One guard short: checking a night-duty roster against the rules
A range office has to detail one forest guard for night duty at the anti-poaching camp on each of seven nights, Monday to Sunday. Four guards are available: A, B, C and D. The Range Officer's standing instructions are: (1) no guard may be detailed on two consecutive nights; (2) each guard must do at least one and at most three nights in the week; (3) A is away at a training camp on Sunday and Monday nights; (4) B must do exactly two nights and they must not be consecutive; (5) C has a fixed weekly off on Thursday and cannot be detailed on Wednesday or Thursday nights. The clerk submits this roster for approval: Monday A, Tuesday B, Wednesday C, Thursday D, Friday B, Saturday C, Sunday A. The Range Officer is due to sign it within the hour, and the clerk says it satisfies every condition.
1. Test the submitted roster against all five instructions. List every violation, and state which instructions the roster does in fact satisfy.
Read the worked answer
Check each night against each rule; do not simply read the roster.
Violations.
- Monday - A. Rule 3 bars A on Sunday and Monday nights. Violation.
- Sunday - A. Rule 3 again. A is away. Violation.
- Wednesday - C. Rule 5 bars C on Wednesday and Thursday nights. Violation.
Rules actually satisfied. Rule 1 holds: no guard appears on consecutive nights - the sequence A, B, C, D, B, C, A has no repetition on adjacent days. Rule 2 holds: A 2, B 2, C 2, D 1, all within 1 to 3. Rule 4 holds: B does exactly two nights, Tuesday and Friday, which are not consecutive.
So three of the seven nights are illegal, and the clerk's assurance is wrong on precisely the rules that are easiest to miss, because they concern individual availability rather than the pattern of the roster. The lesson for approval work is to check a roster rule by rule, person by person, before it goes for signature; a signature on an illegal detail is the Range Officer's responsibility, not the clerk's.
2. Construct a valid roster for the seven nights and demonstrate that it satisfies every instruction.
Read the worked answer
One valid roster is: Monday B, Tuesday A, Wednesday D, Thursday B, Friday C, Saturday A, Sunday D.
Verification, rule by rule:
Rule 1 (no consecutive nights). B, A, D, B, C, A, D - no name repeats on adjacent nights. Satisfied. Rule 2 (1 to 3 nights each). A 2 (Tuesday, Saturday), B 2 (Monday, Thursday), C 1 (Friday), D 2 (Wednesday, Sunday). All within limits. Rule 3 (A not Sunday or Monday). A works Tuesday and Saturday only. Satisfied. Rule 4 (B exactly two, non-consecutive). Monday and Thursday, three nights apart. Satisfied. Rule 5 (C not Wednesday or Thursday). C works only Friday. Satisfied.
Note that the solution is not unique - for example, Monday B, Tuesday D, Wednesday A, Thursday B, Friday D, Saturday A, Sunday C also works. What matters in a case like this is that the roster is built from the restricted guards outwards: fix A away from Sunday and Monday, fix B's two non-adjacent nights, then fill the remaining nights with C and D and re-check the consecutive-night rule at every step. Building from the least flexible guard is the reliable method.
3. D now applies for two nights' leave covering Saturday and Sunday. Determine whether a legal roster still exists; if it does, produce one, and if it does not, state exactly what has to give.
Read the worked answer
A legal roster still exists - D's leave can be absorbed.
Reason it out from Sunday backwards. Sunday can no longer go to D, and rule 3 removes A as well, so Sunday must be B or C. Saturday can go to A, B or C, since D is on leave. Suppose Sunday goes to C: then C's one night is used and C cannot take Saturday either, so Saturday must be A or B. Give Saturday to A. Then A has one night, and may take one more, but not Sunday or Monday. B still needs exactly two non-consecutive nights among Monday to Friday.
One workable result: Monday B, Tuesday D, Wednesday A, Thursday B, Friday D, Saturday A, Sunday C.
Check: no consecutive repeats (B, D, A, B, D, A, C); counts A 2, B 2, C 1, D 2, all within 1 to 3; A not on Sunday or Monday; B exactly two, Monday and Thursday, non-consecutive; C only on Friday, not Wednesday or Thursday; D on leave Saturday and Sunday and detailed only on Tuesday and Friday. All five instructions are met.
So the answer to D is 'leave granted', but the covering note should say that the range is now working with no spare capacity: if any one of A, B or C reports sick, or if an arrest or a fire call-out detains a guard, the roster cannot be repaired within the rules and the Range Officer will have to relax the two-night limit or draw a guard from the neighbouring beat. Flagging that risk in writing is the point of the exercise.