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Solution: Sum of k-Mirror Numbers

Understand how to solve the sum of k-mirror numbers problem by generating base-10 palindromes and checking their palindrome property in base-k. Explore efficient palindrome generation, base conversions, and strategies to handle constraints, equipping you with a method to tackle similar math and geometry coding problems.

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Statement

A k-mirror number is a positive integer without leading zeros that is a palindrome in both base-1010 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≤2 \leq k ≤9\leq 9

  • 1≤1 \leq n ≤30\leq 30

Solution

The key insight is that k-mirror numbers must be palindromes in both base-1010 and base-k. Rather than checking every positive integer, we can drastically reduce the search space by generating only base-1010 palindromes in increasing order and then filtering those that are also palindromes when converted to base-k. We generate base-1010 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 (11 ...