✅ 273546819
• ✅ The string 273546819 includes all digits from 1 to 9 exactly once.
• ✅ 273 is a fascinating number.

Example 3: Non-Fascinating Number (N=190)

• ✅ Given N=190
• ✅ Calculate: N×2=380 N×3=570
• ✅ Concatenate: 190380570
• ✅ Verify: The string 190380570 does not include all digits from 1 to 9. Additionally, some digits are repeated (e.g., 0).
• ✅ Conclusion: 190190190 is not a fascinating number.
• ✅ N×2=380
• ✅ N×3=570
• ✅ 190380570
• ✅ The string 190380570 does not include all digits from 1 to 9. Additionally, some digits are repeated (e.g., 0).
• ✅ 190190190 is not a fascinating number.

Example 4: Non-Fascinating Number (N=123)

• ✅ Given N=123
• ✅ Calculate: N×2=246 N×3=369
• ✅ Concatenate: 123246369
• ✅ Verify: The string 123246369 does not include all digits from 1 to 9 exactly once (missing 7, 8, 0).
• ✅ Conclusion: 123 is not a fascinating number.
• ✅ N×2=246
• ✅ N×3=369
• ✅ 123246369
• ✅ The string 123246369 does not include all digits from 1 to 9 exactly once (missing 7, 8, 0).
• ✅ 123 is not a fascinating number.

Programming Basics for Fascinating Numbers in C++

1. Loops

• ✅ Purpose: It is used to correct the representation of a set of elements in a sequence.
• ✅ Usage: It is used to verify that all the integers between 1 and 8 are present in the unified output and they do not repeat.

Example:

for (int i = 0; i < n; i++) {
    // Perform operations here
}

2. Functions

• ✅ Purpose: Esteeming the whole logic within an encapsulated structure in order to promote code reuse and save time.
• ✅ Usage: Create a function that determines if a given number is fascinating. This organize the code further.

Example:

bool isFascinating(int number) {
    // Function logic here
}

3. String Manipulation

• ✅ Purpose: Sitting appropriately and effectively the tasks of string concatenation in their places.
• ✅ Usage: Use number to string conversion to join the numbers and check out the digits.

Example:

string s = to_string(num1) + to_string(num2) + to_string(num3);

4. Arrays or Maps

• ✅ Purpose: Retaining and keeping track of the number of times particular digits have been displayed.
• ✅ Usage: Use array or a map for the total number of the said digits in the exponential string.

Example:

int digitCount[10] = {0};

Handling input and output for checking fascinating numbers

1. Input:

• ✅ Accept a number from the user to check if it is fascinating.

Example:

int number;
cout << "Enter a number: ";
cin >> number;

2. Output:

• ✅ Display whether the number is fascinating or not.

Example:

if (isFascinating(number)) {
    cout << number << " is a Fascinating Number." << endl;
} else {
    cout << number << " is NOT a Fascinating Number." << endl;
}

Key Features of the Program

• ✅ User Input: It provides an opportunity for users to randomly fix different numbers to be tried out.
• ✅ Validation: It provides that the numbers to be used must appear as three digits on their own and outlines the uniqueness of such numbers.
• ✅ Efficient Logic: Leverage the checking of the digits-based validation techniques, hence self-continually check times.

Step by step breakdown of the algorithm

Step 1: Validation checks of the inputs

• ✅ Receive the number N from the user
• ✅ See whether N has at least three digits. If not, it is not fascinating, as there would be nothing to zoom in on.

Step 2: Number of Multiples

• ✅ Formulate the following two multiples. N×2 N×3

Step 3: Results Combination

• ✅ Combine N, Nx2, and Nx3 all together in one string.

Step 4: Investigating Fascinating Features

• ✅ Set up either an array or a map with the goal of marking all digits that had been encountered for the purpose of easy retrieval.
• ✅ Go through the string in other to obtain the fused characters; Count from one to 9 the number of times each number shows up. However, if any number was recorded more than once or the number 0 appeared in the countings, N is, in this case, a non-fascinating number.
• ✅ Count from one to 9 the number of times each number shows up.
• ✅ However, if any number was recorded more than once or the number 0 appeared in the countings, N is, in this case, a non-fascinating number.

