Problem Description

You are given a positive integer n. Your task is to calculate the sum of all digits in n, but with alternating signs applied to each digit.

The sign assignment follows these rules:

• The leftmost (most significant) digit gets a positive sign
• Every subsequent digit gets the opposite sign of the digit before it

For example, if n = 521:

• The digit 5 (leftmost) gets a positive sign: +5
• The digit 2 gets a negative sign (opposite of previous): -2
• The digit 1 gets a positive sign (opposite of previous): +1
• The result would be: +5 - 2 + 1 = 4

The solution converts the number to a string to access each digit by position. Using enumerate(str(n)), it iterates through each digit with its index i. The expression (-1) ** i generates the alternating sign pattern (+1 for even indices, -1 for odd indices), which is then multiplied by each digit value. The sum() function adds all these signed digits together to produce the final result.

Intuition

When we need to process individual digits of a number with alternating signs, the key insight is recognizing this as a pattern problem. The first digit is positive, second is negative, third is positive, and so on - this creates a clear alternating pattern.

To work with individual digits, we need to extract them somehow. Converting the number to a string is the most straightforward approach since it allows us to iterate through each digit character easily.

For the alternating signs, we observe that:

• Position 0 (first digit): positive sign
• Position 1 (second digit): negative sign
• Position 2 (third digit): positive sign
• Position 3 (fourth digit): negative sign

This pattern matches perfectly with the mathematical expression (-1) ** i, where i is the position index:

• When i = 0: (-1) ** 0 = 1 (positive)
• When i = 1: (-1) ** 1 = -1 (negative)
• When i = 2: (-1) ** 2 = 1 (positive)
• When i = 3: (-1) ** 3 = -1 (negative)

By combining these two ideas - string conversion for digit access and the power of -1 for sign alternation - we can process all digits in a single pass. Each digit gets multiplied by its corresponding sign based on position, and summing all these values gives us the final answer.

Learn more about Math patterns.

Solution Approach

The implementation follows a simulation approach where we process each digit with its corresponding sign.

Starting with the given number n, we convert it to a string representation using str(n). This transformation allows us to access each digit individually as a character.

We use Python's enumerate() function to iterate through the string, which gives us both the index i (position of the digit) and the character x (the digit itself) for each iteration.

For each digit:

1. Convert the character x back to an integer using int(x)
2. Calculate the sign using (-1) ** i:
   • Even indices (0, 2, 4, ...) produce (-1) ** even = 1 (positive)
   • Odd indices (1, 3, 5, ...) produce (-1) ** odd = -1 (negative)
3. Multiply the digit by its sign: (-1) ** i * int(x)

The generator expression ((-1) ** i * int(x) for i, x in enumerate(str(n))) produces all the signed digit values, and the sum() function adds them together to get the final result.

For example, with n = 521:

• Index 0, digit '5': (-1) ** 0 * 5 = 1 * 5 = 5
• Index 1, digit '2': (-1) ** 1 * 2 = -1 * 2 = -2
• Index 2, digit '1': (-1) ** 2 * 1 = 1 * 1 = 1
• Sum: 5 + (-2) + 1 = 4

This single-line solution efficiently handles the alternating sign pattern while processing all digits in one pass.

Example Walkthrough

Let's walk through the solution with n = 7834:

Step 1: Convert to string

• n = 7834 becomes "7834"
• This allows us to access each digit individually

Step 2: Process each digit with enumerate

• enumerate("7834") gives us pairs of (index, character):
  • (0, '7')
  • (1, '8')
  • (2, '3')
  • (3, '4')

Step 3: Apply alternating signs using (-1)^i

• Index 0, digit '7': (-1)^0 * 7 = 1 * 7 = +7
• Index 1, digit '8': (-1)^1 * 8 = -1 * 8 = -8
• Index 2, digit '3': (-1)^2 * 3 = 1 * 3 = +3
• Index 3, digit '4': (-1)^3 * 4 = -1 * 4 = -4

Step 4: Sum all signed digits

• 7 + (-8) + 3 + (-4) = 7 - 8 + 3 - 4 = -2

The final result is -2.

The complete solution in one line:

result = sum((-1) ** i * int(x) for i, x in enumerate(str(7834)))

This efficiently processes all digits in a single pass, applying the alternating sign pattern based on each digit's position.

Solution Implementation

Time and Space Complexity

The time complexity is O(log n), where n is the given number. This is because converting the integer n to a string requires O(log n) time (the number of digits in n is proportional to log₁₀(n)), and then iterating through each digit also takes O(log n) time since there are O(log n) digits.

The space complexity is O(log n). This is due to the string representation str(n) which creates a string of length proportional to the number of digits in n, which is O(log n). Additionally, the generator expression creates intermediate values, but the overall space usage is dominated by the string conversion, resulting in O(log n) space complexity.

Learn more about how to find time and space complexity quickly.

Common Pitfalls

1. Integer Overflow in Sign Calculation

Using (-1) ** i for large index values can be inefficient as exponentiation becomes computationally expensive for large powers. While Python handles arbitrary precision integers, this approach is slower than necessary.

Better Alternative:

sign = 1 if i % 2 == 0 else -1

This modulo operation is more efficient and clearer in intent.

2. Incorrect Sign Pattern Assumption

A common mistake is assuming the pattern starts with negative instead of positive, leading to code like:

# Wrong: This would give -5 + 2 - 1 for n=521
sign = (-1) ** (i + 1)  

Solution: Always verify the problem statement carefully. The leftmost digit should be positive, which means index 0 should have a positive sign.

3. Direct Integer Division Approach Without Reversal

Some might try to extract digits using division and modulo operations:

# This processes digits from right to left
while n > 0:
    digit = n % 10
    n //= 10

This extracts digits in reverse order (least significant first), requiring additional logic to handle the alternating pattern correctly from the most significant digit.

Solution: Either reverse the extracted digits first or stick with the string conversion approach for simplicity.

4. Off-by-One Error in Manual Indexing

When implementing without enumerate:

# Potential error if forgetting to initialize or increment properly
i = 0
for digit_char in str(n):
    sign = (-1) ** i  # Easy to forget to increment i
    # ... rest of logic

Solution: Use enumerate() to automatically handle index tracking, or ensure proper index management with explicit increment i += 1 in each iteration.

5. Type Confusion with String Digits

Forgetting to convert string characters back to integers:

# Wrong: This would concatenate strings or cause type errors
alternating_sum += sign * digit_char  # digit_char is still a string

Solution: Always convert with int(digit_char) before arithmetic operations.