Major

Mathematics

Anticipated Graduation Year

2020

Access Type

Open Access

Abstract

We consider a modified version of the classic combinatorial gameNIM. First we generated 2.5 million states and determined which are winning and which are losing. In some regions the structure seems chaotic, but in others there are patterns. The most obvious is a region toward the end of the game where losing and winning states form wedges and the optimal strategies are fairly simple. But there are also regions where more unexpected wave-like patterns show up. In some cases these persist for a while but eventually disappear, but at least one instance seems to repeat indefinitely. The waves in that case are almost perfectly described by quadratic equations, a phenomenon that we can explain, at least in the limit.

Faculty Mentors & Instructors

Peter Tingley

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.

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Structures From Chaos In a NIM Game

We consider a modified version of the classic combinatorial gameNIM. First we generated 2.5 million states and determined which are winning and which are losing. In some regions the structure seems chaotic, but in others there are patterns. The most obvious is a region toward the end of the game where losing and winning states form wedges and the optimal strategies are fairly simple. But there are also regions where more unexpected wave-like patterns show up. In some cases these persist for a while but eventually disappear, but at least one instance seems to repeat indefinitely. The waves in that case are almost perfectly described by quadratic equations, a phenomenon that we can explain, at least in the limit.