Solution: Sum of k-Mirror Numbers

Statement

A k-mirror number is a positive integer without leading zeros that is a palindrome in both base-$10$ and base-k.

Given an integer k representing the base and an integer n, return the sum of the n smallest k-mirror numbers.

Note: A palindrome is a number that reads the same both forward and backward.

Constraints:

• $2 \leq$ k $\leq 9$

• $1 \leq$ n $\leq 30$

Solution

The key insight is that k-mirror numbers must be palindromes in both base-$10$ and base-k. Rather than checking every positive integer, we can drastically reduce the search space by generating only base-$10$ palindromes in increasing order and then filtering those that are also palindromes when converted to base-k. We generate base-$10$ palindromes efficiently by constructing them from their “prefix” (the first half of the digits), mirroring it to form the full palindrome. We start with single digit palindromes ($1$ ...