Lychrel Number Implementation in C++

In number theory, a Lychrel number is a natural number which cannot form a palindromic number using the repeated process of reversing its digits and adding that number to the original number. A number is a Lychrel number if it never becomes a palindrome in such a procedure after many iterations. It was named after the mathematician Lynch, who first coined the term in the early 20th century.

In this article, we are going to discuss how to implement a C++ program to determine if the given number is a Lychrel number.

Problem Understanding:

In order to test whether a number is a Lychrel number, we proceed as follows:

1. Reverse the digits of the number.
2. Add the reverse number to the original number.
3. In other words, the sum is a palindrome that reads the same both forward and backward; in that case, stop there and not a Lychrel number.
4. When the given number does not result in being a palindrome through a certain number of iterations, then it is said to be a Lychrel number.

Key Concept: Palindrome

A palindrome is a number or a sequence that reads the same if followed forward and backward. For example, 121 is a palindrome, and 123 is not.

Approach to Solve the Problem:

1. Check if a number is a palindrome: We will create a function to check if a number is a palindrome. We will reverse the number’s digits and compare the reversed number with the original number.
2. Reverse a number: A supporting function will reverse the digits of the number, which are used within the process for finding a Lychrel number.
3. Iteration method: We will reverse and add in an iteration fashion. When the sum happens to be a palindrome in the iteration limit, say 50 times, the number is not a Lychrel number. The number will be considered a Lychrel number if the above process fails in these number of iterations.

C++ Code Implementation:

Let us take an example to illustrate the Lychrel number in C++.

Output:

Explanation of the Code:

1. isPalindrome() Function:
   This function checks whether any number is a palindrome by reversing all the digits; if the original number and reversed one are equal, then it returns a true value of being a palindrome; otherwise, it returns false.
2. reverseNumber() Function:
   It is the helper function. It takes an integer, reverses its digits, and returns the reversed number. It is basically important in the iterative process of reversing the number’s digits and adding it to the original number.
3. isLychrel() Function:
   This function implements the check for the Lychrel number. It will just keep on going into an infinite loop of reversals and additions of the original number with the reversed number, peeking in each iteration to check if the resulting number is a palindrome. If it does so in any of the steps, within 50 iterations in this case, the function will return False, which indicates that the number is not a Lychrel number. If the number doesn’t turn into a palindrome in that set number of iterations, it returns true, which means the number is a Lychrel number.
4. Main Function:
   In the main function, it asks for a number from the user. After that, the application calls isLychrel() to find whether the given number is a Lychrel number and prints the result.

Example Runs

Example 1:

Input:

Enter a number: 47

Output:

47 is not a Lychrel number.

Explanation:

• 47 reversed is 74.
• 47 + 74 = 121, which is a palindrome. Thus, 47 is not a Lychrel number.

Example 2:

Input:

Enter a number: 196

Output:

196 is a Lychrel number.

Explanation:

• 196 reversed is 691.
• 196 + 691 = 887, which is not a palindrome.
• 887 reversed is 788.
• 887 + 788 = 1675, which is not a palindrome.
• This process continues, and no palindrome is formed even after 50 iterations, so 196 is considered a Lychrel number.

Time Complexity:

In this solution, the number of steps taken for the problem can be approximated as O(d), where d is the number of digits in the number. It is because reversing a number has to do with iterating over its digits, and the number of iterations in the isLychrel function is capped at a maximum of 50, which is constant.

Conclusion:

In conclusion, this C++ implementation succeeds in checking whether a number is a Lychrel number through the reverse and add process iteratively. Though it is relatively simple in concept, implementing it in C++ helps someone understand number manipulation very well, as well as the concept of recursion along with algorithmic design.