Four-Day Inspection Roster: A Deduction Puzzle
The Vidarbha Regional Office must complete four branch inspections between Monday and Thursday of next week - one inspection per day, one officer per day, each officer visiting a different branch. The officers are Kavya, Lalit, Manoj and Nisha. The Regional Manager's written instructions are: (i) neither Lalit nor Manoj can inspect on Wednesday, because both sit on the credit-audit committee that day; (ii) Nisha must inspect on the day immediately after Kavya, so that the two reports can be consolidated; (iii) Lalit must inspect before Manoj, because Manoj is to review Lalit's observations; (iv) Manoj cannot inspect on Tuesday, when he attends the district-level bankers' committee. The inspection report of the officer inspecting on the last day is due the same evening. An apprentice has prepared a draft roster with Kavya on Monday, Nisha on Tuesday, Lalit on Wednesday and Manoj on Thursday, and has asked the manager to approve it.
1. Derive the only roster that satisfies all four conditions. Show the elimination, not just the final answer.
Read the worked answer
Step 1 - fix the Kavya-Nisha block. Condition (ii) forces the pair into one of three slots: Kavya-Nisha on Mon-Tue, on Tue-Wed, or on Wed-Thu.
Step 2 - test each slot.
- Mon-Tue: Wednesday and Thursday are left for Lalit and Manoj. Condition (i) bars Lalit from Wednesday, so Manoj would take Wednesday and Lalit Thursday - which breaks condition (iii), since Lalit must be before Manoj. Reject.
- Wed-Thu: Monday and Tuesday are left for Lalit and Manoj, and (iii) fixes Lalit on Monday and Manoj on Tuesday. But (iv) bars Manoj on Tuesday. Reject.
- Tue-Wed: Monday and Thursday remain for Lalit and Manoj; (iii) gives Lalit Monday and Manoj Thursday, and (iv) is satisfied because Manoj is not on Tuesday.
Step 3 - result. Monday Lalit, Tuesday Kavya, Wednesday Nisha, Thursday Manoj. Every condition is satisfied and the path above shows each rejected branch fails on a named condition, so the solution is unique.
2. The apprentice's draft has Kavya on Monday, Nisha on Tuesday, Lalit on Wednesday and Manoj on Thursday. Which conditions does it violate, and why is checking one violated condition enough to reject the draft?
Read the worked answer
Test the draft against each condition in turn:
- (i) Lalit is placed on Wednesday - violated. Manoj is on Thursday, which is permitted, but the condition bars Lalit on Wednesday.
- (ii) Nisha (Tuesday) is the day immediately after Kavya (Monday) - satisfied.
- (iii) Lalit (Wednesday) is before Manoj (Thursday) - satisfied.
- (iv) Manoj is on Thursday, not Tuesday - satisfied.
So the draft fails exactly one condition, (i). That single failure is decisive: the roster is a conjunction of hard constraints, not a weighted score, so an arrangement that breaks any one of them is inadmissible even if it looks tidy. A second, independent reason to reject it is that the draft cannot be the unique solution - the Tuesday-Wednesday slot is the only one that survives, and the draft does not use it.
Practical point: in a scheduling dispute, always name the specific clause breached and the specific placement that breaches it, rather than saying the roster is wrong overall. Managers can act on a cited clause; they cannot act on a general objection.
3. If condition (iv) alone is withdrawn, how many rosters satisfy the remaining conditions? List them and explain what the change tells you about how tightly the conditions interact.
Read the worked answer
Re-run the three-slot elimination with (i), (ii) and (iii) only:
- Kavya-Nisha on Mon-Tue: Wednesday-Thursday go to Lalit and Manoj. Condition (i) forces Manoj to Wednesday and Lalit to Thursday, which breaks (iii). Reject - unchanged by the withdrawal.
- Kavya-Nisha on Tue-Wed: Lalit Monday, Manoj Thursday. Valid.
- Kavya-Nisha on Wed-Thu: Monday and Tuesday go to Lalit and Manoj. With (iv) gone, both Lalit-Monday with Manoj-Tuesday, and Manoj-Monday with Lalit-Tuesday, are now open; (iii) keeps only Lalit-Monday, Manoj-Tuesday. Valid.
Two rosters: (Lalit, Kavya, Nisha, Manoj) and (Lalit, Manoj, Kavya, Nisha).
Interpretation: condition (iv) is doing real work - it is the only constraint that separates the two surviving shapes, so it is the binding condition, not decoration. In puzzle design and in real rostering, a condition that eliminates exactly one alternative is fragile: if the Tuesday committee meeting is cancelled, the manager silently loses uniqueness and two officers may turn up at the same branch. Tight rosters should be locked in writing with the version number, so that a later relaxation is visibly re-solved rather than assumed.