Maximum Nim and Josephus Problem algorithm

In this study, we study a Josephus problem algorithm. Let $n,k$ be positive integers and gₖ(n) = n/k-1 +1, where \ \ is a floor function. Suppose that there exists p such that gₖᵖ⁻¹(0) < n(k-1) ≤ gₖᵖ(0), where gₖᵖ is the p-th functional power of gₖ. Then, the last number that remains is nk-h2ₖᵖ(0) in the Josephus problem of n numbers, where every k-th numbers are removed. This algorithm is based on Maximum Nim with the rule function fₖ(n)= n/k. Using the present article's result, we can build a new algorithm for Josephus problem.