An international basketball tournament will be held in either Nation QN or Nation RN. Exactly six nations, including the host,...

OWNING THE DATASET

Visual Representation

Nations Pool: QN, RN, SN, TN, UN, VN, WN (Host nation to be determined)

Conditions:

1. \(\mathrm{SN} \rightarrow \mathrm{TN}\) (If SN participates, then TN must participate)
2. \((\mathrm{VN} \rightarrow \neg\mathrm{UN}) \land (\mathrm{WN} \rightarrow \neg\mathrm{UN})\) (If VN or WN participates, UN cannot)
3. \(\mathrm{WN} \rightarrow \mathrm{RN}\) as host (WN participates only if RN hosts)

Given: WN participates

Immediate Inferences

From WN participating:

• Condition 3: \(\mathrm{WN} \rightarrow \mathrm{RN}\), so tournament must be in RN
• Condition 2: \(\mathrm{WN} \rightarrow \neg\mathrm{UN}\), so UN cannot participate

UNDERSTANDING THE QUESTION

Task Analysis

• Select one nation that must participate
• Select one nation that will not participate
• Given constraint: WN participates

PROCESSING THE SOLUTION

Critical Insight

With WN participating, we immediately know:

1. RN hosts (from condition 3)
2. UN cannot participate (from condition 2)

Strategic Reasoning

Current Status

• Participating: WN (given), RN (host)
• Not participating: UN
• Remaining nations: QN, SN, TN, VN

The Key Deduction

We need exactly 6 nations total:

• 2 already participating (WN, RN)
• 1 definitely not participating (UN)
• 4 spots remaining for exactly 4 remaining nations

Therefore: ALL remaining nations (QN, SN, TN, VN) MUST participate to reach exactly 6.

Verification

Final configuration:

• Participating: WN, RN, QN, SN, TN, VN (6 nations)
• Not participating: UN

This satisfies all conditions:

• check SN participates with TN (Condition 1)
• check UN doesn't participate when WN does (Condition 2)
• check Tournament in RN with WN participating (Condition 3)
• check Exactly 6 nations participate

FINAL SOLUTION SYNTHESIS

Solution Path

1. \(\mathrm{WN}\) participates → \(\mathrm{RN}\) must host
2. \(\mathrm{WN}\) participates → \(\mathrm{UN}\) cannot participate
3. Need exactly 6 nations → All remaining must participate

Final Answer

• Column 1 (Must participate): Any of QN, RN, SN, TN, or VN
• Column 2 (Will not participate): UN

Key Insights

• The "exactly 6 nations" constraint combined with elimination of UN creates a forcing function
• Once we know 2 participate and 1 doesn't, the math dictates all others must participate
• This is a counting problem disguised as a logic puzzle

Exam Strategy

• Look for numerical constraints ("exactly 6") that can force unique solutions
• Chain implications quickly: given fact → immediate consequences → count remaining spots
• When spots equal remaining options, no testing needed—all must fill those spots