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研究生： 李峻丞
Li, Chun-Cheng
論文名稱： 中國跳棋的性質及其相關問題之研究
Several Properties and Problems regarding Chinese Checkers
指導教授： 韓永楷
Hon, Wing-Kai
口試委員: 李哲榮
Lee, Che-Rung
蔡孟宗
Tsai, Meng-Tsung
學位類別： 碩士
Master
系所名稱： 電機資訊學院 - 資訊工程學系
Computer Science
論文出版年： 2022
畢業學年度： 110
語文別： 英文
論文頁數： 48
中文關鍵詞： 中國跳棋 、圖論 、六角座標系統 、廣度優先搜尋
外文關鍵詞： Chinese Checkers, Graph theory, Hexagonal coordinate system, Breadth-first search
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    • 本研究致力於探討數個與中國跳棋相關的問題。基於中國跳棋的規則，我們透過圖論針對六角星形棋盤進行研究，引入一個特殊的斜角坐標系統，並得到數個性質，這使我們能夠解決與中國跳棋有關的問題。我們的主要問題是計算在各種情況下將棋子從棋盤的一個角移動到對角所需的最小移動次數。其中一些問題可以通過適當的數學模型來解決，而其他問題則可以通過修改後的BFS演算法來解決。

    This study aims to explore several problems related to Chinese Checkers. Based on the rules of Chinese Checkers, the hexagram-shaped board was investigated through graph theory. A particular oblique coordinate system is introduced, and several properties are given, enabling us to address the problems concerning Chinese Checkers. Our primary issue is to calculate the minimum number of moves required to move a piece from one corner of the board to the opposite corner under various scenarios. Some of these problems are solved by an appropriate mathematical model, while others by a modified BFS algorithm.

    摘要 i Abstract ii Contents iii Chapter 1 Introduction 1 Chapter 2 Chinese Checkers 3 2.1 History 3 2.2 Rules 4 Chapter 3 Board 5 3.1 Definition of Terms 5 3.2 Properties 6 3.2.1 Number of Holes and Edges 6 3.2.2 Giant Clusters 8 3.2.3 Triangular Grid and Hexagonal Grid 9 3.3 More Settings 10 3.3.1 Starting Point and Destination 10 3.3.2 Coordinate System 10 3.4 Graph Representations 12 3.5 Movement Directions 13 Chapter 4 Minimum Number of Moves — Without Other Pieces 15 4.1 Graph Algorithms 15 4.2 Minimum Number of Moves 16 4.2.1 Distance between (0, 0) and (2n, 2n) 16 4.2.2 Number of Shortest Paths 17 Chapter 5 Minimum Number of Moves — Multiplayer Version 18 5.1 Two Players 19 5.1.1 3n-move Solution 20 5.1.2 Correctness of the 3n-move Solution 20 5.2 Three Players 21 5.2.1 Main Issue 22 5.2.2 Optimal Configuration for Three Pieces 22 5.2.2.1 Mathematical Model 22 5.2.2.2 Sequences 24 5.2.2.3 Subsequences 25 5.2.3 Minimum Number of Moves for a Three-player Game 31 5.3 Four Players 33 5.3.1 Optimal Configuration for Four Pieces 33 5.3.2 Minimum Number of Moves for a Four-player Game 36 5.3.3 Correctness of the 2n-move Solution 37 5.4 Six Players 38 Chapter 6 Minimum Number of Moves — With Fixed Pieces 39 6.1 Input and Output 39 6.2 Definition of Terms 40 6.3 Basic Concepts 42 6.4 Algorithm 43 6.4.1 Phase 1: Cluster Formation 43 6.4.2 Phase 2: Graph Traversal 43 6.5 Performance Analysis 44 Chapter 7 Discussion and Conclusions 46 Bibliography 47

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