How to Calculate the Sum of K-Mirror Numbers?

If you are preparing for a tech interview and encounter a question about the sum of k-mirror numbers, it’s important to understand what these numbers are and how to find their sum efficiently. In most cases, the problem involves generating numbers that are palindromic in both their decimal and k-ary representations, called k-mirror numbers, and then summing the first N of these numbers.

What Are K-Mirror Numbers?

K-mirror numbers are numbers that read the same forward and backward in normal decimal form and also in base k. For example, if k=10, classic palindromic numbers like 121, 1331, and 9449 are 10-mirror numbers. For a different k, such as 2, these numbers are palindromic in binary form, often called binary palindromes.

Example of K-Mirror Numbers

Suppose k=2, and we look at numbers that are palindromic in binary. Numbers like 1 (binary 1), 3 (binary 11), 5 (binary 101), and 9 (binary 1001) are binary palindromes. Among these, some also happen to be palindromic in decimal.

Approach to Finding the Sum

To solve a problem asking for the sum of the first N k-mirror numbers, you should:

Generate candidates for k-mirror numbers.
Verify the k-mirror property for each candidate.
Continue this process until you find N such numbers.
Sum these numbers.

This process involves generating palindromes efficiently, checking their base-k representation, and summing up.

Generating Palindromes

One common and efficient way to generate palindromic numbers is by constructing the number halves and mirroring them:

For odd-length palindromes, generate the first half, then mirror all but the middle digit.
For even-length palindromes, generate the first half, then mirror it to form the full number.

Here is an example implementation in Python:

python

1def generate_palindromes(limit):
2    palindromes = []
3    # Generate odd length palindromes
4    for half in range(1, limit):
5        s = str(half)
6        pal = int(s + s[-2::-1])
7        palindromes.append(pal)
8    # Generate even length palindromes
9    for half in range(1, limit):
10        s = str(half)
11        pal = int(s + s[::-1])
12        palindromes.append(pal)
13    return palindromes

Checking the K-Representation

To verify whether a number is also palindromic in base k, convert the number to base k, then check if the string is a palindrome:

python

1def is_palindrome_in_base(n, k):
2    representation = ''
3    while n > 0:
4        representation = str(n % k) + representation
5        n //=k
6    return representation == representation[::-1]

Combining the Logic

Now, combining the generation and verification steps, you can iterate over generated palindromes, check their base-k form, and sum the first N verified numbers.

Here's a simplified code example to find the sum:

python

1def sum_k_mirror_numbers(k, N):
2    total = 0
3    count = 0
4    limit = 1
5    while count < N:
6        palindromes = generate_palindromes(limit)
7        for p in palindromes:
8            if is_palindrome_in_base(p, k):
9                total += p
10                count += 1
11                if count == N:
12                    return total
13        limit += 1
14    return total

This code generates palindromes, checks if they are also palindromic in base k, and maintains a count and sum until reaching N. Adjustments may be needed to optimize, especially for large N.

Understanding how to generate palindromic numbers and verify their base-k palindrome property is key in solving sum of k-mirror numbers problems. Efficient code can utilize string manipulations and symmetry properties to generate candidates quickly, then filter for the k-base palindromes. This approach allows you to find the sum with reasonable performance, fulfilling the typical requirements of a technical interview question.