The table shows the Six members of a certain chess club, each of whom will play against exactly one other...

Owning the Dataset

Let's start by understanding our chess pairing preferences table with the goal of "owning the dataset" completely.

This table shows six chess players (Priyanka, Raj, Sneha, Dhruv, Aditi, and Vijay) and their ranked preferences for playing partners. Each player has ranked the other five players from 1 (most preferred) to 5 (least preferred).

Key insights from our dataset:

• We have exactly 6 players who must form exactly 3 pairs
• Each player has expressed preferences ranking all other players
• Dhruv has a tie in his preferences (Aditi and Sneha are both ranked 1st)
• Dhruv also has a tie for his 2nd choices (Priyanka and Raj)
• Aditi has a significant preference gap - she strongly prefers Vijay (1st) with her next preference dropping to 3rd

These observations will be crucial for our efficient solving approach, especially when we need to check constraints against specific pairings.

Question Translation

Original: "If the following conditions must be satisfied:

1. Priyanka must play with Raj
2. Sneha must play with Dhruv
3. No player can play with their first choice, and in Dhruv's case, he cannot play with either his first or second choices. Similarly, Raj cannot play with his first choice.

Can these conditions be satisfied?"

What we're looking for:

• Whether three specific constraints can be met simultaneously
• Whether the required pairings (Priyanka-Raj and Sneha-Dhruv) are possible given the constraints
• By extension, whether the third pairing (Aditi-Vijay) would be possible

In other words: We need to determine if the preference constraints prevent any of our required pairings from happening.

Analyzing the Constraints

Rather than checking every possibility, let's use a powerful pattern recognition insight:

Key Insight: When 6 players must form 3 pairs, fixing 2 pairs automatically determines the 3rd pair!

If Priyanka must play with Raj, and Sneha must play with Dhruv, then by simple elimination, Aditi MUST play with Vijay. This transforms our approach completely - we just need to check if any constraints directly prevent these specific pairings.

Let's analyze each constraint directly against our required pairings:

Constraint Analysis for Dhruv

Statement Translation:
Original: "Dhruv cannot play with either his first or second choices"
What we're looking for:

• Who are Dhruv's first and second choices?
• Is Sneha (his required partner) among these choices?

According to our data, Dhruv's preferences show:

• 1st choices (tied): Aditi and Sneha
• 2nd choices (tied): Priyanka and Raj

Direct Contradiction Found: Sneha is one of Dhruv's 1st choices, but the constraint says he cannot play with his 1st choices. This directly prevents the required Dhruv-Sneha pairing.

Constraint Analysis for Raj

Statement Translation:
Original: "Raj cannot play with his first choice"
What we're looking for:

• Who is Raj's first choice?
• Is Priyanka (his required partner) his first choice?

According to our data:

• Raj's 1st choice: Priyanka

Direct Contradiction Found: Priyanka is Raj's 1st choice, but the constraint says he cannot play with his 1st choice. This directly prevents the required Raj-Priyanka pairing.

Constraint Analysis for Aditi

Statement Translation:
Original: "No player can play with their first choice"
What we're looking for:

• Who is Aditi's first choice?
• Is Vijay (her forced partner) her first choice?

According to our data:

• Aditi's 1st choice: Vijay

Direct Contradiction Found: Vijay is Aditi's 1st choice, but the constraint says no player can play with their 1st choice. This prevents the Aditi-Vijay pairing that would be forced by the other requirements.

Final Answer Compilation

We've identified three direct contradictions:

1. Dhruv cannot play with Sneha (she's his 1st choice)
2. Raj cannot play with Priyanka (she's his 1st choice)
3. Aditi cannot play with Vijay (he's her 1st choice)

Any one of these contradictions alone would make it impossible to satisfy all conditions. Since we have three separate contradictions, we can confidently conclude that Could NOT achieve the stated goal.

Learning Summary

Skills We Used

• Structural Pattern Recognition: We immediately identified that with 6 players forming 3 pairs, fixing 2 pairs automatically determines the 3rd pair
• Direct Contradiction Testing: Instead of exploring multiple arrangements, we checked if constraints directly prevented required pairings
• Efficient Constraint Analysis: We focused only on the specific preferences relevant to our required pairings

Strategic Insights

• Start with Structure: In any matching problem, first recognize the structural constraints before diving into details
• Constraint-First Approach: When checking feasibility, look for direct contradictions first rather than trying to build a valid solution
• Forced Pairing Recognition: Identifying that some pairs force others can drastically reduce your analysis time

Efficiency Gains

• We avoided checking every possible pairing arrangement
• We skipped detailed preference analysis for players whose constraints weren't relevant
• We stopped as soon as we found our first contradiction (though we continued to demonstrate all three)

Common Mistakes We Avoided

• Getting lost in analyzing all possible arrangements when direct contradiction testing is faster
• Overlooking the structural insight that fixing two pairs forces the third pair
• Failing to translate constraints into direct tests against requirements

Remember: In GMAT Table Analysis questions, the key is often recognizing a structural insight that makes the solution nearly instant. Always check constraints directly against requirements rather than exploring all possibilities.