Step 5: Justification of available signs

• ✅ Make sure that every figure from one to 9 was used and only used once in the string.

Step 6: Display the answer

• ✅ If the above conditions are met, N is fascinating; otherwise, it is not.

Pseudocode for better understanding

Algorithm: IsFascinating(N)

• ✅ Input: A number N If N < 100, return "NOT fascinating" (must have at least 3 digits)
• ✅ Calculate: multiply2 = N 2 multiply3 = N 3
• ✅ Concatenate: concatenated = String(N) + String(multiply2) + String(multiply3)
• ✅ Initialize: digitCount[10] = {0} // Array to track digit occurrences
• ✅ For each character in concatenated: digit = character - '0' If digit == 0 OR digitCount[digit] > 0:
• ✅ return "NOT fascinating" (repeated digit or zero found) Increment digitCount[digit]
• ✅ For i from 1 to 9: If digitCount[i] != 1:
• ✅ return "NOT fascinating" (missing or repeated digit)
• ✅ Return "Fascinating Number"
• ✅ If N < 100, return "NOT fascinating" (must have at least 3 digits)
• ✅ multiply2 = N * 2
• ✅ multiply3 = N * 3
• ✅ concatenated = String(N) + String(multiply2) + String(multiply3)
• ✅ digitCount[10] = {0} // Array to track digit occurrences
• ✅ digit = character - '0'
• ✅ If digit == 0 OR digitCount[digit] > 0:
• ✅ Increment digitCount[digit]
• ✅ If digitCount[i] != 1:

Example 1:

Example
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#include <iostream>
#include <string>
using namespace std;

// Function to check if a number is fascinating
bool isFascinating(int number) {
    // Step 1: Ensure the number has at least three digits
    if (number < 100) return false;

    // Step 2: Calculate N * 2 and N * 3
    int multiply2 = number * 2;
    int multiply3 = number * 3;

    // Step 3: Concatenate the numbers into a single string
    string concatenated = to_string(number) + to_string(multiply2) + to_string(multiply3);

    // Step 4: Check for the fascinating number condition
    int digitCount[10] = {0}; // Array to count occurrences of digits

    // Step 5: Traverse the concatenated string
    for (char ch : concatenated) {
        int digit = ch - '0'; // Convert character to integer
        if (digit == 0 || digitCount[digit] > 0) {
            return false; // Digit 0 or duplicate digits are not allowed
        }
        digitCount[digit]++; // Increment count for the digit
    }

    // Step 6: Verify if all digits from 1 to 9 appear exactly once
    for (int i = 1; i <= 9; i++) {
        if (digitCount[i] != 1) {
            return false;
        }
    }

    // If all conditions are satisfied, the number is fascinating
    return true;
}

int main() {
    // Input: Get the number from the user
    int number;
    cout << "Enter a number: ";
    cin >> number;

    // Output: Check and display the result
    if (isFascinating(number)) {
        cout << number << " is a Fascinating Number." << endl;
    } else {
        cout << number << " is NOT a Fascinating Number." << endl;
    }

    return 0;
}
Try it Yourself

Output 1:

Enter a number: 190
190 is NOT a Fascinating Number.

=== Code Execution Successful ===

Output 2:

Enter a number: 273
273 is a Fascinating Number.

