Question
If a doughnut shaped house has two doors to the outside and three doors to the inner courtyard, then it's possible to end up back at your starting place by walking through all five doors of the house without ever walking through the same door twice.
True
False
True
False
Answers
GPT-4o mini
To determine if it's possible to walk through all five doors (two to the outside and three to the inner courtyard) without repeating any, we can represent this as a graph problem, where each door is a vertex and each connection between doors is an edge.
In this case, we start with two sets of doors:
1. **Outside Doors**: A and B
2. **Inner Courtyard Doors**: C, D, and E
The structure can be described as follows:
- Doors A and B connect to the outer world.
- Doors C, D, and E connect to each other and to the outside doors.
To ensure that you can create a path that visits all five doors exactly once (an Eulerian path), the main criteria are:
- The graph either has all vertices of even degree, or exactly two vertices of odd degree.
### Degree of each Door:
- Doors A and B (outside doors) have a degree of 1 (odd).
- Doors C, D, and E (inner doors) have a degree of 2 (even).
### Conclusion:
- There are exactly two doors that have an odd degree (A and B).
- Therefore, it is possible to walk through all doors without retracing any steps, starting at one of the odd-degree doors (either A or B) and finishing at the other (if necessary).
So the statement is **True**: it is possible to walk through all five doors of the house without walking through the same door twice, ending up back at your starting place.
In this case, we start with two sets of doors:
1. **Outside Doors**: A and B
2. **Inner Courtyard Doors**: C, D, and E
The structure can be described as follows:
- Doors A and B connect to the outer world.
- Doors C, D, and E connect to each other and to the outside doors.
To ensure that you can create a path that visits all five doors exactly once (an Eulerian path), the main criteria are:
- The graph either has all vertices of even degree, or exactly two vertices of odd degree.
### Degree of each Door:
- Doors A and B (outside doors) have a degree of 1 (odd).
- Doors C, D, and E (inner doors) have a degree of 2 (even).
### Conclusion:
- There are exactly two doors that have an odd degree (A and B).
- Therefore, it is possible to walk through all doors without retracing any steps, starting at one of the odd-degree doors (either A or B) and finishing at the other (if necessary).
So the statement is **True**: it is possible to walk through all five doors of the house without walking through the same door twice, ending up back at your starting place.