A research team maps 7 lab and field sites connected by roads. To analyze travel efficiency, they calculate the total number of unique two-site routes (visiting one site, then another) between consecutive locations. How many such two-site connections are possible if sites are arranged linearly? - Coaching Toolbox

Understanding the Context

The sites are arranged in a straight line: Site 1—Site 2—Site 3—Site 4—Site 5—Site 6—Site 7. Travel between consecutive locations forms the basis for efficient routing. The core challenge is determining how many unique two-site routes are possible when visiting one site and then proceeding to another, only visiting adjacent sequenced locations.

Calculating Unique Two-Site Routes

With sites linearly connected—meaning Travel from Site i to Site j is only defined if j = i+1—the problem simplifies: each site connects directly to its immediate neighbor in both directions. However, since the direction matters in route analysis (visiting Site A then B ≠ Site B then Site A), we count undirected adjacent pairs compounded by directional movement.

Because each site links to the next, and movement is restricted to immediate neighbors:

Key Insights

• There are 6 connections between consecutive sites:
  Site 1 → Site 2
  Site 2 → Site 3
  Site 3 → Site 4
  Site 4 → Site 5
  Site 5 → Site 6
  Site 6 → Site 7

Each of these 6 one-way adjacent segments defines a unique two-site route: visiting one, then moving to the next. Since the study focuses on radiation-safety personnel and equipment transport requiring precise routing, analyzing directional efficiency matters.

But note: A two-site route means proceeding from one specified site to the next in sequence—only the immediate next. Therefore, for any ordered pair of consecutive sites (i, i+1), there is one unique directional route.

Since only 6 such consecutive site pairs exist in the linear chain, and each generates one direct, efficient two-site connection, the total number of unique two-site routes between consecutive locations is:

6 unique two-site routes

Final Thoughts

This metric, foundational in modeling travel time, fuel consumption, and staff deployment planning, enhances logistical precision across the research network.

Practical Implications

By mapping these linear site connections, the research team establishes a foundational model for optimizing travel efficiency in distributed scientific operations. The 6 directional two-site routes enable real-time tracking of movement patterns and inform decisions on staffing coverage, transport scheduling, and safety protocol deployment.

Conclusion

In this on-site study of linear lab connectivity, the research team identifies 6 unique two-site routes—each representing a direct, adjacent transition between sequential locations. This elegant spatial analysis underscores how simple geometric arrangements can yield powerful insights for operational efficiency in large-scale, multi-site scientific environments.