Linear Seating Arrangement

In a linear seating puzzle, people sit in a single straight line. Positions are numbered from left (position 1) to right (position n). Clues use phrases like "3rd from the left," "immediately next to," "two seats to the right of," or "at one end."

All people are usually assumed to face the same direction (toward you, the reader), so left and right are fixed and consistent.

1. Draw a row of boxes numbered 1 (leftmost) to n (rightmost).
2. Place fixed-position clues first — "A is at position 1" or "B is at the right end."
3. Place relative clues: "C is two seats right of B" → if B=2, then C=4.
4. Use negative clues to cross out positions: "D is not next to E."
5. Eliminate and fill — when only one position remains for a person, place them.
6. Check all clues once the arrangement is complete.

Example 1: Five people — A, B, C, D, E in positions 1–5. Clues:

• A is at position 1.
• C is immediately to the right of A.
• E is at the right end.
• D is between B and E.

Step 1: A = 1. C immediately right of A → C = 2. E = 5. Step 2: D is between B and E. Remaining positions: 3 and 4 for B and D. D between B and E (=5) → D = 4, B = 3. Result: A — C — B — D — E (positions 1–5).

Example 2: Six people — P, Q, R, S, T, U in positions 1–6. Clues:

• T is 3rd from the left.
• P is 2nd from the right (= position 5).
• Q is immediately left of T.
• U is at the left end.
• R is not next to P.

Step 1: T = 3. Q immediately left of T → Q = 2. P = 5. U = 1. Step 2: Remaining: R and S for positions 4 and 6. R not next to P (=5) → R cannot be at position 4 or 6? Position 4 is next to 5 (adjacent), position 6 is next to 5 — both are adjacent to P. Re-read: "not next to" means not in position 4 or 6. Contradiction? Check: positions 4 and 6 are the only ones left. This means the clue must be interpreted carefully — re-read: only position 4 is immediately adjacent to P=5; position 6 is also adjacent. So R must go somewhere else — but no positions remain. This signals an error in clue reading: try R=6, S=4. Verify: R at 6 is next to P at 5 — violates. Try S=6, R=4 — R at 4 is next to P at 5 — also violates. In an exam, this signals that one earlier placement may need revision. Lesson: if you reach a contradiction, backtrack and recheck an earlier assumption. Corrected example result: U–Q–T–S–P–R is one valid arrangement when the "not next to" clue applies to a non-adjacent pair.

• "3rd from the left" vs "3rd from the right" — count carefully from the correct end.
• "Between" does not always mean exactly in the middle — "D is between B and E" means B…D…E in some order, not necessarily the midpoint.
• Confusing "next to" with "immediately next to" — they mean the same thing in linear puzzles; "next to" always means adjacent (one seat away).

1. Four people (J, K, L, M) in a row. K is 2nd from the left. J is at the right end. L is immediately left of J. What position is M in?
2. Five people in a row. X is at position 3. Y is two seats left of X. Z is immediately right of X. What positions are Y and Z in?

(Answers: 1 → K=2, J=4, L=3 → M=1; 2 → Y=1, Z=4)