Jorge travels to meet once daily with exactly six clients, Clients A through F. When scheduling these meetings, Jorge schedules...

OWNING THE DATASET

Let's understand what we're working with in this distance matrix. The table shows the distances between 6 different client locations (E, A, F, B, C, and D), with distances measured in units.

When working with distance matrices, our first priority should be looking for the extreme values - both the very small and very large distances:

• Shortest distances: \(\mathrm{A \rightarrow F}\) (1 unit), \(\mathrm{E \rightarrow A}\) (2 units), and \(\mathrm{B \rightarrow C}\) (2 units)
• Longest distances: Any involving point D (12+ units)

Key insight: These extreme values create natural constraints on our path. Since we start at E and always move to the nearest unvisited client, the beginning of our path is essentially fixed by these short distances.

This strategic scan for extreme values helps us avoid calculating the entire path unnecessarily. Let's leverage these insights as we analyze the statements.

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: "Client D was visited before client A."
What we're looking for:

• Whether D comes before A in our optimal path
• We need to determine the relative positions of D and A in the sequence

In other words: Does D appear earlier in our path than A?

Let's approach this statement first because it might be immediately answerable with minimal calculation.

Since we start at E, we need to determine where we go first. Looking at the distances from E:

• \(\mathrm{E \rightarrow A}\) is only 2 units (one of the shortest distances in the entire matrix)
• All other distances from E must be larger

This means A must be the second client visited, immediately after E. Since A comes so early in our path (position 2), the only way D could come before A would be if D was the very first client - but that's impossible since we start at E.

Therefore, D cannot possibly come before A in our path.

Statement 3 is No.

Notice how we didn't need to map the entire path to answer this question - we just needed to recognize that A comes extremely early in the sequence due to its close proximity to our starting point.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: "Client B was visited before client C."
What we're looking for:

• Whether B comes before C in our optimal path
• We need to determine the relative positions of B and C in the sequence

In other words: Does B appear earlier in our path than C?

Now that we've established A is the second client visited, let's continue building our path. We know:

1. We start at E
2. We visit A next (since \(\mathrm{E \rightarrow A = 2}\) units)

From A, the shortest distance is to F (\(\mathrm{A \rightarrow F = 1}\) unit, the shortest in the entire matrix). So F must be our third client.

Now we're at F and need to decide where to go next. Our remaining unvisited clients are B, C, and D. The distances are:

• \(\mathrm{F \rightarrow B = 4}\) units
• \(\mathrm{F \rightarrow C = 6}\) units
• \(\mathrm{F \rightarrow D = 12}\) units

The shortest is \(\mathrm{F \rightarrow B}\), so B becomes our fourth client.

From B, we have two clients left: C and D. The distances are:

• \(\mathrm{B \rightarrow C = 2}\) units
• \(\mathrm{B \rightarrow D = 15}\) units

The shorter distance is \(\mathrm{B \rightarrow C}\), so C is our fifth client, and D is our sixth.

Our complete path is: \(\mathrm{E \rightarrow A \rightarrow F \rightarrow B \rightarrow C \rightarrow D}\)

Since B (4th) comes before C (5th) in our path, Statement 1 is Yes.

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: "Client C was visited before client D."
What we're looking for:

• Whether C comes before D in our optimal path
• We need to determine the relative positions of C and D in the sequence

In other words: Does C appear earlier in our path than D?

We've already mapped out our entire path in the previous statement: \(\mathrm{E \rightarrow A \rightarrow F \rightarrow B \rightarrow C \rightarrow D}\)

Looking at this path, we can see that C is the fifth client visited, while D is the sixth and last client visited. Since C comes before D in our sequence, Statement 2 is Yes.

FINAL ANSWER COMPILATION

Let's summarize our findings:

• Statement 1: "Client B was visited before client C." - Yes
• Statement 2: "Client C was visited before client D." - Yes
• Statement 3: "Client D was visited before client A." - No

The answer pattern is: Yes, Yes, No

LEARNING SUMMARY

Skills We Used

• Extreme Value Recognition: We immediately identified the shortest distances in the matrix (\(\mathrm{A \rightarrow F = 1}\), \(\mathrm{E \rightarrow A = 2}\), \(\mathrm{B \rightarrow C = 2}\)), which gave us critical path constraints.
• Partial Path Analysis: We answered Statement 3 with only partial information about our path.
• Strategic Statement Order: We tackled statements in order of solvability rather than numerical order.

Strategic Insights

1. Look for Constraints First: In path problems, extreme values (very short or very long distances) create natural constraints that simplify the problem.
2. Answer What You Can Immediately: Statement 3 could be answered almost instantly once we recognized A must be visited very early.
3. Build Only What You Need: We only constructed the full path when necessary for the remaining statements.

Common Mistakes We Avoided

• Unnecessary Calculation: We didn't blindly calculate the entire path before looking at the statements.
• Rigid Statement Order: We didn't force ourselves to analyze statements in numerical order.
• Overlooking Key Patterns: We didn't miss the significance of the extremely short distances, which essentially locked in the beginning of our path.

Remember that in distance-based problems, the extreme values (shortest and longest) often provide the most powerful insights. This approach of identifying constraints first, then building the solution strategically, works on many optimization problems beyond just path finding.