Departure Order at the Bus Depot
A depot has to publish the departure order for five long-route buses - H1, H2, H3, H4 and H5 - which will leave at 06:00, 08:00, 10:00, 12:00 and 14:00, one bus to each slot. The traffic manager's handwritten note gives four rules: (i) H3 must not take the first or the last departure; (ii) H5 must leave exactly two slots after H2; (iii) H1 must leave immediately before H4; (iv) the 06:00 departure must not be H1. A new clerk has produced a chart in which H3 leaves at 12:00, H5 at 08:00 and H2 at 10:00, and insists it is valid because every bus gets a different slot. The manager also wants to know whether all four rules are really needed, since the note is faint and one line may be a duplicate of a rule that was already dropped.
1. Work out the only departure order that satisfies all four rules. Show how the possibilities are eliminated, not just the final answer.
Read the worked answer
Number the slots 1 to 5 = 06:00, 08:00, 10:00, 12:00, 14:00.
Start with rule (ii): H5 is exactly two slots after H2, so H2 can only be in slot 1, 2 or 3.
- H2 = slot 1 (06:00), H5 = slot 3 (10:00). Slots 2, 4 and 5 remain for H1, H3, H4. Rule (iii) allows only the pair (slot 4, slot 5), so H1 = 12:00 and H4 = 14:00, leaving H3 = 08:00, which satisfies rule (i). Valid.
- H2 = slot 2 (08:00), H5 = slot 4 (12:00). Remaining slots 1, 3, 5 are all two apart, so no immediate pair exists for H1-H4. Rule (iii) fails. Eliminated.
- H2 = slot 3 (10:00), H5 = slot 5 (14:00). Remaining slots 1, 2, 4; the only adjacent pair is (1, 2), forcing H1 = 06:00, which rule (iv) forbids. Eliminated.
Unique order: 06:00 H2, 08:00 H3, 10:00 H5, 12:00 H1, 14:00 H4.
2. The clerk's chart puts H3 at 12:00, H5 at 08:00 and H2 at 10:00. Identify exactly what is wrong with it and what minimum correction makes it consistent.
Read the worked answer
Check the chart against each rule in turn.
- Rule (ii): H5 must be exactly two slots after H2. If H2 is at 10:00, H5 must be at 14:00, but the chart puts H5 at 08:00 - before H2. This is the fatal error; the chart does not merely misplace one bus, it reverses the required order.
- Rule (iii): with H2 at 10:00 and H5 wrongly at 08:00, the remaining slots are 06:00, 12:00 and 14:00. H1 immediately before H4 forces 12:00 and 14:00, leaving 06:00 for H3 - which is fine, but the whole arrangement then collapses because H5 is in the wrong slot.
- "Every bus gets a different slot" is not a rule the manager wrote; it is only the basic one-per-slot condition, and satisfying it proves nothing.
Minimum correction: move H5 to 14:00 and H2 to 06:00, keeping H1 at 12:00 and H4 at 14:00 is not possible because 14:00 is now H5 - so the correct repair is simply to replace the chart with 06:00 H2, 08:00 H3, 10:00 H5, 12:00 H1, 14:00 H4.
3. Is any one of the four rules redundant? Decide, and prove it by testing the solution with that rule removed.
Read the worked answer
Test each rule by dropping it and re-solving.
- Drop rule (i) (H3 not first or last). Re-run the three cases. H2 = slot 1 still forces H1 = 12:00, H4 = 14:00 and therefore H3 = 08:00. H2 = slot 2 still fails on rule (iii). H2 = slot 3 still fails on rule (iv). The answer is unchanged, so H3 is never at either end anyway.
- Drop rule (iv) (06:00 is not H1). Now H2 = slot 3, H5 = slot 5, H1 = 06:00, H4 = 08:00, H3 = 12:00 becomes a second valid order. The rule is essential.
- Drop rule (iii) or rule (ii). Several orders become possible at once; both are essential.
Conclusion: rule (i) is redundant - it is a conclusion of the other three, not an independent constraint. It can be struck from the note without changing the published order, and the manager should keep the note's other three lines intact.