Let's solve the problem step by step using the information and the image: --- **Step 1: Draw the Graph** From the table and the graph in the image, each hike is represented by a vertex (with labels like L, T, G, etc.). Edges connect hikes (vertices) that overlap, based on this table: - Overlap 1: L and T - Overlap 2: L and G - Overlap 3: T and G - Overlap 4: T and P - Overlap 5: P and S - Overlap 6: G and S So, draw a vertex for each hike (L, T, G, P, S) and connect them according to the overlaps. **Explanation:** Each edge means two hikes can't have the same guide. This is a classic *graph coloring* problem. --- **Step 2: Color the Graph** Color the graph so that no two adjacent vertices (connected by an edge) have the same color. We want the *fewest colors* (guides). Let's try to color: - Start with L. Color it **Red**. - T is connected to L, so use a different color: **Blue**. - G is connected to L and T, so needs a new color: **Green**. - P is connected to T, but not L or G, so can be **Red**. - S is connected to P and G, so can't be Red or Green. Needs **Blue**. So, the coloring could be: - L: Red - T: Blue - G: Green - P: Red - S: Blue **Explanation:** No two connected vertices share a color, and we used only **3 colors**. --- **Final Answer:** **The smallest number of guides needed is 3.** This is because the graph can be properly colored with just 3 colors, meaning 3 guides are sufficient so that no guide leads overlapping hikes.