Scheduling Five Interviews in Five Slots
A five-member recruitment panel at a Gurgaon firm must interview five shortlisted candidates - P, Q, R, S and T - in five one-hour slots on the same day: 9:00, 10:00, 11:00, 12:00 noon and 1:00 p.m. One candidate per slot; no slot is left empty. After a scheduling clash in the previous round, the HR head has fixed four conditions that cannot be altered: Q must be interviewed immediately after S; T must be interviewed before R; R must not take the first or the last slot; and P must be interviewed after R. Two coordinators have made competing claims. The first says that placing Q at 11:00 settles the entire day. The second wants to place P at 11:00 and has been told by a colleague that this is impossible. Neither claim has been tested against the conditions, and the panel will not relax any of them to accommodate a preference.
1. If Q is interviewed at 11:00, who is interviewed at 12:00 noon? Show the deduction step by step.
Read the worked answer
Q at 11:00 is slot 3. Because Q comes immediately after S, S must occupy slot 2 (10:00).
Slots 1, 4 and 5 remain for T, R and P, and the conditions on them are T before R, and P after R.
R cannot take slot 5 (last slot is barred) and cannot take slot 1, so R must take slot 4. Slot 4 is 12:00 noon.
Check the remainder: T must be before R, and the only free slot before slot 4 is slot 1, so T is at 9:00; P is then left at slot 5 (1:00 p.m.), which satisfies P after R.
Full order: T, S, Q, R, P. The answer to the question is R, at 12:00 noon, and the schedule is forced, not merely one option among several.
2. If S is interviewed at 9:00, write out the complete order of interviews and say whether any other order is possible.
Read the worked answer
S at 9:00 is slot 1. Q must follow immediately, so Q is at slot 2 (10:00).
Slots 3, 4 and 5 remain for T, R and P. R cannot be in slot 5 (last slot barred) and cannot be in slot 1 (already occupied), so R is in slot 3 or slot 4.
If R were in slot 3, T would need a slot before R, but slots 1 and 2 are already taken. So R cannot be in slot 3; R must be in slot 4 (12:00 noon).
T must then take the only remaining slot before R, namely slot 3 (11:00), leaving P at slot 5 (1:00 p.m.), which satisfies P after R.
Order: S (9:00), Q (10:00), T (11:00), R (12:00), P (1:00 p.m.). It is the only order compatible with the conditions, so no alternative exists.
3. A coordinator claims that P can never be interviewed at 11:00. Test the claim and state all the positions P can occupy.
Read the worked answer
The claim is false. A single valid counterexample defeats it: T (9:00), R (10:00), P (11:00), S (12:00), Q (1:00 p.m.).
Check it condition by condition: Q immediately follows S (slots 4 and 5) ✓; T is before R (1 before 2) ✓; R is neither first nor last (slot 2) ✓; P is after R (3 after 2) ✓. All four conditions hold, so P at 11:00 is feasible.
Enumerating the four schedules consistent with the conditions gives: S,Q,T,R,P; T,S,Q,R,P; T,R,S,Q,P; and T,R,P,S,Q. P sits in slot 5 in the first three and in slot 3 in the last. So P can be interviewed only at 11:00 or 1:00 p.m. - never at 9:00, 10:00 or 12:00 noon.
The error in the claim is the assumption that R's constraints transfer to P; they do not, because P is constrained only to come after R.