The Volunteer's Single Seating Plan
Five colleagues - A, B, C, D and E - are to be seated in five numbered chairs in a row, 1 (far left) to 5 (far right). The organiser has circulated only four rules: (i) A will not sit at either end; (ii) B sits immediately to the right of C; (iii) D sits somewhere to the left of A; (iv) E is not adjacent to A. A junior volunteer claims the rules are enough to fix the arrangement and starts drafting a single seating plan. Before that plan is finalised, the organiser wants to know how many arrangements actually satisfy all four rules, what is forced in every valid arrangement, and what extra fact would pin the arrangement to one plan. No rule has been overlooked, and every chair is used.
1. How many arrangements satisfy all four rules? Show the elimination, do not guess.
Read the worked answer
Rule (ii) makes CB a block, which can occupy chairs (1,2), (2,3), (3,4) or (4,5).
- CB at (1,2): A must be 3 or 4. If A = 3, D has no chair to the left of A; if A = 4, then D = 3 and E = 5, but E then touches A. Dead.
- CB at (2,3): A cannot be 1 or 5 and cannot be 2 or 3, so A = 4; then D = 1 and E = 5, and E touches A. Dead.
- CB at (3,4): A must be 2 (1 and 5 banned, 3 and 4 taken), so D = 1 and E = 5. E does not touch A. D A C B E survives.
- CB at (4,5): A is 2 or 3. A = 2 forces D = 1, E = 3, and E touches A. A = 3 forces D and E into chairs 1 and 2; taking E = 1 avoids the adjacency, giving E D A C B. Survives.
Exactly two arrangements work, so the volunteer's 'single plan' is unsafe.
2. What is forced in every valid arrangement? In particular, is the volunteer right that A must occupy chair 3?
Read the worked answer
Comparing the two survivors, E always sits at an end (chair 5 in D A C B E, chair 1 in E D A C B). That is the only obvious invariant, and it is a consequence of the four rules rather than a separate fact.
The claim 'A must be chair 3' is false: A sits in chair 2 in D A C B E. The mistake is a plausible one - in three of the four CB positions A is squeezed toward the middle, and the solver generalises from the first surviving case. The fix is to notice that CB can also take the far-right pair (4,5), which frees chair 3 and lets A slide to 2. A claim that holds in one arrangement is not a deduction; only claims that hold in both qualify.
3. Give one additional fact that would reduce the possibilities to exactly one arrangement, and explain what kind of fact would be useless.
Read the worked answer
Any fact true in exactly one of the two survivors works. Examples:
- 'D sits in chair 2' - true only in E D A C B, so the answer is forced to E D A C B.
- 'E sits to the left of D' - same effect.
- 'B sits in chair 4' - true only in D A C B E, forcing that plan.
Useless facts are those already implied by the rules, or true in both survivors. For instance, 'C sits somewhere to the right of A' is true in D A C B E (A 2, C 3) and in E D A C B (A 3, C 4), so it adds nothing. Equally, 'E sits in an end chair' is already a consequence of the rules. A useful extra fact must discriminate between the surviving cases, not restate the givens.