Five AFOs, Five Villages, Five Days
Five Agricultural Field Officers-A, B, C, D and E-must visit five villages-P, Q, R, S and T-from Monday to Friday, one officer per day and one village per day. The district office has issued the following conditions:
- B must visit Q on Tuesday.
- C must visit S, and C's visit must be on the day immediately after A's visit.
- D must visit T on Friday.
- E must visit P on Monday.
- A cannot visit on Monday or Friday.
The office also wants a contingency plan because E may be called for a training programme on Monday. The senior officer asks you to work out the schedule and test whether the contingency is feasible without changing any of the five conditions.
1. Determine the complete day-wise schedule.
Read the worked answer
Fix E on Monday (P), B on Tuesday (Q), and D on Friday (T). The remaining days for A and C are Wednesday and Thursday because C must be immediately after A and A cannot be Monday or Friday. The only consecutive pair left is Wednesday–Thursday, so A visits on Wednesday and C on Thursday. C's village is S, so A must take the remaining village R. Schedule: Monday E–P; Tuesday B–Q; Wednesday A–R; Thursday C–S; Friday D–T.
2. If E is unavailable on Monday, can all five conditions still be met? Explain.
Read the worked answer
No. With B on Tuesday and D on Friday, the days left for A, C and E are Monday, Wednesday and Thursday. A cannot take Monday or Friday. If A takes Thursday, C would have to take Friday, but D already has Friday. Therefore A must take Wednesday and C Thursday. That leaves Monday for E. So E's Monday visit is forced, not optional; if E is unavailable on Monday, no schedule satisfies all five conditions. The contingency plan cannot be met without relaxing a condition, for example moving D from Friday or allowing C not to be immediately after A.
3. Which single condition is redundant in the original set, and why?
Read the worked answer
'A cannot visit on Monday or Friday' is redundant. E is fixed on Monday and D on Friday, so A cannot take either of those days anyway under the rule that one officer visits per day. Also, C must be immediately after A: if A were Friday, C would need Saturday; if A were Monday, C would need Tuesday, which B already occupies. Thus the explicit prohibition on A's Monday/Friday visit does not add a new restriction. The conditions doing the real work are B–Tuesday, D–Friday, E–Monday, and C immediately after A.