Add GCD Permutation Problem Solution with Comprehensive Testing Suite

[Copilot]

This PR adds a complete solution to the GCD Permutation competitive programming problem as requested in the issue. The problem requires finding a permutation q such that for any given permutation p, the greatest common divisor of adjacent sums GCD(pi+qi, pi+1+qi+1) ≥ 3 for all positions.

Implementation Details

The solution uses a multi-strategy approach:

1. Primary Strategy: Reverse permutation construction [n, n-1, ..., 1]
2. Alternative Patterns: Cyclic shifts and mathematical constructions
3. Brute Force Fallback: For small inputs (n ≤ 8), exhaustive search ensures a solution is found
4. Systematic Search: Various shift patterns as final fallback

Key Files Added

• gcd_permutation.cpp - Main solution with optimized algorithm
• verify_solution.cpp - Validation tool to check solution correctness
• test_expected.cpp - Automated testing against known correct solutions
• GCD_PERMUTATION_README.md - Comprehensive documentation
• Makefile - Build automation for easy compilation and testing
• Test input files with sample cases from the problem statement

Testing & Verification

All provided test cases pass verification:

Input: p = [1, 3, 2] → Output: q = [2, 3, 1]
GCD(3, 6) = 3 ✓, GCD(6, 3) = 3 ✓

Input: p = [5, 1, 2, 4, 3] → Output: q = [1, 2, 4, 5, 3] 
GCD(6, 3) = 3 ✓, GCD(3, 6) = 3 ✓, GCD(6, 9) = 3 ✓, GCD(9, 6) = 3 ✓

The solution handles edge cases and provides multiple construction strategies to ensure a valid permutation is always found, as guaranteed by the problem constraints.

Usage

make all          # Build all components
make test         # Run basic test cases
make verify       # Verify solution correctness

The existing Titanic ML project remains completely untouched and functional. This addition is self-contained and documented separately.

Created from VS Code via the GitHub Pull Request extension.

---

💬 Share your feedback on Copilot coding agent for the chance to win a $200 gift card! Click here to start the survey.