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list and mark a possible route that the race can follow startining and finishing at lake bolac, with only one road being travelled twice. (eulerian circult)

To solve the problem of finding a possible route that starts and ends at Lake Bolac while only traveling one road twice (an Eulerian Circuit), we need to look at the properties of the graph formed by the towns and the roads connecting them.

### Steps to Solve:
1. **Identify Degrees of Each Node**: The nodes are the towns, and the degree is the number of roads connected to each town. 
2. **Check for Eulerian Circuit**: An Eulerian Circuit exists in a graph if every vertex has an even degree. However, since we are allowed to travel one road twice, we can find a circuit with exactly two vertices of odd degree.

### Graph Analysis:
Let's calculate the degree of each town:

- **Lake Bolac**: Connected to Wickliffe, Woorndoo, and Stratham. Degree = 3 (odd)
- **Wickliffe**: Connected to Lake Bolac, Chatsworth, and Glenthompson. Degree = 3 (odd)
- **Glenthompson**: Connected to Wickliffe, Chatsworth, and Stratham. Degree = 3 (odd)
- **Stratham**: Connected to Glenthompson, Lake Bolac, and Nerrin Nerrin. Degree = 3 (odd)
- **Nerrin Nerrin**: Connected to Stratham and Woorndoo. Degree = 2 (even)
- **Woorndoo**: Connected to Nerrin Nerrin, Lake Bolac, and Mortlake. Degree = 3 (odd)
- **Mortlake**: Connected to Woorndoo, Hexham, and Chatsworth. Degree = 3 (odd)
- **Hexham**: Connected to Chatsworth and Mortlake. Degree = 2 (even)
- **Chatsworth**: Connected to Hexham, Mortlake, Wickliffe, and Glenthompson. Degree = 4 (even)

### Route Construction:
Since Lake Bolac and one other town should have an odd degree and be used twice, we have the following:

1. **Start at Lake Bolac.**
2. **Travel to Wickliffe (11 km).**
3. **Wickliffe to Chatsworth (21 km).**
4. **Chatsworth to Glenthompson (18 km).**
5. **Glenthompson to Stratham (36 km).**
6. **Stratham to Nerrin Nerrin (24 km).**
7. **Nerrin Nerrin to Woorndoo (23 km).**
8. **Woorndoo to Mortlake (23 km).**
9. **Mortlake to Hexham (18 km).**
10. **Hexham to Chatsworth (16 km).**
11. **Chatsworth to Wickliffe (21 km).** (This road will be traveled twice.)
12. **Return to Lake Bolac (11 km).**

This route travels every road exactly once except for the road between Wickliffe and Chatsworth, which is traveled twice.

If you need further explanation or have more questions, feel free to ask! Here are five follow-up questions:

1. What are the criteria for a graph to have an Eulerian Circuit?
2. How would the route change if no road could be traveled twice?
3. What is the total distance of the proposed route?
4. Can we simplify the graph to identify circuits more easily?
5. How do odd and even degrees of vertices affect possible routes in such problems?

**Tip**: In graph theory problems, checking the degree of vertices is a crucial step in determining the possibility of Eulerian paths and circuits.

Explore how to find an Eulerian Circuit starting and ending at Lake Bolac while traveling each road exactly once, except one which is traveled twice. Learn about graph theory concepts like Eulerian Circuits and vertex degrees.

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