Scheduling Four Escorted Appointments
Four inmates - Alder, Boone, Cruz, and Diaz - each have one 45-minute appointment that requires an escort, and all four must be completed between 0800 and 1200. After each appointment, 15 minutes is required to return the inmate to the unit, so appointments can begin only at 0800, 0900, 1000, or 1100. Officer Kwan is the only escort available before 0930. Diaz is the subject of a standing no-contact order that prohibits assignment to Kwan. Alder's attorney is available only between 0800 and 0930. Boone must be escorted before Cruz. All other pairings and assignments are unrestricted.
1. Which inmate must occupy the 0800 slot, and which inmate cannot occupy it? Justify each answer from the constraints.
Read the worked answer
0800 must be Alder. Diaz cannot be at 0800.
Alder at 0800: Alder's attorney is available only from 0800 to 0930. A 0900 start would run to 0945, which is already past 0930, so 0900 and later are impossible for Alder. That leaves 0800.
Diaz not at 0800: Officer Kwan is the only escort available before 0930, so any 0800 appointment is escorted by Kwan. The standing no-contact order prohibits pairing Diaz with Kwan. Therefore Diaz cannot be the 0800 appointment.
Why these two conclusions combine: the 0800 slot is forced to Alder by the attorney window, and the no-contact order independently bars Diaz from it. Both lines of reasoning point the same way, which is why the answer is certain rather than merely likely.
Note the clock arithmetic: a 45-minute appointment plus a 15-minute return means each start consumes a full hour. An appointment that starts at 0900 does not end at 0900.
2. List every valid sequence of the four appointments, using the four start times. Show how you eliminated the invalid ones.
Read the worked answer
Fixed: 0800 = Alder.
Remaining: Boone, Cruz, and Diaz fill 0900, 1000, and 1100, with the single constraint Boone before Cruz.
Total arrangements of three inmates = 3 × 2 × 1 = 6. Half of them put Cruz before Boone, which is forbidden, leaving 3:
- 0800 Alder - 0900 Boone - 1000 Cruz - 1100 Diaz
- 0800 Alder - 0900 Boone - 1000 Diaz - 1100 Cruz
- 0800 Alder - 0900 Diaz - 1000 Boone - 1100 Cruz
Eliminated: any order with Cruz ahead of Boone → Alder, Cruz, Boone, Diaz; Alder, Cruz, Diaz, Boone; Alder, Diaz, Cruz, Boone.
Reasoning note: "Boone before Cruz" is an ordering constraint, not a timing constraint. It removes exactly half of the remaining arrangements rather than fixing a position. If a supervisor needs one schedule, pick any of the three; if a supervisor says the answer is unique, an unstated constraint is missing.
3. At 0745, Kwan calls in sick and no replacement escort arrives until 1000. What should happen to the 0800 appointment, and why can it not simply be moved to 1000?
Read the worked answer
The 0800 appointment cannot proceed and cannot simply slide to 1000.
- No authorized escort exists at 0800. Escorting without a qualified officer is not a permitted shortcut; the appointment must be delayed or cancelled.
- Moving Alder to 1000 does not work. Alder's attorney is available only 0800–0930. A 1000 appointment falls outside that window entirely, so the appointment cannot be rescheduled that morning on the attorney's availability. The failure is the attorney's window, not the escort's arrival time.
Correct actions, in order: notify the shift supervisor immediately; attempt to identify another qualified escort before 0930; if none is found, cancel the Alder appointment and document the reason, the time Kwan's absence was reported, and the time the supervisor was notified; notify the attorney or classification staff so the appointment can be re-set. Then re-run the schedule for the 1000 arrival - Diaz still cannot be paired with Kwan if Kwan returns, but Diaz may now be placed in any slot that does not require Kwan.
Reasoning note: do not solve a staffing problem by violating a custody requirement, and do not assume a cancelled slot can be refilled by simply shifting everyone one hour later.