Placement Week Scheduling Under Five Frozen Conditions
Six recruiters - P, Q, R, S, T and U - will visit a Kolkata B-school during placement week, Monday to Saturday, exactly one recruiter per day. The placement committee has frozen five conditions after negotiation with the student council: U visits on Saturday; R visits on Wednesday; Q visits on the day immediately before S; T does not visit on Monday; and P visits before R. The committee secretary has already printed a draft calendar showing Q on Thursday, but the faculty coordinator refuses to sign until the conditions are formally checked. Three further requests arrive the same afternoon. The training and placement officer wants the entire week compressed to Monday–Friday with U still on Friday and all other conditions unchanged. Separately, one recruiter may withdraw if its hiring freeze continues. The student council also asks whether the condition "T does not visit on Saturday" should be added for the sake of clarity.
1. Derive the day on which each of the six recruiters visits, showing how the alternatives are eliminated. Is the secretary's draft date for Q correct?
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Step 1 - anchors. U = Saturday, R = Wednesday. Free days for P, Q, S, T: Monday, Tuesday, Thursday, Friday.
Step 2 - adjacency. "Q the day before S" can only use the adjacent free pairs (Mon, Tue) or (Thu, Fri).
Step 3 - test (Mon, Tue). Q = Mon, S = Tue leaves Thursday and Friday for P and T, but P must come before R (Wednesday). Both remaining days are after Wednesday, so this branch fails.
Step 4 - test (Thu, Fri). Q = Thu, S = Fri leaves Monday and Tuesday for P and T. T cannot take Monday, so T = Tue and P = Mon. Check P before R: Mon precedes Wed ✓.
Result: Mon P, Tue T, Wed R, Thu Q, Fri S, Sat U. Exactly one arrangement survives, so the schedule is fully determined and the secretary's day for Q (Thursday) is correct - though the draft still needs checking against the other four slots.
2. Should the committee add "T does not visit on Saturday"? Assess what the added condition does to the solution set and to the committee's risk.
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It does nothing to the solution set, but it is not pointless.
Logic test: U occupies Saturday and the rules allow exactly one recruiter per day, so no other recruiter - T included - can ever sit on Saturday. The added condition is therefore redundant: it is implied by "U visits on Saturday" plus the one-per-day structure. Adding it leaves the schedule Mon P, Tue T, Wed R, Thu Q, Fri S, Sat U unchanged and creates no contradiction.
Why the committee may still add it: redundancy is cheap insurance against a change of premise. If U's visit slips and Saturday is re-opened, the condition stops being vacuous and protects T's stated unavailability. The cost is that the rule set now contains a clause that can never be used to eliminate a branch, which invites the false belief that the puzzle is more constrained than it is - a candidate reasoning from it may feel justified in skipping the branch analysis in Step 3, which is precisely where the only real work lies.
Judgement: record it as a note, not as a numbered condition; keep the five operative conditions clean.
3. The TPO wants a Monday–Friday week with U still on Friday. If exactly one recruiter withdraws, which single withdrawal makes that compressed week feasible, and why do the others fail?
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First, the counting trap. Six recruiters into five days, one per day, is impossible before any logic is applied. So a withdrawal is mandatory, and the compressed week cannot be signed as it stands.
With one withdrawal: U = Fri and R = Wed, leaving Monday, Tuesday and Thursday for three of the remaining five.
Adjacency: Q immediately before S. Thursday's successor is U, so Q cannot be on Thursday. The only possible pair is Q = Mon, S = Tue. Thursday then holds whichever of P or T remains.
Test each withdrawal:
- Q or S withdraws → the adjacency condition cannot be satisfied at all. Fail.
- T withdraws → P must take Thursday, but P must visit before R (Wednesday). Fail.
- P withdraws → T takes Thursday, T ≠ Monday ✓, and "P before R" becomes vacuous because P is absent. Valid: Mon Q, Tue S, Wed R, Thu T, Fri U.
Answer: P's withdrawal. The lesson is that a condition about an absent entity silently stops constraining anything - the coordinator must confirm whether "P before R" should survive P's exit.