Asked in: Walmart
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All Testcases Passed ✔
Solution
from functools import lru_cache
n1,n2 = map(int, input().split())
@lru_cache(None)
def solve(n):
// ... rest of solution available after purchaseExplanation
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To solve the problem of identifying Mystic Numbers between two given boundaries N1 and N2, it's important to first clearly understand the definition and constraints, and then devise an approach that is efficient enough given the potentially large input range (up to 10^9).
Step 1: Understanding the Mystic Number Criteria
- The number must be even. This is straightforward: any number ending with 0, 2, 4, 6, or 8 qualifies.
- The repeated sum of the digits (digital root) must be a single-digit even number. The process of repeatedly summing digits until a single digit remains is commonly known as calculating the "digital root."
- The possible digital roots range from 1 to 9 (0 only if the number itself is zero, but here our numbers are positive).
- Among these, the even single-digit numbers are 2, 4, 6, and 8.
Step 2: Understanding Digital Root and Its Properties
- The digital root of a positive integer can be calculated using a known formula based on modulo 9 arithmetic:
- digital_root = 9 if number mod 9 == 0 else number mod 9
- This allows us to avoid repeatedly summing digits which would be computationally expensive for large numbers.
- We want the digital root to be one of {2,4,6,8}.
Step 3: Formulating the Problem Efficiently
- Since numbers can be as large as 10^9, checking every number in the range by brute force is not feasible.
- Instead, think about how to leverage arithmetic properties:
- We know all Mystic Numbers must be even.
- Among these, the digital root must be in the set {2,4,6,8}.
- Given these constraints, the problem reduces to identifying all even numbers in the range whose digital root is one of these four values.
Step 4: Mapping Digital Root Values Back to Numbers
- To find such numbers, consider iterating over the range but skip unnecessary computations:
- Since only even numbers qualify, iterate in steps of 2.
- Calculate the digital root using the modulo 9 property.
- Check if the digital root is in {2,4,6,8}.
Step 5: Optimizing the Iteration Further
- Even stepping reduces the number of candidates roughly by half.
- However, iterating over up to 10^9/2 numbers might still be slow.
- Look for patterns or cycles in digital roots for numbers modulo 18 or other multiples:
- Digital roots repeat in a pattern when numbers increase by 9.
- Since we restrict to even numbers, investigate how digital roots behave modulo 18.
- Using this, the problem reduces to:
- Identify the pattern of digital roots for numbers modulo 18.
- Count and enumerate the Mystic Numbers in the range using the discovered pattern.
- This way, instead of checking every number, you can jump through the range in chunks, computing counts and positions efficiently.
Step 6: Generating the List of Mystic Numbers
- After counting, for reporting all Mystic Numbers, apply the modular arithmetic to reconstruct actual Mystic Numbers in the range.
- For each number fitting the modular pattern and digital root condition, print or store it.
Step 7: Edge Cases and Validations
- Check the order of N1 and N2, swap if necessary to handle any input.
- Ensure that the range includes boundary values and the logic handles them correctly.
- Validate that single-digit numbers are handled properly, especially small ranges.
- Confirm that zero and odd numbers are correctly excluded.
Step 8: Summary of Approach
- Use the digital root formula to avoid expensive digit summations.
- Limit candidates to even numbers.
- Exploit modulo 18 cyclicity to identify numbers whose digital roots are {2,4,6,8}.
- Calculate starting points and increments to efficiently enumerate or count Mystic Numbers.
- Carefully handle input boundaries and output formatting.
This approach balances mathematical insight and computational efficiency, ensuring that the solution scales well even for very large input ranges.
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