Is it possible to take a walk that crosses each bridge once and returns to the starting point without crossing any bridge twice? If not, can you do it if you do not end at the starting point? Explain how you know.
Is it possible to take a walk that crosses each bridge once and returns to the starting point without crossing any bridge twice? If not, can you do it if you do not end at the starting point? Explain how you know.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:**Problem Statement:**
Is it possible to take a walk that crosses each bridge once and returns to the starting point without crossing any bridge twice? If not, can you do it if you do not end at the starting point? Explain how you know.
**Diagrams:**
The image depicts a simplified version of the Seven Bridges of Königsberg problem. It shows a river with two landmasses and seven bridges connecting them. The arrangement is as follows:
- Three bridges connect the left landmass to the right landmass.
- Two bridges connect the top of the left landmass to the middle area.
- Two bridges connect the right landmass to the middle area.
**Analysis:**
In the context of graph theory, each landmass can be regarded as a vertex and each bridge as an edge. The question aligns with Euler's study on graph theory regarding Eulerian circuits and paths.
To solve this:
- An Eulerian circuit (path beginning and ending at the same vertex) exists if and only if every vertex in the graph has an even degree (each vertex connects to an even number of edges).
- An Eulerian path (starts and ends at different vertices) requires exactly two vertices of odd degree.
**Conclusion:**
- In this problem, each landmass is connected by an odd number of bridges. Thus, all vertices have an odd degree.
- Therefore, it is not possible to start at one point, cross each bridge once, and return to the starting point (no Eulerian circuit).
- However, since all vertices have an odd degree, it is neither possible to have an Eulerian path, where you might start and end at different vertices.
This problem highlights the mathematical principles behind Euler's work on solving the Königsberg bridge problem, demonstrating the fundamental aspects of graph theory.
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