Count Pairs of Cyclically Equivalent Integers
Company: Capital One
Role: Software Engineer
Category: Coding & Algorithms
Difficulty: medium
Interview Round: Take-home Project
# Count Pairs of Cyclically Equivalent Integers
Two non-negative integers are **cyclically equivalent** when their standard decimal strings have the same length and one string can be obtained by moving zero or more leading characters to the end of the other.
Given a list `numbers`, count unordered index pairs `(i, j)` with `i < j` whose values are cyclically equivalent. Equal values count as cyclically equivalent, and duplicate occurrences are distinct list elements.
## Function Signature
```python
def count_cyclic_pairs(numbers: list[int]) -> int:
...
```
## Constraints
- `0 <= len(numbers) <= 200_000`
- `0 <= numbers[i] <= 1_000_000_000`
- Standard decimal notation has no leading zero, except that zero is represented as `"0"`.
- The answer may exceed 32-bit range.
## Examples
```text
Input: numbers = [123, 312, 231, 45]
Output: 3
```
```text
Input: numbers = [10, 1, 10, 100]
Output: 1
```
```text
Input: numbers = [7, 7, 7]
Output: 3
```
Quick Answer: Count unordered index pairs whose non-negative integers have decimal strings equivalent under cyclic rotation. The algorithm challenge emphasizes a precise equivalence relation, duplicate occurrences, string-length and leading-zero semantics, scalability to 200,000 values, and pair counts beyond 32-bit range.
