Five Loan Files, Five Days: Sequencing the Credit Desk
The credit desk of a district central cooperative bank must clear five loan files next week - Home (H), Gold (G), Vehicle (V), Personal (P) and MSME (M). Exactly one file is cleared per working day, Monday to Friday. The processing rules fixed for that week are: the Gold file cannot be cleared on Monday or on Friday; the Vehicle file must be cleared exactly two working days after the Gold file; the MSME file must be cleared on the working day immediately before the Home file; the Personal file cannot be cleared on Monday; and the Home file cannot be cleared on Friday. A clerk has already written “Wednesday – Gold” on the board and claims Wednesday is the only workable slot for Gold. The assistant manager wants the full week verified before any file is marked cleared. A second clerk insists that if the Personal file also had to be cleared before the Gold file, the week could simply be rearranged.
1. Work out the only possible day-wise schedule for the five files. Show the elimination, not just the final order.
Read the worked answer
Step 1 - place Gold. Vehicle must be exactly two days after Gold, so Gold can only be Monday, Tuesday or Wednesday. Gold is barred on Monday and Friday, so Gold is Tuesday or Wednesday.
Step 2 - test Gold = Tuesday. Then Vehicle = Thursday. The days left for MSME, Home and Personal are Monday, Wednesday and Friday. MSME must sit immediately before Home, and no two of Monday, Wednesday and Friday are consecutive. Impossible.
Step 3 - therefore Gold = Wednesday, Vehicle = Friday. Days left: Monday, Tuesday, Thursday for MSME, Home, Personal. The only consecutive pair available is Monday–Tuesday, and MSME comes first, so MSME = Monday and Home = Tuesday. Personal takes Thursday.
Step 4 - check every rule. Personal is not on Monday (it is Thursday) ✔; Home is not on Friday (it is Tuesday) ✔; Vehicle is exactly two days after Gold ✔.
Schedule: Monday – MSME, Tuesday – Home, Wednesday – Gold, Thursday – Personal, Friday – Vehicle. The schedule is unique because Step 2 was the only rival branch and it failed.
2. The clerk says “Wednesday is the only workable day for Gold.” Is that conclusion forced by the rules, or is it an accident of the two-day gap? Explain what happens if Vehicle must be cleared exactly three working days after Gold.
Read the worked answer
The conclusion is not a property of the rules in general - it is forced only by the two-day gap combined with the other four conditions. Change the gap and the answer changes.
With a three-day gap, Vehicle = Gold + 3, and Vehicle must fall on or before Friday, so Gold can only be Monday or Tuesday. Monday is barred, so Gold = Tuesday and Vehicle = Friday. The free days are then Monday, Wednesday and Thursday for MSME, Home and Personal. The only consecutive pair is Wednesday–Thursday, so MSME = Wednesday and Home = Thursday, leaving Personal = Monday.
That breaches the rule that Personal cannot be cleared on Monday, so no valid schedule exists at all with a three-day gap.
The lesson for the desk: the placement of Gold has to be derived from the whole constraint set each week. Writing it on the board in advance and working around it invites a plan that silently violates a rule.
3. A second clerk claims that if the Personal file also had to be cleared before the Gold file, the week could simply be rearranged. Test this claim.
Read the worked answer
Test both surviving positions for Gold.
Case Gold = Tuesday. Vehicle = Thursday. MSME and Home must occupy two consecutive days out of Monday, Wednesday and Friday - impossible, exactly as before. This branch dies irrespective of where Personal goes.
Case Gold = Wednesday. Vehicle = Friday. MSME = Monday and Home = Tuesday are forced as the only consecutive pair, leaving Personal = Thursday. But Thursday comes after Wednesday, so Personal is after Gold, which is the opposite of the new condition.
Every legal position for Gold has now been tested and both fail, so the claim is false - the week cannot be rearranged. Adding one more ordering condition to a puzzle that is already tight is exactly the kind of instruction a clerk should test on paper before promising a customer a disposal date.