欧几里得游戏是基于辗转相除法(欧几里得算法)的数学游戏，其背后数学原理丰富又兼具趣味性，是数学游戏化教学的良好资源。辗转相除法相关内容作为数论的基础无论是在数学还是其他领域都占据着重要地位，但其在初中教学中并未受到足够的重视，欧几里得游戏则正是将辗转相除法引入初中数学教学的良好载体。本研究主要由以下两部分内容构成：

第一部分采用文献分析法详细梳理了欧几里得游戏背后的数学理论，分析了游戏的获胜位置、必胜策略、游戏的长度、游戏相关性质、游戏与连分数之间的关联以及游戏的推广等，补充完善了基于连分数的游戏必胜策略和游戏长度定理的证明。

第二部分以辗转相除法为切入点，将欧几里得游戏应用在初中数学教学中。在分析了将辗转相除法引入初中教学的重要意义后，将“辗转相除法求最大公因数”作为初中一年级的拓展课进行了游戏化的教学设计。最后，将教学设计加以实践，设计教学实验以探究游戏化教学方式对学生数学学习兴趣、课堂表现情况及对课堂学习内容掌握程度的影响。实验分为实验班与对照班，实验班的教学中引入游戏，而对照班则进行常规模式教学，利用问卷调查等方法反馈实验结果。通过对两班前后测问卷和后测试卷成绩等数据的分析对比，得到了游戏化教学不仅可以提升学生数学学习兴趣、提高学生课堂参与度与积极性，还能够促进学生对课堂知识的掌握的结论。

The Euclidean game is a mathematical game based on the Euclidean algorithm. Its underlying mathematical principles are rich and engaging, making it a valuable resource for gamifying mathematics education. As a fundamental concept in number theory, the vital role that Euclidean algorithm plays is not restricted in mathematics. However, teachers and scholars have not paid sufficient attention to its significance in middle school education. The Euclidean game serves uniquely as an effective medium for introducing the Euclidean algorithm into secondary school mathematics education.

This study comprises two parts. The first part entails a detailed analysis of the mathematical theory of the Euclidean game. This thesis will analyze the winning positions, strategies, lengths, and properties of the game, as well as its connection with continued fractions and its generalization. Furthermore, this article will provide a proof for the game winning strategy and the game length theorem by utilizing continued factions.

The second part focuses on the application of the Euclidean game in secondary school mathematics education, introducing the Euclidean algorithm as a starting point. After analyzing the significance of introducing the Euclidean algorithm in middle school, this thesis proposes a gamified teaching design that can be carried out as an extension course “Calculating the Greatest Common Divisor by Euclidean Algorithm” for the first-year middle-schoolers. Finally, this project put the teaching design into practice to explore the influence of gamification teaching methods on students' interest in mathematics learning, classroom performance and mastery of classroom learning content. Subjects of the teaching experiment were divided into an experimental class and a control class. In the experimental class, students were exposed to the gamified teaching design, as opposed to the conventional teaching design used in the control class. The results of the experiment are collected through questionnaires. Through the analysis and comparison of the data of the pre- and post-test questionnaires and post-test exam papers in the two classes, the thesis draws the conclusion that gamification teaching can not only improve students' interest in mathematics learning, increase students' participation and enthusiasm in class, but also promote students' mastery of classroom knowledge.