=== Code Execution Successful ===

Example 2:

Here's an optimized implementation with reduced memory usage and early termination:

#include <iostream>
using namespace std;

bool isFascinating(int number) {
    if (number < 100) return false; // Must have at least 3 digits

    // Concatenate digits using bitmask
    int bitmask = 0;
    int count = 0;

    auto addDigits = [&](int num) {
        while (num > 0) {
            int digit = num % 10;
            if (digit == 0 || (bitmask & (1 << digit)) != 0) return false; // Invalid
            bitmask |= (1 << digit); // Mark digit as seen
            num /= 10;
            count++;
        }
        return true;
    };

    // Check digits from N, N*2, and N*3
    if (!addDigits(number) || !addDigits(number * 2) || !addDigits(number * 3)) {
        return false;
    }

    // Ensure all digits 1 to 9 are used exactly once
    return count == 9 && (bitmask == 0b1111111110);
}

int main() {
    int number;
    cout << "Enter a number: ";
    cin >> number;

    if (isFascinating(number)) {
        cout << number << " is a Fascinating Number." << endl;
    } else {
        cout << number << " is NOT a Fascinating Number." << endl;
    }

    return 0;
}

Output 1:

Enter a number: 679
679 is NOT a Fascinating Number.

=== Code Execution Successful ===

Output 2:

Enter a number: 273
273 is a Fascinating Number.

=== Code Execution Successful ===

Fascinating Numbers Applications

However, primarily a mathematical interest, fascinating numbers can be employed in diverse fields such as patterns, puzzles and theoretical studies. Here is an attempt to delv into these numbers' applications as regards:

What the concept is in terms of mathematical patterns and puzzles

• ✅ Pattern recognition: Fascinating numbers have a permutation of digits connected to it. They create interesting puzzles where the winner is the one with the different digits placed in unique genes.
• ✅ Recreational Mathematics: Math often comes in handy when inventing puzzles or riddles to get the students or enthusiasts interested in the subject. If the above conditions are met, N is fascinating; otherwise, it is definitely no more than a 'non-fascinating number.
• ✅ Teaching Tool: Aids in teaching concepts like: Multiplication and concatenation. Properties of digits and permutations. Logical reasoning and problem-solving skills.
• ✅ Base Systems: The concept can be extended to other number bases (e.g., binary, octal), exploring how fascinating-like properties manifest in those systems.
• ✅ Fascinating numbers have a permutation of digits connected to it.
• ✅ They create interesting puzzles where the winner is the one with the different digits placed in unique genes.
• ✅ Math often comes in handy when inventing puzzles or riddles to get the students or enthusiasts interested in the subject.
• ✅ If the above conditions are met, N is fascinating; otherwise, it is definitely no more than a 'non-fascinating number.
• ✅ Aids in teaching concepts like: Multiplication and concatenation. Properties of digits and permutations. Logical reasoning and problem-solving skills.
• ✅ Multiplication and concatenation.
• ✅ Properties of digits and permutations.
• ✅ Logical reasoning and problem-solving skills.
• ✅ The concept can be extended to other number bases (e.g., binary, octal), exploring how fascinating-like properties manifest in those systems.

Captivating integers in cryptography and number theory

• ✅ Unique Digit Properties The condition that every digit from 1 to 9 shows once and only once seems to follow such principles from cryptography as: Uniqueness of a key. Avoidance of repeats and patterns in an encryption scheme.
• ✅ Key Generation As simple systems of ciphers, some numbers could serve as a source of generation of keys with numeric codes.
• ✅ Hash Functions The ideas of fascinating numbers may also affect hash function : Creating keys with integrity properties. Protecting the sequence of digits during cryptographic procedures.
• ✅ The condition that every digit from 1 to 9 shows once and only once seems to follow such principles from cryptography as: Uniqueness of a key. Avoidance of repeats and patterns in an encryption scheme.
• ✅ Uniqueness of a key.
• ✅ Avoidance of repeats and patterns in an encryption scheme.
• ✅ As simple systems of ciphers, some numbers could serve as a source of generation of keys with numeric codes.
• ✅ The ideas of fascinating numbers may also affect hash function : Creating keys with integrity properties. Protecting the sequence of digits during cryptographic procedures.
• ✅ Creating keys with integrity properties.
• ✅ Protecting the sequence of digits during cryptographic procedures.

