Maximum Nim and Josephus Problem algorithm

In this study, we study a Josephus problem algorithm. Let \(n,k\) be positive integers and \(g_k(n) = \left\lfloor \frac{n}{k-1} \right\rfloor +1\), where \( \left\lfloor \ \ \right\rfloor\) is a floor function. Suppose that there exists \(p\) such that \(g_{k}^{p-1}(0) < n(k-1) \leq g_{k}^{p}(0)...

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Bibliographic Details
Published inarXiv.org
Main Authors Takahashi, Shoei, Manabe, Hikaru, Miyadera, Ryohei
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.04.2024
Subjects
Algorithms
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Summary:In this study, we study a Josephus problem algorithm. Let \(n,k\) be positive integers and \(g_k(n) = \left\lfloor \frac{n}{k-1} \right\rfloor +1\), where \( \left\lfloor \ \ \right\rfloor\) is a floor function. Suppose that there exists \(p\) such that \(g_{k}^{p-1}(0) < n(k-1) \leq g_{k}^{p}(0)\), where \(g_{k}^p\) is the \(p\)-th functional power of \(g_k\). Then, the last number that remains is \(nk-h2_{k}^{p}(0)\) 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 \(f_k(n)=\left\lfloor \frac{n}{k} \right\rfloor\). Using the present article's result, we can build a new algorithm for Josephus problem.
ISSN:2331-8422