A Variant on the Feline Josephus Problem

TL;DR

This paper introduces a new variant of the Feline Josephus problem where multiple soldiers are affected simultaneously, providing formulas for survivors and exploring solution methods for this complex elimination process.

Contribution

It proposes a novel variant of the Feline Josephus problem with multiple soldiers affected at once and offers closed-form solutions for certain cases.

Findings

Closed formulas for survivor in specific cases

New variant with consecutive removals and skips

Potential methods for solving other cases

Abstract

In the Feline Josephus problem, soldiers stand in a circle, each having $\ell$ `lives'. Going around the circle, a life is taken from every $k$th soldier; soldiers with 0 lives remaining are removed from the circle. Finding the last surviving soldier proves to be an interesting and difficult problem, even in the case when $\ell=1$. In our variant of the Feline Josephus problem, we instead remove a life from $k$ consecutive soldiers, and skip 1 soldier. In certain cases, we find closed formulas for the surviving soldier and hint at a way of finding such solutions in other cases.