The Warden's de Bruijn Sequence

TL;DR

This paper analyzes the Warden's Game, a two-player coin transfer game, deriving optimal strategies and exploring variations, with connections to de Bruijn sequences.

Contribution

It introduces a novel analysis of the Warden's Game, linking it to de Bruijn sequences and providing strategies for both players.

Findings

Optimal strategies for both players are derived.

The game dynamics relate to de Bruijn sequences.

Variations of the game are analyzed for different outcomes.

Abstract

The Warden's Game is a 2-player game, played with a row of coins. One player (the prisoner) wants to get all coins to show tails; the other player (the warden) wants to delay that as long as possible. At each turn, one player transfers the coin on the far right to the far left, and optionally flips that coin; the prisoner transfers heads, and the warden transfers tails. We will find the optimal strategies for both players, and we will also analyze some variations on this game.