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Grokking the Coding Interview Patterns in Python

Solution: Sum of k-Mirror Numbers

Explore how to identify and sum the n smallest k-mirror numbers, which are positive integers palindromic in base 10 and base k. Understand efficient palindrome generation from prefixes, conversion between bases, and palindrome checking to optimize your solution for coding interviews.

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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 ...