Night Duty Roster for Four Clerks
The orderly room of an IAF station must detail one clerk for night duty on each of four consecutive nights: Monday, Tuesday, Wednesday and Thursday. Four clerks, P, Q, R and S, are available, and exactly one of them must be detailed on each night. The adjutant has laid down four conditions after checking leave plans and travel bookings: (i) P cannot be detailed on Monday or on Thursday; (ii) Q must be detailed on the night immediately following R's night; (iii) S is not available on Wednesday; (iv) R cannot be detailed on Monday. The roster must be published before 1600 hours and passed to the security flight. An earlier draft was returned unsigned because it detailed P on Monday. A second clerk now claims that another arrangement, different from the obvious one, also satisfies every condition, and the adjutant wants that claim settled before the roster is signed.
1. Publish the roster: which clerk is detailed on each of the four nights? Show how each condition is used and rule out the alternatives.
Read the worked answer
Step 1 - place R. Condition (iv) rules out Monday, and condition (ii) needs a following night, so R can only be Tuesday, Wednesday or Thursday.
Step 2 - test each case.
- R on Tuesday forces Q on Wednesday; Monday and Thursday are then left for P and S, but (i) forbids P on either night. Rejected.
- R on Thursday leaves no night for Q to follow. Rejected.
- R on Wednesday forces Q on Thursday; Monday and Tuesday are left for P and S, and (i) forbids P on Monday, so P takes Tuesday and S takes Monday. Condition (iii) is satisfied because S sits on Monday, not Wednesday.
Roster: Monday - S; Tuesday - P; Wednesday - R; Thursday - Q.
A good answer should notice that the roster never actually uses condition (iii), rather than pretending it was needed.
2. Is condition (iii) necessary? Decide whether the roster stays unique without it, and use the result to settle the second clerk's claim.
Read the worked answer
Delete condition (iii) and retest. R on Tuesday still fails, because P would then have to fill Monday or Thursday. R on Thursday still fails, because no night follows for Q. R on Wednesday still puts Q on Thursday and leaves Monday and Tuesday for P and S; (i) forces P to Tuesday and S to Monday. So the only roster is still S, P, R, Q, and S never lands on Wednesday: condition (iii) removes no arrangement that the other three conditions allow.
Conclusion: (iii) is redundant, and the second clerk's claim of a second valid arrangement is wrong - the conditions fix exactly one roster. The practical point for the adjutant is that a redundant condition in a standing order is harmless only while the other conditions stand; if P's restriction is ever relaxed, (iii) becomes load-bearing and the roster stops being unique.
3. A suggestion is made to reverse condition (ii) so that R must be detailed on the night immediately after Q. State what happens and what would be needed to restore a single roster.
Read the worked answer
Place Q under the reversed rule:
- Q on Monday → R on Tuesday; P and S fill Wednesday and Thursday; (i) forbids P on Thursday, so P takes Wednesday and S takes Thursday. Valid: Q, R, P, S.
- Q on Tuesday → R on Wednesday; P and S fill Monday and Thursday; (i) forbids P on both nights. Rejected.
- Q on Wednesday → R on Thursday; P and S fill Monday and Tuesday; (i) forces P to Tuesday and S to Monday. Valid: S, P, Q, R.
- Q on Thursday → R would fall on Friday, outside the roster. Rejected.
Two rosters now satisfy every condition, so no single roster can be published from the rules as reversed. An extra condition is required - for example 'S must not be detailed on Monday', which eliminates the first arrangement and leaves S, P, Q, R. Note that the reversed rule changes the answer only for Q on Monday, which is why the check has to be done case by case rather than by intuition.