The Seating Puzzle That Refuses to Settle
Meera is 11 minutes from the end of a 35-question Reasoning Ability sectional. She has four standalone one-mark questions untouched and one five-question seating puzzle half-finished. There is no negative marking. The puzzle: five analysts P, Q, R, S and T sit in chairs numbered 1 to 5 from left to right, all facing north. The clues she has copied are: (i) P does not sit at either end; (ii) Q sits immediately to the right of R; (iii) S sits immediately to the left of P; (iv) T sits at one of the ends. Meera tested clue (i) first, assumed the set must have exactly one solution, and after three minutes has two rival arrangements on her rough sheet. She is about to pick the one that 'looks neater', answer all five questions from it, and move on - which she estimates will take four minutes. Each of the four standalone questions would take roughly 90 seconds.
1. Enumerate every arrangement of P, Q, R, S and T that satisfies clues (i) to (iv). How many are there, and what does the count tell Meera?
Read the worked answer
Anchor on the pair R–Q, because clue (ii) makes them a single block that can sit at (1,2), (2,3), (3,4) or (4,5).
- Block at (1,2): seats 3, 4, 5 remain. P cannot be at an end, so P is 3 or 4. If P = 3, then S must be at 2, which Q occupies - reject. So P = 4, S = 3, T = 5. T is at an end. R Q S P T
- Block at (2,3): seats 1, 4, 5 remain. P can only be 4, and S would then need seat 3, occupied by Q - reject.
- Block at (3,4): seats 1, 2, 5 remain. P must be 2, so S = 1 and T = 5 (an end). S P R Q T
- Block at (4,5): seats 1, 2, 3 remain. If P = 2, S = 1 and T = 3, which is not an end - reject. If P = 3, S = 2 and T = 1 (an end). T S P R Q
Exactly three arrangements survive: R Q S P T, S P R Q T and T S P R Q.
The lesson is diagnostic, not arithmetic. A well-set exam puzzle is built to have one solution; a set that collapses to three interchangeable arrangements usually means Meera has mis-read or mis-copied a clue. She should re-read the clues against the printed page for 20 seconds, and if nothing is amiss, treat the item as defective and stop investing in it.
2. Should Meera answer all five puzzle questions from one arrangement? Justify the decision numerically and state what she should actually do with the 11 minutes.
Read the worked answer
Three arrangements are equally consistent with the printed clues and the paper gives her no way to rank them, so picking one is a one-in-three guess. Expected marks from five questions = 5 × (1/3) ≈ 1.67. The four standalone questions are near-certain marks if attempted: 4 questions at about 90 seconds each is 6 minutes for close to 4 marks.
Her plan is therefore upside-down. She is proposing to spend 4 minutes to secure an expected 1.67 marks while four easy marks sit untouched, and she has 11 minutes - enough for both. Correct sequence:
- Take the four standalone questions first (about 6 minutes). That is roughly 2.4 times the expected yield of the puzzle for 1.5 times the time.
- Return to the puzzle with about 5 minutes, re-read the clues for a transcription error, and only then fill the five options.
Two habits are worth extracting. First, an under-determined puzzle is a signal to audit the clues, not to beautify the answer key. Second, 'neatness' has no evidentiary value - in a linear puzzle every valid arrangement is equally tidy, so a preference for the symmetrical-looking one is pure noise. Choosing on noise converts a solvable puzzle into a coin toss.
3. Meera later recalls that the printed puzzle also fixed one person's exact chair, but the word is smudged. Using the three valid arrangements, state whose exact chair would have made the puzzle unique and whose would not, and explain why.
Read the worked answer
Tabulate the three surviving arrangements by seat number:
| Seat | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| A | R | Q | S | P | T |
| B | S | P | R | Q | T |
| C | T | S | P | R | Q |
Read down each column:
- R sits at 1, 3, 4 - all different. Fixing R's seat identifies the arrangement.
- Q sits at 2, 4, 5 - all different. Fixing Q's seat works.
- P sits at 4, 2, 3 - all different. Fixing P's seat works.
- S sits at 3, 1, 2 - all different. Fixing S's seat works.
- T sits at 5, 5, 1 - the same in A and B. A clue about T narrows the field to two arrangements but cannot separate A from B.
So any absolute clue naming the chair of R, Q, P or S restores a unique solution; a clue about T does not.
The general point: the printed clues are purely relational and pin the people into a chain, but the chain can still slide between three positions. One absolute anchor is enough to lock it - provided the anchor falls on a person whose position actually varies across the candidate solution set. Meera should test any extra clue against the surviving arrangements before trusting it, rather than assuming 'any additional clue will do'.