Rescheduling the Inter-House Quiz
St Xavier's School holds an inter-house quiz with five rounds: A (Science), B (Mathematics), C (Literature), D (Current Affairs) and E (Logical Reasoning). Each round is held in a different one-hour slot on the same day: 9:00, 10:00, 11:00, 12:00 and 1:00. The notice board lists four rules: (i) Round A is held immediately before Round D. (ii) Round B is held at 10:00 or at 12:00. (iii) Round C is held after Round D but not in the last slot. (iv) Round E is not held in the slot immediately before or immediately after Round A. The quiz coordinator has already printed the hall passes when the judge for Round E telephones to say that, because of a train delay, she can reach school only by 12:00. The coordinator wants to keep every rule on the printed notice unchanged and simply reshuffle the rounds. A member of the student council meanwhile claims that Rule (iv) is the one that is 'doing no work anyway'.
1. Using rules (i) to (iv), determine the full schedule, showing why each alternative is eliminated.
Read the worked answer
Number the slots 1 to 5 (9:00, 10:00, 11:00, 12:00, 1:00).
Step 1 - place A and D. They occupy consecutive slots. If A, D were at 3, 4 or at 4, 5, then C must come after D, but the only slot after D is 5 (forbidden by rule iii) or none at all. So A, D must be at 1, 2 or at 2, 3.
Step 2 - test A, D at 2, 3. C must then be at 4. The remaining slots for B and E are 1 and 5, but rule (ii) forces B into 2 or 4, both already taken. Impossible.
Step 3 - so A = 1, D = 2. C is after D and not last, so C is at 3 or 4.
Step 4 - test C at 4. Remaining slots for B and E are 3 and 5; B needs 2 or 4, both taken. Impossible.
Therefore C = 3, B = 4, E = 5.
Schedule: 9:00 A (Science), 10:00 D (Current Affairs), 11:00 C (Literature), 12:00 B (Mathematics), 1:00 E (Logical Reasoning). Check rule (iv): E at 5 is not adjacent to A at 1.
2. The judge for Round E can arrive only by 12:00, so E must move to the 12:00 slot. Keeping all four rules unchanged, is any valid schedule possible? Justify your answer.
Read the worked answer
No valid schedule exists under the unchanged rules.
Put E at slot 4 (12:00). Rule (ii) requires B at slot 2 or slot 4; slot 4 is now taken by E, so B must be at slot 2 (10:00).
The remaining slots for A, C and D are 1, 3 and 5. Rule (i) demands that A and D be in consecutive slots, and none of 1, 3, 5 is adjacent to another of them - 1 and 3 are separated by 2, and 3 and 5 by 4. So A and D cannot be placed at all.
The coordinator's plan of 'just reshuffling' therefore cannot work. Something on the notice must give way: the simplest fixes are to allow B at a non-even hour, or to run Round E at 1:00 and have the judge's session recorded, or to swap the E round to a different day. Recognising that a constraint set is unsatisfiable, rather than forcing a wrong timetable, is the point of the question.
3. The council member says Rule (iv) is 'doing no work'. Is Rule (iv) redundant? Justify your answer by checking whether the unique schedule depends on it.
Read the worked answer
Yes, Rule (iv) is redundant - but it is also consistent.
Rework the deduction using only (i), (ii) and (iii):
- A, D must be at 1, 2 (the placements at 3, 4 and 4, 5 leave C nowhere legal, and 2, 3 leaves B with no even slot).
- C cannot be at 4 for the same reason, so C = 3, B = 4, E = 5.
So rules (i)–(iii) already force a single arrangement, and in that arrangement E at slot 5 is nowhere near A at slot 1 - the condition in (iv) is satisfied automatically. Removing (iv) admits no extra timetable.
A rule earns its place only if dropping it would allow a second arrangement. Here it adds no restriction, so it could be deleted from the notice without any change to the quiz - a useful test of whether an added condition is actually carrying weight.