A Five-Day Help-Desk Roster and a Sudden Transfer
The ward office runs a single help-desk counter from Monday to Friday. Five clerks are available - Anil, Bhavna, Chirag, Divya and Ela - and in a normal week each clerk mans the counter on exactly one day. The roster rules are: (1) Anil can be rostered only on Monday or Friday, because he attends cash reconciliation on the other three days; (2) Bhavna must be rostered on the day immediately after Anil's day, because she takes over the pending-file bundle from him; (3) among Chirag, Divya and Ela, Chirag must be rostered last in the week, and Divya must be rostered before Ela; (4) no clerk is rostered twice in the week. On Friday morning Chirag reports that he has been transferred with immediate effect and will not attend. The counter cannot remain closed. A substitute clerk can be borrowed from the zonal office for one day only, and rules (1) to (4) continue to apply to the four ward clerks.
1. Build the roster for a normal week and show how each rule forced the arrangement.
Read the worked answer
Step 1 - place Anil. Rule (1) allows Monday or Friday. Rule (2) needs a day immediately after Anil. Friday has nothing after it in a Monday-Friday week, so Anil cannot be on Friday. Anil = Monday.
Step 2 - Bhavna. Rule (2) then fixes her on the next day. Bhavna = Tuesday.
Step 3 - the remaining three days. Wednesday, Thursday and Friday are left for Chirag, Divya and Ela. Rule (3) puts Chirag last in the week, and the last available day is Friday. Chirag = Friday.
Step 4 - Divya and Ela. Rule (3) also says Divya before Ela, and only Wednesday and Thursday remain, so the order is forced. Divya = Wednesday, Ela = Thursday.
Final roster: Mon Anil, Tue Bhavna, Wed Divya, Thu Ela, Fri Chirag.
Check against every rule: Anil on Monday (1 satisfied); Bhavna immediately after Anil (2 satisfied); Chirag last of the three and Divya before Ela (3 satisfied); each name appears once (4 satisfied). No rule is left doing no work - each one removed a genuine alternative, which is why the roster is unique rather than merely one of several.
2. Chirag is gone. Can the counter be staffed all five days using only the four remaining ward clerks? Prove your answer.
Read the worked answer
No. The proof does not depend on the rules at all, only on arithmetic. There are five days to cover, one clerk per day, and rule (4) allows each clerk on exactly one day. Four clerks can therefore cover at most four days. One day will be unmanned whatever the arrangement.
The tempting but wrong answers
- Put someone on two days. That breaks rule (4) directly. It is also the point at which double duty on a public counter starts producing errors in receipts, which is where the real cost lands.
- Drop Chirag's rule and rearrange the other four. Rule (3) concerned only Chirag, Divya and Ela; removing it frees nothing, because Anil and Bhavna are already pinned by rules (1) and (2) and there are only Wednesday-to-Friday slots for three names, of which only two remain.
- Close the counter on Friday. Excluded by the stated constraint, and Friday is typically the heaviest day for tax and certificate enquiries before the weekend.
So the answer is a borrowing, not a rearrangement. The one-day substitute from the zonal office is not a convenience here; it is required by the arithmetic. State the conclusion that way in the file: the requirement follows from the number of days, so no amount of internal adjustment will meet it.
3. With one borrowed substitute from the zonal office, produce one valid roster and state how many valid rosters the rules leave open.
Read the worked answer
One valid roster: Mon Anil, Tue Bhavna, Wed Divya, Thu Ela, Fri Substitute.
How it is built. Rules (1) and (2) are untouched by Chirag's departure and still force Anil on Monday and Bhavna on Tuesday. Chirag's rule now has no subject, so the only live ordering rule is Divya before Ela. Wednesday to Friday must be filled by Divya, Ela and the substitute, with Divya ahead of Ela.
Enumerating the possibilities
- Wed Divya, Thu Ela, Fri Substitute
- Wed Divya, Thu Substitute, Fri Ela
- Wed Substitute, Thu Divya, Fri Ela
Any other arrangement either puts Ela before Divya or leaves Anil and Bhavna out of place. So three valid rosters remain, i.e. two degrees of freedom.
How to choose among the three, rather than picking arbitrarily.
- Put the substitute on the day with the least handover load. A borrowed clerk does not know the ward's files, pending bundles or the register format.
- Keep Ela on Friday if Friday is the heavy enquiry day and she is the most experienced, since a stranger on Friday is the worst of the three options.
- Record the reason for the choice in the roster file. All three are legal; only one is defensible six weeks later when someone asks why the substitute was placed where he was.