Scheduling Five Loan Files When One Rule Is Redundant and Another Is Not
A branch has five loan files - P, Q, R, S and T - to be cleared on five consecutive working days, Monday to Friday, exactly one file per day. The branch head sets four conditions:
- Q must be cleared on Tuesday.
- R must be cleared on the working day immediately after S.
- T must not be cleared on Friday.
- P must not be cleared on a day adjacent to Q's day.
You are asked to produce the schedule. After you do, the branch head argues that condition 4 is dead weight: 'P ended up on Friday, nowhere near Tuesday, so that condition never did any work and we can drop it from the written procedure.' A colleague disagrees and offers a different candidate for deletion - condition 3 - saying that one really is doing nothing. You must produce the schedule and settle both claims with explicit reasoning rather than intuition.
1. Determine the one schedule that satisfies all four conditions.
Read the worked answer
Q is fixed on Tuesday.
R immediately follows S, and Q already occupies Tuesday, so S cannot be Monday (that would put R on Tuesday, clashing with Q) and cannot be Tuesday itself. Two placements survive on the face of it: S on Wednesday with R on Thursday, or S on Thursday with R on Friday.
Test S on Thursday, R on Friday. Monday and Wednesday remain for P and T. Q is on Tuesday, so the days adjacent to Q are Monday and Wednesday - and those are precisely the two days left. P would have to occupy one of them, which condition 4 forbids. This branch is impossible.
Test S on Wednesday, R on Thursday. Monday and Friday remain for P and T. Condition 3 bars T from Friday, so T takes Monday and P takes Friday. P on Friday is not adjacent to Tuesday, so condition 4 is satisfied.
Schedule: Monday T, Tuesday Q, Wednesday S, Thursday R, Friday P. The key move is that the impossible branch was eliminated by condition 4, not by arithmetic - testing both consecutive-pair placements is what surfaces this.
2. Is the branch head right that condition 4 did no work?
Read the worked answer
No. The reasoning confuses 'not binding in the surviving schedule' with 'not needed to reach it'.
When S is placed on Thursday, R lands on Friday and the only free days for P and T are Monday and Wednesday - both adjacent to Q on Tuesday. Without condition 4, that branch would yield valid-looking schedules, for example Monday P, Tuesday Q, Wednesday T, Thursday S, Friday R, and the mirror version with T on Monday and P on Wednesday. Condition 4 is exactly what kills the entire S-on-Thursday branch and forces S onto Wednesday.
So condition 4 performs the decisive elimination. What is true is narrower: once the schedule is fixed, condition 4 is satisfied by the outcome and does not visibly constrain the final row. A rule can be invisible in the answer and still have done all the work. Dropping it from the written procedure would leave three schedules instead of one, which is a real defect in a procedure meant to produce a single agreed order.
3. Is the colleague right that condition 3 is the one doing nothing? Test it by removing condition 3 and counting the valid schedules.
Read the worked answer
Yes - condition 3 is the redundant one, and the count proves it.
Remove condition 3, keep conditions 1, 2 and 4.
Branch S on Wednesday, R on Thursday: Monday and Friday remain for P and T, freely assignable. Option 1: P on Monday, T on Friday - P on Monday is adjacent to Q on Tuesday, so condition 4 rejects it. Option 2: P on Friday, T on Monday - accepted. One schedule survives.
Branch S on Thursday, R on Friday: Monday and Wednesday remain for P and T, again freely assignable. But both days are adjacent to Q on Tuesday, so whichever day P takes, condition 4 rejects it. Zero schedules survive.
Total: one valid schedule, identical to the schedule obtained with condition 3 in force. Removing condition 3 changes nothing, because the T-on-Friday possibility it was designed to block is already blocked by the interaction of conditions 2 and 4.
Verdict: the colleague is correct on the facts but has the wrong rule in mind for the branch head's claim - condition 3 can be struck as redundant, condition 4 cannot. The general lesson is to test redundancy by deleting a rule and re-solving, never by inspecting the final answer.