The Mathematics behind an Elimination Game
Zhichun Cao
Abstract
The Josephus problem has been studied in depth over the years. We propose an elimination game that keeps the killing process from the Josephus problem but positions the soldiers in a straight line such that a soldier on an odd-numbered position is uniformly chosen and eliminated each time. After the modification, who survives the longest? Which soldier becomes the last to die? In this article, we employ induction and Markov chain method to model each step of elimination and algebraically derive probability formulae for a soldier to be the last survivor. Subsequently, we investigate how survival probability changes during the elimination game by interpreting graphs generated by experiments. Eventually, we simulate the game to show that, contrary to our intuition, the soldier who has the longest life expectancy does not necessarily emerge as the last survivor.
