A Seating Plan That Cannot Be Solved - Choosing Which Condition to Break
Six officers - Anil, Bina, Chetan, Divya, Esha and Farhan - must occupy desks 1 to 6, one each, at the single-window counter the zonal office is piloting in Bilaspur. Desks run left to right, 1 at the far left. The zonal office sent five conditions: (1) Anil must occupy desk 1; (2) Chetan must occupy the desk immediately to the right of Divya's desk; (3) Bina must be exactly two desks to the left of Esha; (4) Farhan must not sit next to Bina; (5) Chetan must not sit at desk 6. Every branch that tried the arrangement reported that no seating satisfied all five, and two branches circulated different 'final' rosters. The file is due tomorrow. The zonal office will accept the relaxation of exactly one condition and nothing more, and wants the reasoning recorded so that the same error is not repeated in the next pilot.
1. Determine which single condition must be relaxed. Show the elimination, not just the conclusion.
Read the worked answer
Take the conditions in the order that constrains most.
From (1): Anil takes desk 1, so desks 2–6 are open.
From (3): Bina and Esha can only be at (1,3), (2,4), (3,5) or (4,6). Desk 1 is taken, so three pairs survive: (2,4), (3,5), (4,6).
Test (2,4): remaining desks 3, 5, 6. Condition (2) needs Divya and Chetan adjacent, Divya on the left. The only adjacent free pair is (5,6), so Divya = 5, Chetan = 6, Farhan = 3. But (5) forbids Chetan at desk 6. Fails.
Test (3,5): remaining desks 2, 4, 6. No two of these are adjacent, so (2) cannot be met. Fails.
Test (4,6): remaining desks 2, 3, 5. The only adjacent free pair is (2,3), so Divya = 2, Chetan = 3, Farhan = 5. Check (5): Chetan at 3, allowed. Check (4): Farhan at 5 sits next to Bina at 4. Violates (4) only.
So exactly one seating satisfies conditions (1), (2), (3) and (5), and condition (4) is the one that must be relaxed. Because the elimination is exhaustive, the conclusion is not a guess - every other branch pair was tested and failed for a stated reason.
2. State the final roster and explain what you would place on the file so the relaxation cannot be reopened later.
Read the worked answer
Final roster: desk 1 Anil, desk 2 Divya, desk 3 Chetan, desk 4 Bina, desk 5 Farhan, desk 6 Esha.
Verification: (1) Anil at 1; (2) Chetan at 3 is immediately right of Divya at 2; (3) Bina at 4 is two desks left of Esha at 6; (5) Chetan is not at 6. Condition (4) is the single breach - Farhan at 5 is adjacent to Bina at 4 - and it is unavoidable.
File note, in four parts:
- The four conditions retained and the exact desk numbers that satisfy them, so a reviewer can check in ten seconds.
- The three candidate Bina–Esha placements and why two of them fail. This is the part that stops next year's clerk from repeating the exercise: the elimination is the evidence.
- The statement that condition (4) was relaxed, with the authority for the relaxation and the date.
- The operational consequence, stated plainly: Farhan and Bina are adjacent, so if the separation in (4) existed to prevent collusion between two cash-handling desks, the counter supervisor must be told to rotate the two at mid-day instead.
A roster is only a roster until someone asks why. The file is what makes the answer survive a transfer of the officer who wrote it.
3. The zonal office refuses to drop (4) and offers to relax (1) instead. Show what this produces and advise.
Read the worked answer
Re-test with (1) dropped, keeping (2), (3), (4), (5). Anil is now free, so all four Bina–Esha pairs are live.
- (1,3): remaining desks 2, 4, 5, 6. Adjacent pair for (2) is (4,5) → Divya 4, Chetan 5, leaving 2 and 6 for Anil and Farhan. Farhan at 2 would be adjacent to Bina at 1, so Farhan = 6, Anil = 2. Valid.
- (2,4): remaining 1, 3, 5, 6. Adjacent pair (5,6) puts Chetan at 6, forbidden by (5). Fails.
- (3,5): remaining 1, 2, 4, 6. Adjacent pair (1,2) → Divya 1, Chetan 2, leaving 4 and 6. Farhan at 4 sits next to Bina at 3, so Farhan = 6, Anil = 4. Valid.
- (4,6): remaining 1, 2, 3, 5. Both candidate desks left for Farhan (3 and 5) are adjacent to Bina at 4. Fails.
So relaxing (1) yields two valid rosters, not one: (Anil 2, Bina 1, Esha 3, Divya 4, Chetan 5, Farhan 6) and (Divya 1, Chetan 2, Bina 3, Anil 4, Esha 5, Farhan 6).
Advice: decline the offer. Relaxing (4) yields a single, unambiguous roster; relaxing (1) yields two, and two is worse than one - a branch will pick whichever it sees first, the two branches will drift apart, and the pilot stops producing comparable data. If the zonal office insists on keeping (4), accept it only with a tie-breaker rule written into the pilot notes (for example, the officer with the longest service takes the lower-numbered desk), so that the choice between the two rosters is made by a rule rather than by preference.