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Maximum Nim and Josephus Problem

Shoei Takahashi, Hikaru Manabe, Ryohei Miyadera

Abstract

In this study, we study the relation between Grundy numbers of a Maximum Nim and Josephus problem. Let f(x) = floor(x/k), where floor( ) is the floor function and k is a positive integer. We prove that there is a simple relation with a Maximum Nim with the rule function f and the Josephus problem in which every k-th numbers are to be removed.

Maximum Nim and Josephus Problem

Abstract

In this study, we study the relation between Grundy numbers of a Maximum Nim and Josephus problem. Let f(x) = floor(x/k), where floor( ) is the floor function and k is a positive integer. We prove that there is a simple relation with a Maximum Nim with the rule function f and the Josephus problem in which every k-th numbers are to be removed.
Paper Structure (3 sections, 4 theorems, 20 equations, 3 figures)

This paper contains 3 sections, 4 theorems, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Maximum Nim
  3. Maximum Nim with the rule function $f(x)=\left\lfloor \frac{x}{k} \right\rfloor$ for $k \in \mathbb{N}$ and Josephus Problem.

Key Result

Theorem 2.2

For any position $(x)$, $\mathcal{G}(x)=0$ if and only if $(x)$ is the $\mathcal{P}$-position.

Figures (3)

  • Figure 1:
  • Figure 3:
  • Figure 5:

Theorems & Definitions (9)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Example 3.3
  • Theorem 3.4
  • proof