Josephus problem

The Josephus problem is a classic puzzle in mathematics and computer science, describing a scenario where $n$ people stand in a circle and are systematically eliminated by counting every $k$-th person (with $k\ge 2$) until only one survivor remains; the goal is to determine the initial position of that survivor, denoted as $Jk(n)$. This elimination process forms a permutation of the positions, and the problem has been generalized to various counting rules and moduli.