Discrete Math Projects – Logic, Algorithms, and Combinatorics in Practice

Vision

These projects were designed to connect discrete mathematics theory with hands-on algorithm development. Spanning satisfiability, combinatorics, game theory, and data structure implementation, the projects reinforce critical thinking, mathematical modeling, and computational problem solving.

Approach

• ValidDates (Combinatorics & Recursion)
  Implemented a recursive and mathematical model to count valid "date" pairings between unique individuals, each with exclusivity constraints. The solution combined brute-force search with optimized recurrence relations and cycle enumeration to reduce computational complexity.

• SquareGame (Game Theory & Nim Strategy)
  Developed a competitive two-player game based on the principles of Nim. Implemented optimal computer strategy using XOR-sum logic to guarantee winning outcomes under ideal conditions, with randomized game board generation and human-interactive gameplay.

• 2-CNF Satisfiability
  Created an algorithm to verify the satisfiability of 2-CNF Boolean expressions using implication graphs and strongly connected components, mimicking the foundation of SAT solvers.

• SkipList Data Structure
  Built a probabilistic alternative to balanced trees with average-case O(log n) operations for search, insert, and delete. Simulated skip levels and random promotions to ensure realistic performance behavior.

Challenges

• Translating theoretical mathematical concepts into efficient and testable code.

• Managing deep recursive logic for exhaustive search and backtracking.

• Ensuring correctness under all game and edge conditions in probabilistic and combinatorial models.

Conclusion

These projects demonstrated the real-world applicability of discrete mathematics in software development. They highlight proficiency in recursion, logical reasoning, probabilistic design, and algorithm analysis—skills foundational to computer science and critical for building optimized systems.