Rebuilding a Lost Shift Assignment
Four troopers - Alvarez, Boone, Chen, and Doss - must each be assigned to one of four patrol zones (1, 2, 3, 4) and one of four shifts (days, evenings, nights, relief). Each zone and each shift is used exactly once, and no trooper holds more than one of each. The assignment file contains five rules:
- Chen is assigned to zone 2.
- Alvarez works days.
- Boone's zone number is exactly one greater than Alvarez's zone number, and Boone works evenings.
- Doss works relief.
- The trooper on relief works a zone with a lower number than the trooper on nights.
The watch commander needs the complete assignment before roll call. A new trooper asks whether all five rules are actually needed, and whether losing a rule from the file would change the answer.
1. Determine the complete assignment, showing the order in which the constraints force each conclusion.
Read the worked answer
Work zones first, because they constrain each other.
- Rule 3 says Boone = Alvarez + 1, so Alvarez cannot be zone 4 (no zone 5).
- Alvarez cannot be zone 2, because Chen holds zone 2 (rule 1).
- Alvarez cannot be zone 1, because that would force Boone into zone 2, which Chen holds.
- Therefore Alvarez = 3 and Boone = 4, leaving zones 1 and 2 to Chen and Doss. Rule 1 gives Chen = 2, so Doss = 1.
Now shifts. Rule 2: Alvarez = days. Rule 3: Boone = evenings. That leaves nights and relief to Chen and Doss, with zones 2 and 1 respectively. Rule 5 requires the relief zone to be lower than the night zone, so relief must be zone 1 and nights must be zone 2.
Final: Alvarez - zone 3, days; Boone - zone 4, evenings; Chen - zone 2, nights; Doss - zone 1, relief. Verify: rule 4 holds, and 1 < 2 satisfies rule 5.
2. Is any of the five rules redundant? Demonstrate by deriving the same assignment without it.
Read the worked answer
Rule 4 (Doss works relief) is redundant. Remove it and re-derive.
Zones are fixed by rules 1 and 3 alone: Chen 2, Alvarez 3, Boone 4, Doss 1. Shifts: rules 2 and 3 fix Alvarez on days and Boone on evenings, leaving nights and relief to Chen (zone 2) and Doss (zone 1). Rule 5 requires the relief trooper's zone to be lower than the night trooper's zone. The only arrangement satisfying that is Doss in zone 1 on relief and Chen in zone 2 on nights, since the reverse would require 2 < 1.
So rule 4's conclusion follows from the other four; it is a check, not a constraint. Keeping it in the file is harmless and even useful as an arithmetic check, but the answer is unchanged if it is dropped.
3. Suppose rule 1 were lost from the file. How many distinct assignments would satisfy the remaining four rules? Justify each.
Read the worked answer
Exactly two. Without rule 1, Alvarez may sit in zone 1 or zone 3.
Case A - Alvarez zone 1, Boone zone 2. Remaining zones 3 and 4 go to Chen and Doss. Rule 4 makes Doss relief, so rule 5 forces Doss into the lower of the two zones: Doss zone 3 relief, Chen zone 4 nights (3 < 4 holds). This is a valid solution.
Case B - Alvarez zone 3, Boone zone 4. Remaining zones 1 and 2 go to Chen and Doss. Again Doss is relief and must take the lower zone: Doss zone 1 relief, Chen zone 2 nights (1 < 2 holds). This is the original assignment.
Alvarez in zone 2 is barred by nothing except Boone's needing zone 3 - which is actually available, but then Chen and Doss would compete for 1 and 4 and rule 5 could not be satisfied. So two solutions survive, and it is rule 1 that eliminates Case A.