Laboratory Access Logic
Four engineers-Tan, Lim, Raj, and Wong-must each be assigned to exactly one of four laboratories: Alpha, Bravo, Charlie, and Delta. Each laboratory receives exactly one engineer. The security officer imposes these rules: (1) Tan cannot be assigned to Alpha or Delta. (2) Lim is assigned to Bravo if and only if Raj is assigned to Charlie. (3) Wong is assigned to Delta only if Tan is assigned to Bravo. (4) Raj is not assigned to Delta. (5) Charlie must be assigned to either Lim or Wong. The roster clerk claims there are several valid assignments and that no engineer's laboratory is fixed. You must test that claim, determine every assignment that satisfies all five rules, and identify any assignment that is forced. No other constraints apply.
1. Determine every valid assignment and state which engineer-laboratory placements are forced.
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Tan cannot be in Alpha or Delta, so Tan is in Bravo or Charlie. If Tan were in Charlie, rule 5 would fail because Charlie must go to Lim or Wong, not Tan. Therefore Tan must be in Bravo. Rule 2 says Lim is in Bravo iff Raj is in Charlie. Bravo is occupied by Tan, so Lim is not in Bravo, hence Raj is not in Charlie. Raj also cannot be in Delta by rule 4, and Bravo is taken, so Raj must be in Alpha. That leaves Lim and Wong for Charlie and Delta. Rule 3 is satisfied either way because its condition 'Tan in Bravo' is true. The two valid assignments are: Tan–Bravo, Raj–Alpha, Lim–Charlie, Wong–Delta; and Tan–Bravo, Raj–Alpha, Lim–Delta, Wong–Charlie. The forced placements are Tan in Bravo and Raj in Alpha.
2. Is the clerk correct that no engineer's laboratory is fixed? Justify your answer.
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No. The deduction shows that Tan must be in Bravo and Raj must be in Alpha in every valid assignment. The clerk has probably noticed that Lim and Wong can swap between Charlie and Delta, producing two solutions, and inferred that nothing is fixed. But variability in part of the roster does not mean variability everywhere. A logical search should test each placement against all rules. Tan in Charlie is eliminated by rule 5, and Tan in Alpha or Delta is eliminated directly by rule 1. Once Tan is fixed in Bravo, rule 2 and rule 4 force Raj to Alpha. So the clerk's claim is false: two of the four placements are fixed, and only the Lim–Wong pair is interchangeable.
3. Which single rule is essential to the conclusion that Tan must be in Bravo? Demonstrate by removing that rule.
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Rule 5 is essential. Without rule 5, Tan could be in Charlie. Suppose Tan is in Charlie. Rule 2 then forces Lim not to be in Bravo, because Raj cannot be in Charlie (Charlie is taken by Tan). Raj cannot be in Delta, so Raj is in Alpha or Bravo. If Raj is in Alpha, Lim must be in Delta and Wong in Bravo, which satisfies rule 3 because Wong is not in Delta. If Raj is in Bravo, then Wong cannot be in Delta by rule 3, so Wong is in Alpha and Lim in Delta. Both arrangements satisfy rules 1–4 without rule 5. Thus rule 5 is the constraint that eliminates Tan–Charlie and forces Tan–Bravo.