The Penney's Game with Group Action

TL;DR

This paper extends Penney's game to patterns defined by group actions on words, analyzing non-transitivity, expected wait times, and bounds under various group symmetries.

Contribution

It introduces a framework for Penney's game with group action-based patterns, providing new insights into non-transitivity and pattern-based probabilities.

Findings

Non-transitivity persists as pattern length increases.

Derived bounds on Conway leading number and wait times.

Analyzed game behavior under cyclic and symmetric group actions.

Abstract

Consider equipping an alphabet $\mathcal{A}$ with a group action that partitions the set of words into equivalence classes which we call patterns. We answer standard questions for the Penney's game on patterns and show non-transitivity for the game on patterns as the length of the pattern tends to infinity. We also analyze bounds on the pattern-based Conway leading number and expected wait time, and further explore the game under the cyclic and symmetric group actions.