Five Slots, Four Rules: Scheduling the Science Exhibition
Five project teams - P, Q, R, S and T - must present in five consecutive 20-minute slots beginning at 9:00 a.m., that is, at 9:00, 9:20, 9:40, 10:00 and 10:20.
The organising committee has fixed these rules, and no rule may be broken:
- Exactly two presentations lie between P and T (so P and T are separated by three slots).
- Q presents immediately before R.
- S does not present in the last slot.
- R does not present first.
The student secretary must publish the running order on the notice board tonight. A teacher asks her whether all four rules are really needed, and warns her that a sixth team may withdraw or be replaced at short notice, so she should know how the schedule behaves if the rules are altered.
1. Determine the complete running order and show the deduction, including why exactly one arrangement survives.
Read the worked answer
Final order: S, P, Q, R, T (S at 9:00, T at 10:20).
Step 1 - place P and T. Rule 1 requires three slots between them, so within five slots the only pairs are (1, 4) or (2, 5).
Step 2 - test P = 1, T = 4. The free slots are 2, 3 and 5. Rule 2 needs Q and R in consecutive slots; the only consecutive free pair is (2, 3), forcing Q = 2, R = 3 and leaving S = 5. But rule 3 forbids S in the last slot. This branch dies.
Step 3 - test P = 2, T = 5. The free slots are 1, 3 and 4. The only consecutive pair is (3, 4), so Q = 3, R = 4, and S takes the remaining slot, 1. Rule 3 is satisfied (S is first) and rule 4 is satisfied (R is fourth).
Step 4 - uniqueness. Every alternative was eliminated by a stated rule, so only one order exists. Reading the times: S at 9:00, P at 9:20, Q at 9:40, R at 10:00, T at 10:20.
2. The teacher asks: "Is every one of the four rules actually doing work?" Identify the rule that can be deleted without changing the unique schedule, and justify your answer by showing where R can possibly sit.
Read the worked answer
Rule 4 (R does not present first) is redundant.
To see this, look only at rules 1 and 2, which independently constrain R:
- If P = 1 and T = 4, the free slots are 2, 3, 5. Q and R must be consecutive, and the only consecutive free pair is 2–3, so R is forced to slot 3.
- If P = 2 and T = 5, the free slots are 1, 3, 4. The only consecutive free pair is 3–4, so R is forced to slot 4.
In both branches R lands at slot 3 or slot 4. Rule 4 says R cannot be first - a situation that rules 1 and 2 already make impossible. The rule is therefore true but uninformative; deleting it leaves the same unique schedule.
By contrast, rules 1, 2 and 3 are all load-bearing: remove rule 1 and P, T become unconstrained; remove rule 2 and Q, R no longer occupy a fixed adjacent pair; remove rule 3 and the branch P = 1, T = 4 with S last becomes legal, giving two schedules instead of one.
The practical lesson is that a redundant constraint is not harmless: it wastes checking time and can mislead a solver into thinking it carries information about the answer.
3. Rule 1 is relaxed to "at least two presentations lie between P and T". All other rules stay. How many valid running orders now exist? List them and show why no others are possible.
Read the worked answer
Three valid orders.
"At least two between" means the gap may be two or three presentations, so P and T may now sit at (1, 4), (1, 5) or (2, 5).
Case (1, 4): free slots 2, 3, 5. Q and R must be consecutive, so Q = 2, R = 3, leaving S = 5 - barred by rule 3. No order.
Case (1, 5): free slots 2, 3, 4. Q and R consecutive can be (2, 3) or (3, 4):
- Q = 2, R = 3 → S = 4 → P, Q, R, S, T
- Q = 3, R = 4 → S = 2 → P, S, Q, R, T
Case (2, 5): free slots 1, 3, 4. Only consecutive pair is (3, 4), so Q = 3, R = 4, S = 1 → S, P, Q, R, T.
All three satisfy rules 2, 3 and 4 (S is never last; R is third, third and fourth respectively). No other arrangements exist because these are all possible placements of the P–T pair and, in each, all valid placements of the Q–R block have been exhausted.
Note the consequence: relaxing one constraint turned a single publishable schedule into three, which is exactly the situation the secretary needed to anticipate before printing the notice board.