Research
Research Presentation
I am currently working on research in Combinatorial Game Theory at Oglethorpe University with Dr. Bill Shillito. My research focuses on winning strategies in an impartial combinatorial game called the Sparse Ruler Game.
A sparse ruler is a ruler with as few marks as possible while still being able to measure every integer distance using the differences between those marks. Think of it as an optimized ruler. We transformed this concept into a two-player game where players take turns placing integer marks on a fixed-length ruler. The goal is simple: be the last player who can make a legal move.
To understand winning strategies, I developed a Python program that constructs a complete game tree and classifies every possible game state. Each state is labeled as either a P-position (losing position — every legal move leads to a winning position for the opponent) or an N-position (winning position — at least one legal move leads to a losing position for the opponent).
To verify our results, we also computed Grundy numbers for each position. A Grundy number of zero indicates a P-position, while any nonzero value indicates an N-position. We analyzed ruler sizes up to 20 using this approach.
Key findings from our research:
- For ruler sizes less than 8, the pattern is irregular. But starting at size 9, a stable pattern emerges.
- Under perfect play, odd ruler sizes consistently favor the second player, while even ruler sizes favor the first player.
- For ruler sizes 9-20, the parity of the ruler size alone determines which player has the winning advantage.
The next phase of this research involves developing a mathematical proof to explain why this parity pattern emerges and whether it holds for ruler sizes beyond 20. This work beautifully combines theoretical mathematics with computational programming, bridging abstract reasoning with practical algorithm design.
You can find the code on GitHub.
Research poster