Generalizations of the Feline and Texas Chainsaw Josephus Problems

Abstract

We define and study the Extended Feline Josephus Game, a game in which n players, each with ℓ lives, stand in a circle. The game proceeds by alternating between hitting k consecutive players—each of whom will consequently lose a life—and skipping s consecutive players. This cycle continues until every player except one loses all of their lives. Given the nonnegative integer parameters n , k , s and ℓ , the goal of the game is to identify the surviving player. In this paper, we show how the defining parameters n , k , s , and ℓ affect the survivor of games with specific constraints on those parameters and our main results provide new closed formulas to determine the survivor of these Extended Feline Josephus Games. Moreover, for cases where these formulas do not apply, we provide recursive formulas for reducing the initial game to other games with smaller parameter values. For the interested reader, we present a variety of directions for future work in this area, including an extension which considers players lying on a general graph, rather than on a circle.