2081. Sum of k-Mirror Numbers

2081. Sum of k-Mirror Numbers

Difficulty: Hard

Topics: Math, Enumeration

A k-mirror number is a positive integer without leading zeros that reads the same both forward and backward in base-10 as well as in base-k.

• For example, 9 is a 2-mirror number. The representation of 9 in base-10 and base-2 are 9 and 1001 respectively, which read the same both forward and backward.
• On the contrary, 4 is not a 2-mirror number. The representation of 4 in base-2 is 100, which does not read the same both forward and backward.

Given the base k and the number n, return the sum of the n smallest k-mirror numbers.

Example 1:

• Input: k = 2, n = 5
• Output: 25
• Explanation: The 5 smallest 2-mirror numbers and their representations in base-2 are listed as follows:

base-10	base-2
1	1
3	11
5	101
7	111
9	1001

• Their sum = 1 + 3 + 5 + 7 + 9 = 25.

Example 2:

• Input: k = 3, n = 7
• Output: 499
• Explanation: The 7 smallest 3-mirror numbers are and their representations in base-3 are listed as follows:

base-10	base-2
1	1
2	2
4	11
8	22
121	11111
151	12121
212	21212

• Their sum = 1 + 2 + 4 + 8 + 121 + 151 + 212 = 499.

Example 3:

• Input: k = 7, n = 17
• Output: 20379000
• Explanation: The 17 smallest 7-mirror numbers are: 1, 2, 3, 4, 5, 6, 8, 121, 171, 242, 292, 16561, 65656, 2137312, 4602064, 6597956, 6958596

Constraints:

• 2 <= k <= 9
• 1 <= n <= 30

Hint:

1. Since we need to reduce search space, instead of checking if every number is a palindrome in base-10, can we try to "generate" the palindromic numbers?
2. If you are provided with a d digit number, how can you generate a palindrome with 2*d or 2*d - 1 digit?
3. Try brute-forcing and checking if the palindrome you generated is a "k-Mirror" number.

Solution:

We need to find the sum of the n smallest k-mirror numbers. A k-mirror number is a positive integer that reads the same both forward and backward in both base-10 and base-k. The solution involves generating palindromic numbers in base-10 and checking if they are also palindromic in base-k.

Approach

1. Generate Palindromic Numbers in Base-10:

   • We generate palindromic numbers in increasing order by their digit lengths. For each digit length L:
     • Even Length (L is even): The palindrome is formed by concatenating a number x with its reverse. For example, if x is 12, the palindrome is 1221.
     • Odd Length (L is odd): The palindrome is formed by concatenating a number x with the reverse of x without its last digit. For example, if x is 12, the palindrome is 121.
   • We start generating palindromes from the smallest possible length (1 digit) and proceed to longer lengths until we collect n k-mirror numbers.

2. Check Palindrome in Base-k:

   • For each generated base-10 palindrome, convert it to base-k.
   • Check if the base-k representation is a palindrome. This involves:
     • Converting the number to base-k by repeatedly dividing the number by k and collecting remainders.
     • Comparing the generated base-k string with its reverse to determine if it reads the same backward as forward.

3. Sum the Valid Numbers:

   • Collect the first n valid k-mirror numbers and return their sum.

Let's implement this solution in PHP: 2081. Sum of k-Mirror Numbers

<?php
/**
 * @param Integer $k
 * @param Integer $n
 * @return Integer
 */
function kMirror($k, $n) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

/**
 * @param $num
 * @param $k
 * @return bool
 */
function is_basek_palindrome($num, $k) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo kMirror(2, 5) . PHP_EOL;   // 25
echo kMirror(3, 7) . PHP_EOL;   // 499
echo kMirror(7, 17) . PHP_EOL;  // 20379000
?>

Explanation:

1. Generating Palindromic Numbers:

   • For each digit length L, we generate palindromic numbers:
     • Even L: The palindrome is created by mirroring the entire string of digits. For example, for x = 12, the palindrome is 1221.
     • Odd L: The palindrome is created by mirroring the string except the last digit. For example, for x = 12, the palindrome is 121.
   • We start with the smallest digit length (1) and proceed to longer lengths until we have collected n k-mirror numbers.

2. Checking Base-k Palindrome:

   • For each generated palindrome, we convert it to base-k by repeatedly dividing the number by k and collecting remainders. The remainders form the base-k representation in reverse order, which we then reverse to get the correct string.
   • We check if this base-k string is a palindrome by comparing it with its reverse.

3. Summing Valid Numbers:

   • We collect the first n numbers that are palindromic in both base-10 and base-k. The sum of these numbers is returned as the result.

This approach efficiently generates palindromic numbers in increasing order and checks their validity in base-k, ensuring optimal performance for the given constraints.

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