Five Speakers, Five Days, One Inconsistent Proposal
A debating society must schedule exactly one visiting speaker per day, Monday to Friday. Five speakers are available: P, Q, R, S and T. The patron has fixed these conditions: (i) one speaker per day, no day empty; (ii) S speaks exactly two days after P, so P on Monday puts S on Wednesday, P on Tuesday puts S on Thursday, and P on Wednesday puts S on Friday; (iii) R does not speak on Monday or Friday; (iv) T speaks before Q; (v) Q does not speak on Friday. A committee member proposes adding a sixth condition: "Q speaks on the day immediately after R." The patron says the added condition is harmless because it simply makes the week more orderly. Meanwhile two members are arguing about which extra clue would settle the remaining uncertainty, and one has suggested "T speaks on Tuesday" as the tie-breaker.
1. Determine which days can be fixed for P, S and T, and show why the other two speakers remain open.
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Test the three possible placements of P, since condition (ii) allows only three.
If P is Monday, S is Wednesday. Monday and Wednesday are filled, leaving Tuesday, Thursday and Friday for Q, R and T. Condition (iii) puts R on Tuesday or Thursday. If R is Tuesday, Q and T occupy Thursday and Friday; (iv) forces T Thursday and Q Friday, which (v) forbids. If R is Thursday, Q and T occupy Tuesday and Friday; (iv) forces T Tuesday and Q Friday, again forbidden. So P cannot be Monday.
If P is Tuesday, S is Thursday. The remaining days are Monday, Wednesday and Friday for Q, R and T. Condition (iii) puts R on Wednesday. Then Q and T occupy Monday and Friday; (iv) forces T Monday and Q Friday, forbidden. So P cannot be Tuesday either.
Only P on Wednesday, S on Friday survives. The remaining days, Monday, Tuesday and Thursday, go to Q, R and T. R must be Tuesday or Thursday. Either way T is Monday, because if T were placed on Thursday or Tuesday, Q would be pushed to a day forbidden by (iv) or (v). So P is Wednesday, S is Friday, T is Monday, and R and Q trade Tuesday and Thursday.
2. Is the patron's description of the added condition as "harmless" correct? Assess the proposal against the existing five conditions.
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The patron is wrong, and the label "harmless" is the error to catch. A condition that merely narrows the options is harmless; a condition that removes every schedule is fatal, and this one is fatal.
From the first question we know only two schedules exist, and both fix P on Wednesday, S on Friday and T on Monday, with R and Q on Tuesday and Thursday.
Case one: R on Tuesday. "Q immediately after R" puts Q on Wednesday, but Wednesday belongs to P. Contradiction.
Case two: R on Thursday. The rule puts Q on Friday, which condition (v) expressly forbids.
There is no third placement for R. The added condition therefore contradicts the set rather than refining it; the correct response is to report that the six conditions are jointly unsatisfiable, not to drop a speaker or invent a second session. The practical lesson is procedural: before a new constraint is adopted, it should be tested against the schedules the existing constraints already permit, not against an imagined tidier version of the week.
3. Three extra clues are proposed: (a) Q speaks on Thursday; (b) R speaks on Tuesday; (c) T speaks on Tuesday. Which, if added, would fully determine the week, and why?
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Recall the settled core: P Wednesday, S Friday, T Monday, with R and Q alternating between Tuesday and Thursday.
Clue (a), Q on Thursday, does determine the week. Thursday is taken, so R must fall on Tuesday, which is the only remaining slot from the pair. Full schedule: T Monday, R Tuesday, P Wednesday, Q Thursday, S Friday.
Clue (b), R on Tuesday, also determines the week, by the same logic from the other end: Q takes Thursday. Full schedule identical to the one above. So (a) and (b) are two routes to a single outcome, not competitors.
Clue (c), T on Tuesday, determines nothing because it cannot be satisfied. T is already forced to Monday; placing T on Tuesday would leave Monday for R, Q or P, and Monday is forbidden to R and impossible for P under the surviving placement. The proposal is inconsistent, not merely unhelpful.
The useful distinction is between a clue that is informative, one that is redundant, and one that is contradictory. Both (a) and (b) are informative; neither is redundant; (c) is neither, because it is impossible.