Maximum Nim and Josephus Problem algorithm
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913584193208320 |
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| author | Takahashi, Shoei Manabe, Hikaru Miyadera, Ryohei |
| author_facet | Takahashi, Shoei Manabe, Hikaru Miyadera, Ryohei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06112 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximum Nim and Josephus Problem algorithm Takahashi, Shoei Manabe, Hikaru Miyadera, Ryohei Combinatorics 91A46, 91A05 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. |
| title | Maximum Nim and Josephus Problem algorithm |
| topic | Combinatorics 91A46, 91A05 |
| url | https://arxiv.org/abs/2404.06112 |