Five Workshops, One Week, Two What-Ifs
The Pune office of a consulting firm must run five half-day workshops - Operations (O), Finance (F), Marketing (M), Human Resources (H) and Analytics (A) - from Monday to Friday, exactly one per day. The learning team has fixed these rules. HR must run on Wednesday because the external facilitator is available only that day. Analytics must run the day immediately after Marketing, because the same case pack is used in both. Finance cannot be on Monday, when the audit closes, and cannot be on Friday, when the finance head travels. Operations must run before HR, so that new analysts complete it before their HR induction. Marketing cannot be on Friday. Twenty-two analysts are enrolled, and each workshop runs once, so nobody is asked to attend two sessions on the same day. Two of the five facilitators are external and bill by the day; the firm has already paid for all five days, so cancelling a workshop is not an option. The office head wants next week’s calendar published and has two what-if questions before signing off.
1. Build a valid Monday-to-Friday calendar and show why it is the only one.
Read the worked answer
HR is fixed on Wednesday. Finance can be Tuesday or Thursday only, so test both.
- If Finance = Thursday: the free days are Monday, Tuesday and Friday. Marketing and Analytics must be consecutive, and the only consecutive pair available is Monday–Tuesday, so M = Monday, A = Tuesday, leaving O = Friday. That breaks the rule that Operations runs before HR. Rejected.
- If Finance = Tuesday: Operations must precede Wednesday and Tuesday is taken, so O = Monday. Thursday and Friday remain for Marketing and Analytics, and Analytics must follow Marketing, so M = Thursday, A = Friday. The rule that Marketing is not on Friday is satisfied.
Calendar: Monday Operations, Tuesday Finance, Wednesday HR, Thursday Marketing, Friday Analytics.
Uniqueness follows from exhaustion: Finance had only two legal days, one of which fails, so the rest is forced. Any answer that stops at ‘a valid arrangement exists’ has not answered the question.
2. The HR facilitator now offers Thursday instead of Wednesday. Is a valid calendar still possible?
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No, and the proof is short. With HR on Thursday, Finance can only be Tuesday or Wednesday (Monday and Friday are barred).
- Finance = Tuesday: the remaining days are Monday, Wednesday and Friday. Marketing and Analytics must be consecutive, but no two of those three days are consecutive, since Tuesday and Thursday are occupied. Impossible.
- Finance = Wednesday: the remaining days are Monday, Tuesday and Friday. The only consecutive pair is Monday–Tuesday, so Marketing = Monday and Analytics = Tuesday, leaving Operations on Friday. Operations would then fall after the Thursday HR session, breaking the sequencing rule. Impossible.
Both branches fail, so no valid calendar exists if HR moves to Thursday. The correct response is not to reshuffle harder but to identify what must give: either the Operations-before-HR rule, or Marketing-immediately-before-Analytics, or a different HR day entirely. Naming the constraint to relax, rather than forcing a schedule, is the judgement being tested.
3. If the rule that Operations must run before HR is dropped, how many valid calendars remain?
Read the worked answer
Keep HR on Wednesday, Finance on Tuesday or Thursday, and Marketing immediately before Analytics.
- Finance = Tuesday: free days are Monday, Thursday and Friday for O, M and A. The only consecutive pair is Thursday–Friday, so M = Thursday, A = Friday, O = Monday. One calendar.
- Finance = Thursday: free days are Monday, Tuesday and Friday. The only consecutive pair is Monday–Tuesday, so M = Monday, A = Tuesday, O = Friday. One calendar.
Total: two calendars, namely (O, F, H, M, A) and (M, A, H, F, O).
Notice how much work the dropped rule was doing. It cut two arrangements to one, and it was also the single reason the Thursday-HR scenario became impossible. In a scheduling question, counting the calendars that survive one relaxed rule is a quick way to show which constraint is actually binding - and, in a real office, which concession is cheapest to ask for.