Magical Number in Python with O(1) Time Complexity

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

Example 1:

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

Example 2:

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

Problem Definition

The problem is to find the magical number of a given positive integer. A magical number is obtained by repeatedly summing the digits of the number until a single digit is obtained.

Input:

A positive integer N.

Output:

A single digit which is the magical number.

Constraints:

The input number N is a positive integer.

Example:

Input: 39
Output: 3

Understanding the Problem

The core challenge 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.

Potential pitfalls include not handling large numbers efficiently and misunderstanding the repeated summation process.

Approach

To solve this problem, we can use a mathematical property of digital roots. The digital root of a number can be found using modulo 9 arithmetic:

If the number is 0, the digital root is 0.
If the number is not 0, the digital root is 9 if the number is divisible by 9, otherwise, it is the number modulo 9.

Naive Solution

A naive solution involves repeatedly summing the digits until a single digit is obtained. This approach is not optimal for large numbers due to its iterative nature.

Optimized Solution

The optimized solution leverages the properties of digital roots to achieve an O(1) time complexity. This approach is efficient and handles large numbers gracefully.

Algorithm

Here is a step-by-step breakdown of the optimized algorithm:

Check if the number is 0. If yes, return 0.
Check if the number is divisible by 9. If yes, return 9.
Otherwise, return the number modulo 9.

Code Implementation

def magic_number(n):
    # If the number is 0, return 0
    if n == 0:
        return 0
    # If the number is divisible by 9, return 9
    if n % 9 == 0:
        return 9
    # Otherwise, return the number modulo 9
    return n % 9

# Test cases
print(magic_number(39))  # Output: 3
print(magic_number(928435))  # Output: 4

Complexity Analysis

The time complexity of the optimized solution is O(1) because it involves a constant number of operations regardless of the input size. The space complexity is also O(1) as no additional space is required.

Edge Cases

Potential edge cases include:

Input is 0: The output should be 0.
Input is a multiple of 9: The output should be 9.

Input: 0
Output: 0

Input: 18
Output: 9

Testing

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

Simple cases: Single-digit numbers.
Complex cases: Large numbers with multiple digits.
Edge cases: Numbers like 0 and multiples of 9.

# Additional test cases
assert magic_number(0) == 0
assert magic_number(18) == 9
assert magic_number(123456789) == 9
assert magic_number(987654321) == 9
assert magic_number(111111111) == 1

Thinking and Problem-Solving Tips

When approaching such problems, consider mathematical properties and patterns that can simplify the solution. Practice similar problems to develop a deeper understanding of number theory and digital roots.

Conclusion

Understanding and solving the magical number problem helps in grasping the concept of digital roots and their applications. Practice and exploration of similar problems can enhance problem-solving skills.