Sum of k-Mirror Numbers

Sum of k-Mirror Numbers

What is this problem about?

The "Sum of k-Mirror Numbers" problem is a high-level mathematical challenge. A number is defined as "k-mirror" if it is a palindrome in both base 10 (decimal) and base $k$. Given $k$ and an integer $n$, your task is to find the sum of the $n$ smallest positive k-mirror numbers. For example, if $k=2$, you are looking for the smallest numbers that are palindromes in decimal and also palindromes in binary.

Why is this asked in interviews?

Companies like Google, Microsoft, and Meta use this Hard-level problem to test a candidate's ability to handle multi-base conversions and systematic enumeration. Since checking every number sequentially is too slow, the problem requires generating palindromes directly. It evaluates if a candidate can optimize a search by generating only the candidates that satisfy one constraint (decimal palindrome) and then checking them against the second constraint (base-$k$ palindrome).

Algorithmic pattern used

The primary pattern is "Palindromic Enumeration." Instead of checking all numbers 1, 2, 3..., you generate decimal palindromes in increasing order of length (1-digit, 2-digit, 3-digit...). For each generated decimal palindrome, you convert it to base $k$ and check if the resulting string is also a palindrome. To keep the numbers in order, you can use a queue or a length-based generation strategy. You continue until you have found $n$ such numbers.

Example explanation

Suppose $k=2$ and we need the first few k-mirror numbers.

• Decimal 1 is a palindrome. In binary, 1 is "1" (palindrome). Found 1.
• Decimal 3 is a palindrome. In binary, 3 is "11" (palindrome). Found 2.
• Decimal 7 is a palindrome. In binary, 7 is "111" (palindrome). Found 3.
• Decimal 9 is a palindrome. In binary, 9 is "1001" (palindrome). Found 4. The algorithm generates these by constructing palindromes like "1", "2", ..., "9", "11", "22", etc., and performing the base-$k$ check.

Common mistakes candidates make

1. Iterative Checking: Trying to check every single number 1...N. Since k-mirror numbers can be very large, this will never finish within time limits.
2. Base Conversion inefficiency: Writing a slow base-conversion function or repeatedly converting the same values.
3. Missing Palindromes: Failing to correctly generate both odd-length (e.g., 121) and even-length (e.g., 1221) palindromes systematically.

Interview preparation tip

Practice generating palindromes by taking a prefix (e.g., "12") and reflecting it (to get "121" or "1221"). This is a common requirement in "Hard" palindromic problems. Also, ensure you are comfortable with base conversion algorithms using division and modulo.