冷卻骨牌遊戲之組合對局值

Abstract

組合對局理論是一種利用數學方法分析對局遊戲的理論。此理論以對局樹表示遊戲盤面狀態，使遊戲盤面的結果可以對局數值表示，並且由對局值推導盤面優勢及勝敗結果。 傳統骨牌遊戲是一個適用於組合對局理論範疇的對局遊戲，經由研究骨牌遊戲，可以得知此遊戲的許多盤面勝負結果以及對局值。 2010年由高國元等人提出了冷卻骨牌遊戲，它改變了傳統骨牌遊戲的規則，複雜度較高。高國元等人分析了所有3x3大小的冷卻骨牌遊戲盤面，提出一線性公式簡化並計算3x3盤面對局值。 這篇研究的目的是實作分析冷卻骨牌遊戲盤面的程式，驗證3x3大小的冷卻骨牌遊戲對局值。進一步計算 4x4大小盤面的對局值，分析結果，並加以分類，嘗詴找出簡化並計算4x4盤面結果的方法。

Combinatorial Game theory uses mathematical theory to study computer games. The theory describes a game position with a game tree and makes a specific value stand for the result of a game. This game value helps us to analyze the advantage of both players and find the optimum moves. Combinatorial game theory can be applied to Domineering. By studying the game with combinatorial game theory, we know many results of different Domineering game positions. Chilled Domineering is a game invented by Kao et al. in 2010. The rule of chilled domineering is different from domineering and its complexity is higher than domineering. They analyzed all the 3x3 positions of chilled domineering and provided a linear formula to simplify and calculate the position value. The main purpose of this research is to implement a program to analyze Chilled Domineering, generate all 4x4 positions, and calculate the values. Moreover, the research analyzes and categorizes these values and tries to find a way to determine the winner of the position.