📚 The Revolution in Prussia | 普鲁士的数学革命：从柯尼斯堡七桥问题到图论
In 1736, Leonhard Euler solved a puzzle in Konigsberg, a Prussian city built around the river Pregel. The problem asked whether a walker could cross each of the seven bridges exactly once and return to the starting point. Euler’s answer not only settled the puzzle but also launched a silent revolution: the birth of graph theory.
1736 年，莱昂哈德·欧拉解决了普鲁士城市柯尼斯堡的一个难题。该问题问：一位散步者能否恰好经过七座桥各一次并回到起点？欧拉的回答不仅解决了这个谜题，还引发了一场静默的革命：图论的诞生。
1. The Prussian Setting | 普鲁士的历史背景
Konigsberg was the capital of East Prussia, located on both banks of the Pregel River. Two large islands, Kneiphof and Lomse, sat in the river and were connected to the banks and to each other by seven bridges. Local residents enjoyed walking through the city and began to ask whether a route could cross all seven bridges exactly once.
柯尼斯堡是东普鲁士的首府，坐落于普雷格尔河两岸。河中有两座大岛——克奈普霍夫岛和洛姆塞岛，它们与河岸以及彼此之间由七座桥相连。当地居民喜欢在城中散步，于是开始思考：是否存在一条路线，能恰好走过全部七座桥各一次？
This was not a question about distance, speed, or cost. It was a purely structural question about connection. For that reason, it resisted the usual tools of arithmetic and geometry available in the 18th century.
这不是一个关于距离、速度或成本的问题，而是一个纯粹关于连接关系的问题。因此，18 世纪已有的算术和几何工具都无法直接解决它。
2. The Seven Bridges Puzzle | 七桥问题
The puzzle has four pieces of land: the north bank, the south bank, Kneiphof island, and Lomse island. Seven bridges link these land masses. The challenge is to find a closed or open walk that traverses every bridge exactly once.
这个谜题包含四片陆地：北岸、南岸、克奈普霍夫岛和洛姆塞岛。七座桥将这些陆地连接起来。挑战是要找到一条闭合或开放的行走路线，使每座桥恰好被走过一次。
Many residents tried to draw possible routes, but no one could prove whether a solution existed. Some believed it was possible; others suspected it was not, but no rigorous argument had been given.
许多居民尝试画出可能的路线，但没有人能证明解是否存在。有些人认为可能，另一些人怀疑不可能，但始终缺乏严格的论证。
3. Euler’s Breakthrough Abstraction | 欧拉的突破性抽象
Euler’s key idea was to replace each land mass with a single point, or vertex, and each bridge with a line, or edge, connecting the relevant points. The exact shape of the land or the length of the bridge became irrelevant.
欧拉的关键想法是把每片陆地替换为一个点（顶点），把每座桥替换为一条连接相应点的线（边）。陆地的具体形状和桥的长度都变得无关紧要。
This reduction turned a geographical puzzle into a purely combinatorial one: given a collection of vertices and edges, can we trace the whole figure without lifting the pen and without repeating an edge?
这种简化将地理谜题转化为一个纯组合问题：给定一组顶点和边，能否一笔画完整幅图形，且不重复经过任何一条边？
The revolution was not the answer ‘no’, but the method: Euler recognised that only the connection pattern matters. This was the first appearance of topology and graph-theoretic thinking.
这场革命的意义不在于答案是“不能”，而在于方法：欧拉认识到只有连接模式才是关键。这是拓扑学和图论思维的最早出现。
4. Vertices, Edges and Degree | 顶点、边与度数
In modern A-Level Decision Mathematics, a graph consists of vertices (nodes) and edges (arcs). The degree of a vertex is the number of edges incident to it. In the Konigsberg graph, we count the bridges meeting each land mass.
在现代 A-Level 决策数学中，图由顶点（节点）和边（弧）组成。顶点的度数就是与该顶点相连的边的条数。在柯尼斯堡图中，我们统计每片陆地上连接的桥数。
Let the four land masses be A, B, C and D. The degrees are 3, 5, 3 and 3 respectively. Notice that the sum of degrees is 14, exactly twice the number of edges, which is a fundamental handshaking lemma.
设四片陆地为 A、B、C 和 D。它们的度数分别为 3、5、3、3。注意到度数和为 14，恰好是边数 7 的两倍，这正是基本的握手引理。
A vertex with odd degree is called an odd vertex; a vertex with even degree is an even vertex. Euler’s insight links the existence of certain walks to the number of odd vertices.
度数为奇数的顶点称为奇点；度数为偶数的顶点称为偶点。欧拉的洞察把某种行走路径的存在性与奇点的数量联系起来。
5. Euler’s Theorem for Trails and Circuits | 欧拉路径与回
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