A magical number is obtained from a positive number by adding its digits repeatedly until we obtain one digit.

**Example 1:**

Input:N= 39Output:3Explanation:magicNumber(39) = magicNumber(3 + 9) = magicNumber(12) = magicNumber(1 + 2) = 3

**Example 2:**

Input:N= 928435Output:4Explanation:9 + 2 + 8 + 4 + 3 + 5 = 31 => 3 + 1 = 4

The core challenge of this problem is to repeatedly sum the digits of a number until a single digit is obtained. This problem is significant in various applications, such as digital root calculations in number theory. A common pitfall is not recognizing that this can be solved efficiently without iterative summation.

To solve this problem, we can use a mathematical property of numbers known as the digital root. The digital root of a number can be found using the formula:

digital_root(N) = 1 + (N - 1) % 9

This formula works because of the properties of numbers in modular arithmetic. However, for the sake of understanding, we will also discuss a naive approach.

The naive approach involves repeatedly summing the digits of the number until a single digit is obtained. This approach is straightforward but not optimal for large numbers.

The optimized approach uses the digital root formula, which provides a direct way to compute the result without iterative summation. This approach is efficient and has a constant time complexity.

- Initialize a variable to store the sum of digits.
- While the number has more than one digit:
- Sum the digits of the number.
- Update the number to be the sum of its digits.

- Return the single-digit result.

- If the number is 0, return 0.
- Otherwise, return 1 + (N - 1) % 9.

```
public class MagicalNumber {
// Naive approach to find the magical number
public static int magicNumberNaive(int n) {
while (n >= 10) {
int sum = 0;
while (n > 0) {
sum += n % 10;
n /= 10;
}
n = sum;
}
return n;
}
public static void main(String[] args) {
System.out.println(magicNumberNaive(39)); // Output: 3
System.out.println(magicNumberNaive(928435)); // Output: 4
}
}
```

```
public class MagicalNumber {
// Optimized approach to find the magical number using digital root
public static int magicNumberOptimized(int n) {
if (n == 0) return 0;
return 1 + (n - 1) % 9;
}
public static void main(String[] args) {
System.out.println(magicNumberOptimized(39)); // Output: 3
System.out.println(magicNumberOptimized(928435)); // Output: 4
}
}
```

**Naive Approach:** The time complexity is O(d), where d is the number of digits in the number. The space complexity is O(1).

**Optimized Approach:** The time complexity is O(1) because it uses a direct mathematical formula. The space complexity is also O(1).

Consider the following edge cases:

- Single-digit numbers (e.g., 5): The output should be the number itself.
- Zero (e.g., 0): The output should be 0.
- Large numbers (e.g., 123456789): The algorithm should handle large inputs efficiently.

To test the solution comprehensively, consider a variety of test cases:

- Simple cases: 5, 9
- Edge cases: 0, 10
- Complex cases: 39, 928435, 123456789

Use a testing framework like JUnit to automate the testing process.

When approaching such problems, consider the following tips:

- Understand the problem requirements and constraints thoroughly.
- Think about mathematical properties or patterns that can simplify the problem.
- Start with a naive solution to understand the basic approach, then optimize.
- Practice similar problems to improve problem-solving skills.

In this blog post, we discussed how to solve the problem of finding a magical number by summing its digits repeatedly. We explored both a naive and an optimized approach, analyzed their complexities, and provided Java code implementations. Understanding such problems helps in developing strong problem-solving skills and mathematical insights.

For further reading and practice, consider the following resources: