Fitting Five Modules into a Three-Day Induction Programme
A provincial services academy must fit exactly five compulsory modules into a three-day induction programme: Fiscal Policy (F), Public Procurement (P), Communication (C), IT Skills (I) and Ethics (E). No day may carry more than two modules, and Day 3 carries exactly one, because the closing ceremony occupies the rest of that day. Four rules were agreed at the planning meeting: (1) Fiscal Policy must be delivered on an earlier day than Public Procurement; (2) Ethics cannot be placed on Day 1; (3) Communication and Public Procurement must not fall on the same day; (4) Day 1 must carry exactly two modules. Each module is delivered once, in a single slot, and no module spans two days.
The deputy director has circulated a draft timetable: Day 1 - Fiscal Policy and Communication; Day 2 - Public Procurement and Ethics; Day 3 - IT Skills. A guest speaker has since been invited to deliver Ethics, and she can come only on Day 3. The academy also intends to drop rule 4 at the next meeting as 'unnecessary paperwork'.
1. Verify the circulated draft against the four rules and the slot structure. Identify the interpretation error most likely to make a candidate reject a valid draft.
Read the worked answer
Check each rule against the draft:
- Rule 1: F on Day 1, P on Day 2 → F is on an earlier day. Satisfied.
- Rule 2: E on Day 2, not Day 1. Satisfied.
- Rule 3: C on Day 1, P on Day 2 → different days. Satisfied.
- Rule 4: Day 1 carries exactly two modules (F, C). Satisfied.
- Structure: five modules, at most two per day, Day 3 exactly one (I). Satisfied.
The draft is admissible.
The likely error is treating rule 1 as an ordering rule within a day ('F in the morning slot, P in the afternoon slot') rather than as a day-ordering rule. Read that way, the draft looks broken because F and P are on different days. A second, related error is assuming IT Skills must be on Day 3 because it is listed last: nothing forces that. I is on Day 3 only because the other four modules fill Days 1 and 2, and any of the five could occupy the single Day-3 slot subject to rules 1–3.
Note also what the draft does not address: it was built before the Ethics guest fixed her date, so it is valid as a document but already obsolete as a plan.
2. If Ethics must now be delivered on Day 3, is the programme still schedulable? Show the complete arrangement or prove that none exists.
Read the worked answer
With E fixed on Day 3, Days 1 and 2 must absorb F, P, C and I, two modules each (rule 4 fixes Day 1 at two; the total of five with Day 3 at one fixes Day 2 at two). Two further deductions apply: F and P cannot share a day at all, since one must be on an earlier day than the other; and P cannot share a day with C.
Test every possible Day-1 pair drawn from {F, P, C, I}:
| Day 1 | Day 2 forced | Verdict |
|---|---|---|
| F, P | C, I | F and P same day → violates rule 1 |
| F, C | P, I | F Day 1 < P Day 2; C Day 1 ≠ P Day 2 → valid |
| F, I | P, C | C and P same day → violates rule 3 |
| P, C | F, I | violates rule 3 directly |
| P, I | F, C | F on Day 2, P on Day 1 → F is not earlier → violates rule 1 |
| C, I | F, P | F and P same day → violates rule 1 |
Only one arrangement survives: Day 1 - Fiscal Policy and Communication; Day 2 - Public Procurement and IT Skills; Day 3 - Ethics.
The programme is schedulable, but the academy now has no slack. The guest's date consumes the last degree of freedom, so if she postpones, the entire timetable has to be rebuilt rather than patched. That is worth telling the deputy director before the invitation is confirmed.
3. The academy proposes to delete rule 4 as redundant. Is it? What follows for the academy's flexibility if it wants to change the timetable?
Read the worked answer
Rule 4 is redundant - it excludes nothing.
Five modules; each day at most two slots; Day 3 exactly one slot. Days 1 and 2 must therefore carry the remaining four modules between them, and with a ceiling of two per day the only possible split is 2 + 2. 'Day 1 carries exactly two modules' is a consequence of the slot structure, not an independent constraint. Delete it and the set of admissible timetables is unchanged.
Why it matters practically. A planner who believes rule 4 is doing work may feel licensed to relax the real structural assumption - that Day 3 has exactly one slot - in order to accommodate the guest, and will then be surprised when the timetable changes. The binding constraints are rule 1 (strict day-ordering of F before P) and rule 3 (C and P separated); together with the two-slot days they consume all the slack. That is why the Day-3 assignment of Ethics turned out to have exactly one solution.
Implication for flexibility. The academy cannot buy slack by deleting a redundant rule. It must change something physical: add a slot on Day 3 (difficult while the closing ceremony occupies the day), extend to a fourth day, or permit three modules on one day. Any of those creates real options; deleting rule 4 creates only the illusion of them.