The Jan Seva Kendra Roster: Two Valid Answers and One Hidden Constraint
The block office in Gumla has four Jan Seva Kendra counters, numbered 1 to 4 from left to right. Four clerks - Anita, Bipin, Chhaya and Deepak - must be posted one to a counter for the whole day. At the morning meeting three conditions were recorded:
(i) Chhaya must sit immediately to the left of Deepak, that is, on the counter numbered one less. (ii) Bipin will not sit at counter 1. (iii) Anita will not sit at counter 4.
Two clerks produced rosters. Roster P: 1 Chhaya, 2 Deepak, 3 Anita, 4 Bipin. Roster Q: 1 Anita, 2 Chhaya, 3 Deepak, 4 Bipin. The superintendent looked at both and said, 'Only one of these can be correct.'
An hour later a technician reported that the Aadhaar biometric device is installed at counter 3 only, and that among the four clerks only Bipin and Deepak have been trained to use it. Counter 3 must be manned by a trained clerk.
1. Was the superintendent right? Test rosters P and Q against conditions (i)-(iii) and list every arrangement that satisfies all three.
Read the worked answer
Test method: condition (i) makes Chhaya and Deepak a block of two adjacent counters with Chhaya on the left. The block can start at counter 1, 2 or 3 only.
- Block at 1-2 (Chhaya 1, Deepak 2). Counters 3 and 4 remain for Anita and Bipin. Bipin cannot be at 1, which is irrelevant here; Anita cannot be at 4, so Anita = 3 and Bipin = 4. This is Roster P.
- Block at 2-3 (Chhaya 2, Deepak 3). Counters 1 and 4 remain. Bipin cannot be at 1, so Bipin = 4 and Anita = 1. This is Roster Q.
- Block at 3-4 (Chhaya 3, Deepak 4). Counters 1 and 2 remain. Bipin cannot be at 1, so Bipin = 2 and Anita = 1. Call this Roster R: 1 Anita, 2 Bipin, 3 Chhaya, 4 Deepak.
All three satisfy (i), (ii) and (iii). So the superintendent was wrong: P and Q are both valid, and R is valid as well. The three conditions do not determine a unique roster - there are three solutions, not one. The error is a common one: checking that an answer satisfies every condition proves it is allowed, not that it is the only possibility.
2. With the biometric constraint added, which roster must be adopted? Explain why this constraint does more than simply choose between P and Q.
Read the worked answer
The rule is: counter 3 must be held by Bipin or Deepak.
Test all three survivors:
- P - counter 3 is Anita, untrained. Out.
- R - counter 3 is Chhaya, untrained. Out.
- Q - counter 3 is Deepak, trained. Only survivor.
So the forced roster is 1 Anita, 2 Chhaya, 3 Deepak, 4 Bipin.
Why the constraint does more than break a tie: a tie-breaking clue only needs to separate the two candidates someone happened to write down. This rule is a filter applied to the whole solution set, and it eliminates a roster (R) that neither clerk had proposed and that the superintendent had never considered. Had the office reasoned only about P versus Q, it might have accepted P - which is invalid for a reason nobody had yet stated. The general lesson for roster, seating and scheduling puzzles is: enumerate all solutions first, then apply the operational constraint, rather than arguing between the two answers on the table.
3. Suppose the biometric device had never been reported. What single additional condition would make the roster unique, and why do some plausible-sounding conditions fail?
Read the worked answer
A useful condition must be true of exactly one of the three rosters P, Q and R. Anything true of two or three of them cannot isolate an answer.
Works: 'Deepak is at counter 3.' True only of Q (P has Deepak at 2, R has Deepak at 4). The roster becomes unique immediately.
Fails: 'Bipin is at counter 4.' Bipin is at 4 in both P and Q, so it eliminates only R and leaves the original ambiguity - it looks informative because it mentions a named clerk, but it does not discriminate.
Also fails: 'Anita is not at counter 1.' Anita is at 3 in P and at 1 in Q and R, so it kills Q and R but cannot separate P from anything else - it produces a unique answer only by accident of which options were listed.
Method to state in the answer: list all solutions, write down which clerks occupy which counters in each, and choose a condition whose truth value differs across solutions. Checking that a clue is consistent with the intended answer is not enough; it must be inconsistent with every other answer.