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Jorge travels to meet once daily with exactly six clients, Clients A through F. When scheduling these meetings, Jorge schedules the most urgent meeting to be the first of the day and then schedules each subsequent meeting according to which client is the least distance from the meeting immediately prior to it. The table shows the distances between Jorge's clients. On Tuesday, Jorge's most urgent meeting will be with Client E.
| Client | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| A | 0 | 7 | 3 | 4 | 2 | 1 |
| B | 7 | 0 | 2 | 15 | 17 | 4 |
| C | 3 | 2 | 0 | 3 | 18 | 6 |
| D | 4 | 15 | 3 | 0 | 12 | 7 |
| E | 2 | 17 | 18 | 12 | 0 | 9 |
| F | 1 | 4 | 6 | 7 | 9 | 0 |
For each of the following pairs of clients, select Yes if Jorge will schedule meetings with the given clients in the given order. Otherwise, select No.
Client B then Client C
Client C then Client D
Client D then Client A
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:
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.
Statement 3 Translation:
Original: "Client D was visited before client A."
What we're looking for:
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:
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.
Statement 1 Translation:
Original: "Client B was visited before client C."
What we're looking for:
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:
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:
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:
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.
Statement 2 Translation:
Original: "Client C was visited before client D."
What we're looking for:
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.
Let's summarize our findings:
The answer pattern is: Yes, Yes, No
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.
Client B then Client C
Client C then Client D
Client D then Client A