Saturday Interview Slots
A community radio station in Sligo is scheduling four half-hour interviews for its Saturday morning programme, in the slots 9:00, 9:30, 10:00 and 10:30. The guests are Bríd, Cian, Dara and Eimear, one per slot. The producer's booking sheet contains four conditions only: (i) Cian must be interviewed before Dara; (ii) Eimear must not take the first slot; (iii) Bríd must be in a slot immediately before or immediately after Dara; (iv) Cian must not take the last slot. Separately, the presenter has promised Dara that she will not be the final guest of the morning, and the producer says that promise adds nothing to the sheet.
1. List every running order that satisfies the four conditions, and state what, if anything, is forced regardless of which order is chosen.
Read the worked answer
Work through the possible positions of the Bríd–Dara adjacent pair, then place Cian and Eimear.
- Pair in slots 1–2. Bríd then Dara: Dara is second, so Cian must precede her in slot 1 - already occupied. Dara then Bríd: Dara is first, so Cian must be before slot 1 - impossible. No orders.
- Pair in slots 2–3. Bríd(2), Dara(3): Cian must be in slot 1, Eimear in 4 → C, B, D, E. Dara(2), Bríd(3): Cian must be in slot 1, Eimear in 4 → C, D, B, E.
- Pair in slots 3–4. Bríd(3), Dara(4): Cian and Eimear fill 1 and 2, and Eimear cannot be first, so Cian(1), Eimear(2) → C, E, B, D. The alternative Eimear(1), Cian(2) is barred by (ii). Dara(3), Bríd(4): Cian and Eimear fill 1 and 2 with Eimear not first → C, E, D, B.
Four valid orders: CBDE, CDBE, CEBD, CEDB.
Forced: Cian takes the 9:00 slot in every solution. Nothing else is forced - each of Bríd, Dara and Eimear occupies different slots across the four orders, and Dara is never last, which is why the presenter's promise is already implied.
2. Show whether condition (iv) adds anything to conditions (i)–(iii), and justify your answer without listing the orders.
Read the worked answer
It adds nothing; it is entailed by (i). Condition (i) says Cian comes before Dara, so Dara occupies a later slot than Cian. For Cian to be in the last slot there would have to be a slot after the last slot for Dara, which is impossible. Therefore Cian can never be last once (i) holds, and (iv) is redundant.
The listing confirms this - all four solutions begin with C - but the entailment is the stronger argument, because it holds for any number of slots and any arrangement satisfying (i).
Practical consequence: the sheet has three independent conditions plus one restatement. If a fifth guest were added, or if (i) were relaxed to allow Cian and Dara in the same hour on different days, (iv) would suddenly start doing work. Redundant constraints are worth keeping only if they guard against a later amendment removing the condition that implied them; otherwise they create a false impression that four separate things can go wrong.
3. The presenter wants a unique running order and proposes adding 'Dara must not be last'. Show that this does not work, and give one condition that does produce a unique order.
Read the worked answer
The proposed condition is already true in all four solutions - Dara is last in none of CBDE, CDBE, CEBD, CEDB. It is in fact another consequence of (i), since Dara must be after Cian and therefore cannot occupy the single last slot. Adding it leaves four valid orders, so the producer is right that the presenter's promise adds nothing.
A condition that does work: 'Bríd must be in the 10:30 slot.' Bríd is last only in CEDB, leaving one solution.
Other single conditions that also pin it down: 'Dara must be in the 9:30 slot' (CDBE only); 'Eimear must be in the 10:30 slot' (CBDE or CDBE - not unique, so this one fails); 'Bríd must be in the 10:00 slot' (CDBE or CEBD - also fails).
The general lesson: to check a proposed constraint, test it against the full solution set. A constraint that removes exactly three of four solutions gives uniqueness; one that removes none, or one, is either redundant or insufficient, and intuition about what 'feels' restrictive is an unreliable guide.