Captivating integers in number theory

• ✅ Digit-Based Analysis Fascinating numbers belong in digit-based number theory, which looks at the following: Vertices with unique characteristics. Concatenated numbers with permutations and combinations.
• ✅ Multiplicative Properties Facts such as how interesting numbers are related to simple number sequences e.g. N, Nx2, Nx3 are known to help: Arithmetic progression. Residues, modular space, and residues.
• ✅ Generalizations What about expanding the concept: What if there are N×4, N×5, N×2 and so on. What is the evolution of the property with a larger rung or different rules?
• ✅ Prime Fascination Investigating whether fascinating numbers exhibit unique interactions with prime numbers or form new classes of numbers.
• ✅ Fascinating numbers belong in digit-based number theory, which looks at the following: Vertices with unique characteristics. Concatenated numbers with permutations and combinations.
• ✅ Vertices with unique characteristics.
• ✅ Concatenated numbers with permutations and combinations.
• ✅ Facts such as how interesting numbers are related to simple number sequences e.g. N, Nx2, Nx3 are known to help: Arithmetic progression. Residues, modular space, and residues.
• ✅ Arithmetic progression.
• ✅ Residues, modular space, and residues.
• ✅ What about expanding the concept: What if there are N×4, N×5, N×2 and so on. What is the evolution of the property with a larger rung or different rules?
• ✅ What if there are N×4, N×5, N×2 and so on.
• ✅ What is the evolution of the property with a larger rung or different rules?
• ✅ Investigating whether fascinating numbers exhibit unique interactions with prime numbers or form new classes of numbers.

Fascinating numbers in Computational Applications

• ✅ Algorithm Testing Fascinating number algorithms serve as benchmarks for testing: Efficiency in digit validation. Performance in handling permutations and concatenations.
• ✅ Educational Software Used in coding challenges and educational software to teach programming and logical thinking.
• ✅ Data Compression The concept of unique digit usage aligns with compression techniques, where redundancy is minimized.
• ✅ Fascinating number algorithms serve as benchmarks for testing: Efficiency in digit validation. Performance in handling permutations and concatenations.
• ✅ Efficiency in digit validation.
• ✅ Performance in handling permutations and concatenations.
• ✅ Used in coding challenges and educational software to teach programming and logical thinking.
• ✅ The concept of unique digit usage aligns with compression techniques, where redundancy is minimized.

Conclusion:

In the realm of number theory, the perplexing and sprawling territories show many fascinating numbers that present a captivating and unique state of sieving image protrusion formation. When it depends upon certain numbers 2 and 3, whereby sin integral formation multiformity ensures that anticipating each '3' strings must be present within 0-9 recorded alphanumeric formation integral turns out to be interesting numbers. This is a fascinating topic; the study gained much attention not only in the field of mathematics but also in fields such as professional computational, cryptography, and number theorists to be able to optimize and enhance the algorithm.

We learnt by means of equipment employed in logically constructing a series of numbers zero to form the number one. It is sieving by multiplying either number with two in regions and three separate places stooping turns out to be a captivating number. Techniques used in C++ programming such as abstracting and several other relevant technologies, including manipulation of Strings, bit masking and halting. All in a bid to design most of these checks even with large inputs, they will be done fast and accurately are particularly beneficial factors for the precision focused operational major. These techniques were listening for efficiency integral turns out to be very.

The pattern, permutation, and nature exhibit of fascinating numbers indeed have almost no usage but rather have significantly lower portions. Rather the important usage is exploring patterns, permutation, and Z permutations along with Pokémon in the other deep nature of numbers fascinate competition as well the numbers exploration of fascinating enhances computational skills. Exploring all types within even the establishment of fascinating numbers extends beyond a more ordinary level. In the future, we can extend this concept by examining how it evolves in different mathematical contexts, such as when multiplying by more than three or exploring properties in different numeral systems.

Ultimately, fascinating numbers offer a blend of theoretical beauty and practical utility, making them an intriguing subject of study in both mathematical puzzles and computational challenges.