Fixing the Concurrent Audit Schedule
A bank's concurrent audit team must cover five branches - P, Q, R, S and T - from Monday to Friday of next week, exactly one branch each working day. Five officers - Anil (A), Bhavna (B), Chirag (C), Divya (D) and Ehsan (E) - will each visit exactly one branch, and no two officers visit on the same day. The assignment must satisfy all of the following:
- Bhavna visits on Wednesday.
- Ehsan visits on Friday.
- Q is visited on Monday.
- T is visited exactly two days after Q.
- Anil visits on the day immediately after Q's visit.
- Chirag visits P, and P is not visited on Monday or Tuesday.
- Anil does not visit S.
The audit head wants the schedule fixed before the circular goes out, but is unsure whether every condition is doing useful work, and wants to know what happens if one of them is relaxed.
1. Derive the complete schedule - day, officer and branch - showing the chain of inference at each step.
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Step 1 - anchor the branch days. Clue 3 puts Q on Monday. Clue 4 makes T Wednesday. Clue 5 puts Anil on Tuesday.
Step 2 - place the officers. Wednesday's officer is Bhavna (clue 1) and Friday's is Ehsan (clue 2). With Anil on Tuesday, the officers still to be placed - Chirag and Divya - must occupy Monday and Thursday.
Step 3 - fix Chirag. Chirag visits P (clue 6). P cannot be Monday or Tuesday, and Wednesday is already T, so P must be Thursday or Friday. Chirag is available only on Monday or Thursday, and Monday is excluded by the P condition. Therefore Chirag visits P on Thursday, and Divya takes Monday, which is branch Q.
Step 4 - settle the last two branches. Branches left are R and S, for Tuesday (Anil) and Friday (Ehsan). Clue 7 rules out S for Anil, so Anil visits R on Tuesday and Ehsan visits S on Friday.
Final schedule: Monday - Divya, Q; Tuesday - Anil, R; Wednesday - Bhavna, T; Thursday - Chirag, P; Friday - Ehsan, S.
2. Condition 2 is replaced by: 'Ehsan visits before Chirag.' Everything else stays. How many valid schedules now exist, and which one is unchanged?
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Clues 3, 4 and 5 still fix Q on Monday, T on Wednesday and Anil on Tuesday. Bhavna is on Wednesday, so Chirag, Divya and Ehsan fill Monday, Thursday and Friday. Since P cannot be Monday, Tuesday or Wednesday (Wednesday is T), P is Thursday or Friday, and Chirag must be on Thursday or Friday.
Case 1 - Chirag on Thursday (P on Thursday). Divya and Ehsan take Monday and Friday; Ehsan must precede Chirag, so Ehsan is Monday and Divya is Friday. Branch Q is Monday, T is Wednesday, P is Thursday; R and S fall to Tuesday and Friday, and Anil cannot take S, so Tuesday is R and Friday is S. This is the original schedule, unchanged: Ehsan–Q, Anil–R, Bhavna–T, Chirag–P, Divya–S.
Case 2 - Chirag on Friday (P on Friday). Divya and Ehsan take Monday and Thursday, in either order, since both precede Friday. Branches: Monday Q, Wednesday T, Friday P; R and S fall to Tuesday and Thursday, and Anil must take R, leaving S on Thursday. Two schedules: (Ehsan–Q, Divya–S) and (Divya–Q, Ehsan–S).
Total: three valid schedules, one of which is the original.
3. A colleague proposes adding: 'Divya must visit before Chirag, to make the schedule more robust.' Evaluate the proposal. Would it have been useful under a different set of conditions?
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Under the original seven conditions the schedule is already unique (Divya on Monday, Chirag on Thursday), so the proposed rule is satisfied automatically. It adds no information, eliminates no alternative, and cannot detect an error - a redundant constraint. The audit head should not treat its inclusion as extra assurance: over-determination is not verification. In fact, redundancy is mildly harmful, because a reader may believe the rule is load-bearing and relax one of the conditions that actually does the work - for instance condition 6, which is what forces Chirag off Monday.
A constraint earns its place only if at least one alternative arrangement satisfies the other conditions and fails this one. Here, no such arrangement exists, so the rule is decoration.
It would have been useful in a genuinely under-determined version. For example, with condition 2 removed and 'Ehsan before Chirag' in its place, three schedules survive - and adding 'Divya before Chirag' leaves two. In that setting the rule does real filtering work. The lesson: test a proposed condition by dropping it and asking whether the solution count changes; if it does not, it is not a condition, it is a comment.