Question

Imagine a pentagon shaped building. The walls of the building are sides of the pentagon, and there is a watchtower at each vertex. Each watchtower can have guards who can watch two walls connected to the watchtower. What is the minimum number of guards required if each wall needs to be watched by at least 3 guards?

Hint:

When distributing guards or resources across a shared system, it’s essential to calculate the shared responsibility at each intersection or node for optimal efficiency.

Answer 1

The pentagon has 5 walls and 5 vertices. Each wall must be watched by at least 3 guards. Since each guard stationed at a watchtower can watch two walls connected to that vertex, the distribution of guards can be optimized as follows:
1. Distribute guards at the vertices so that each wall shared between two vertices is covered by 3 guards.
2. By carefully assigning guards, it is possible to achieve the coverage of 3 guards per wall with a total of 8 guards distributed across the vertices.
Conclusion: The minimum number of guards required is 8.