The Seating Puzzle That Has No Single Answer
A reasoning trainer gives five officers - P, Q, R, S and T - seated in a single row of five seats, all facing north. Only these clues are given: (i) Q does not sit at either end; (ii) R sits immediately to the right of Q; (iii) T sits at one of the two ends; (iv) P is not adjacent to Q. Two candidates submit different final arrangements. Candidate 1 says the arrangement is T-P-S-Q-R (left to right) and claims it is the only possible one. Candidate 2 says the arrangement is S-P-T-Q-R and also claims uniqueness. The trainer asks you to settle the dispute without adding any clue that is not already written.
1. Test the two claimed arrangements against all four clues, then determine whether the data give a unique arrangement.
Read the worked answer
Check Candidate 1: T-P-S-Q-R (positions 1-5). Q at 4 (not an end) ✓; R at 5, immediately right of Q ✓; T at 1, an end ✓; P at 2, not adjacent to Q at 4 ✓. Valid.
Check Candidate 2: S-P-T-Q-R. Seats: S1, P2, T3, Q4, R5. Clue (iii) fails - T is at 3, not at an end. Invalid.
Is Candidate 1 unique? No. Work the clues systematically. R = Q+1 and Q ∈ {2,3,4}.
- Q=2, R=3: T ∈ {1,5}. P ∉ {1,3}.→ (T1,P4,S5), (T1,P5,S4), (T5,P4,S1). Three solutions.
- Q=3, R=4: P ∉ {2,4}.→ (T1,P5,S2), (T5,P1,S2). Two solutions.
- Q=4, R=5: T=1, P ∉ {3,5} → P=2, S=3 → (T1,P2,S3,Q4,R5). One solution.
Minimum six valid arrangements exist, so the correct conclusion is 'cannot be determined' - Candidate 1 is right about validity but wrong about uniqueness. The trap: a candidate who finds one workable arrangement and stops.
2. What is the smallest additional clue that would make the arrangement unique? Verify it.
Read the worked answer
Proposed clue: 'R sits at the extreme right end.'
Verification: R at seat 5 forces Q to seat 4 (R is immediately right of Q). Q at 4 is not an end ✓. T must then take an end, and seat 5 is occupied, so T = 1. Remaining seats 2 and 3 go to P and S. P cannot be adjacent to Q at 4, which rules out seat 3, so P = 2 and S = 3.
Unique arrangement: T – P – S – Q – R.
Why this clue and not others: a clue that merely repeats information ('Q is not at an end') or narrows without binding ('S sits somewhere left of T') still leaves multiple branches. The efficient clue is one that removes an entire branch of the case tree - here it kills the Q=2 and Q=3 branches outright, leaving a single chain. Testing candidate clues against the branch list before recommending one is the whole skill being examined.