The "Hidden QC Data" problem typically involves analyzing quality control data where defects are not directly observable but must be inferred from indirect evidence or patterns. Below is a step-by-step solution approach and example: A factory produces items, each with a unique ID (1 to 100). The defect status is hidden and determined by a rule:

• Defective if the item ID is a prime number or a perfect square.
• Non-defective otherwise.

Goal: Calculate the total number of defective items and the defect rate.

Solution Steps

1. Identify Defective Items:

   • Prime numbers: IDs divisible only by 1 and themselves.
   • Perfect squares: IDs that are squares of integers (e.g., 1, 4, 9, 16, ..., 100).

2. Generate IDs:

   IDs range from 1 to 100.

3. Check Conditions:

   For each ID, check if it is prime or a perfect square.

4. Count Defectives:

   Sum all IDs meeting either condition.

5. Calculate Defect Rate:

   Defect rate = (Number of defective items) / (Total items).

Example Calculation

Defective IDs (1 to 100):

• Prime numbers:
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
  → 25 primes.

• Perfect squares:
  1, 4, 9, 16, 25, 36, 49, 64, 81, 100
  → 10 squares.

• Overlap: IDs that are both prime and perfect square:
  Only 1 (since 1 is a perfect square but not prime; no other ID satisfies both).
  → No overlap to subtract.

Total Defective Items:

25 (primes) + 10 (squares) = 35 defective items.

Defect Rate:

35 defective / 100 total items = 35%.

Key Code Implementation (Python)

def is_prime(n):
    if n < 2:
        return False
    for i in range(2, int(n**0.5) + 1):
        if n % i == 0:
            return False
    return True
def is_perfect_square(n):
    root = int(n**0.5)
    return root * root == n
def find_defective_items(max_id=100):
    defectives = set()
    for id in range(1, max_id + 1):
        if is_prime(id) or is_perfect_square(id):
            defectives.add(id)
    return defectives
defective_ids = find_defective_items()
total_defective = len(defective_ids)
defect_rate = total_defective / 100  # Since max_id=100
print(f"Defective items: {total_defective}")
print(f"Defect rate: {defect_rate:.1%}")  # Output: 35.0%

Output

Defective items: 35
Defect rate: 35.0%

Explanation

• Prime Check: Efficiently tests divisibility up to the square root of the ID.
• Perfect Square Check: Uses integer square root to verify if an ID is a square.
• Set Usage: Avoids double-counting IDs that meet both conditions (though none exist here).
• Scalability: Works for any max_id (e.g., 1000, 10000).

This approach efficiently uncovers hidden defects by leveraging mathematical properties and systematic verification.