Sum of k-Mirror Numbers - Problem

A k-mirror number is a fascinating mathematical concept - it's a positive integer that reads the same both forward and backward in both base-10 (decimal) and base-k representations.

What makes this interesting? While many numbers are palindromic in one base, finding numbers that are palindromic in multiple bases simultaneously is much rarer and more challenging.

Examples:

  • 9 is a 2-mirror number: In base-10 it's 9, in base-2 it's 1001 - both are palindromes!
  • 4 is NOT a 2-mirror number: In base-10 it's 4 (palindrome), but in base-2 it's 100 (not a palindrome)

Your Goal: Given a base k and number n, find the sum of the n smallest k-mirror numbers. This requires efficiently generating palindromes in base-10, then checking if they're also palindromes in base-k.

Input & Output

example_1.py โ€” Basic k=2 case
$ Input: k = 2, n = 5
โ€บ Output: 25
๐Ÿ’ก Note: The first 5 k-mirror numbers for k=2 are: 1, 3, 5, 7, 9. Their sum is 1+3+5+7+9 = 25. For example, 9 in base-10 is '9', in base-2 is '1001' - both are palindromes.
example_2.py โ€” Larger k value
$ Input: k = 3, n = 7
โ€บ Output: 499
๐Ÿ’ก Note: The first 7 k-mirror numbers for k=3 are: 1, 2, 4, 8, 121, 151, 212. Sum = 1+2+4+8+121+151+212 = 499. Note how 121 works: base-10 '121' and base-3 '11111' are both palindromes.
example_3.py โ€” Edge case n=1
$ Input: k = 7, n = 1
โ€บ Output: 1
๐Ÿ’ก Note: For any k โ‰ฅ 2, the first k-mirror number is always 1, since 1 is a palindrome in any base (represented as '1').

Visualization

Tap to expand
K-Mirror Discovery ProcessBase-10121Palindrome โœ“ConvertBase-3 Conversion121 โ†’ 1111111111 palindrome โœ“VerifyK-MIRROR!Add to sumSystematic Generation vs Brute ForceBrute Force:Check: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22...Palindromes: 1,2,3,4,5,6,7,8,9,___,11,___,___,___,___,___,___,___,___,___,___,22...Optimized:Generate: 1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,101,111,121,131...Example: First 5 2-Mirror Numbers1, 3, 5, 7, 9 โ†’ Sum = 25Time Complexity: O(n ร— log V) where V is the nth k-mirror value
Understanding the Visualization
1
Generate Base-10 Palindromes
Create palindromes systematically: 1,2,3...9, then 11,22,33...99, then 101,111,121...
2
Convert to Base-K
For each base-10 palindrome, convert it to base-k representation
3
Check Base-K Palindrome
Verify if the base-k representation is also palindromic
4
Collect K-Mirrors
Keep valid k-mirror numbers in ascending order until we have n of them
5
Sum Results
Add up the first n k-mirror numbers for the final answer
Key Takeaway
๐ŸŽฏ Key Insight: Generate palindromes systematically rather than checking every number - this reduces the search space from O(V) to O(โˆšV) candidates, where V is the value of the nth k-mirror number.

Time & Space Complexity

Time Complexity
โฑ๏ธ
O(m * log m)

Where m is the value of the nth k-mirror number. We check m numbers, and each check takes O(log m) time for palindrome verification

n
2n
โšก Linearithmic
Space Complexity
O(log m)

Space needed to store the base-k representation string for palindrome checking

n
2n
โœ“ Linear Space

Constraints

  • 2 โ‰ค k โ‰ค 9
  • 1 โ‰ค n โ‰ค 30
  • All k-mirror numbers will fit in a 64-bit signed integer
  • The answer will be the sum of exactly n k-mirror numbers
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