Seating Arrangement

The frame

Number the seats 1 to n from the west (left) end for a north-facing row. Write the numbers under the boxes. Every clue then becomes a small equation on seat numbers, which is far safer than moving letters around in your head.

For north-facing people:

• "B is k-th to the right of A" → B = A + k.
• "B is k-th to the left of A" → B = A − k.
• "A is at an end" → A = 1 or A = n.
• "p-th from the right end" → seat n − p + 1.

For south-facing people the signs flip: "k-th to the right" → B = A − k.

Order of attack

1. Absolute clues (ends, middle, "third from the left") go in first.
2. Distance clues attached to a placed person come next. "D is fourth to the right of C" with C at seat 1 gives D = 5 straight away.
3. Floating chains. If a clue links two people who are not yet placed, write the block (for example "P _ Q") on the side. Then slide it along the free seats to see where it fits.
4. Negative clues last. "Not at an end", "not a neighbour of", "not adjacent to" usually leave exactly one case standing.

Worked example

Seven people C, A, B, D, E, F, G sit in a row facing north. C sits at the left end. F is fourth to the right of C. A is second to the left of F. D sits immediately right of F. G is not at an end. B is a neighbour of C.

• C = 1, F = 1 + 4 = 5, A = 5 − 2 = 3, D = 6.
• B neighbours C, so B = 2.
• Seats 4 and 7 remain for E and G. G is not at an end, so G = 4 and E = 7.
• Row: C B A G F D E. Third to the right of A is seat 6 = D.

Case splitting

When a clue allows two places ("A sits at one of the ends"), draw two rows side by side, one per case, and carry both until a clue breaks one. Never guess one case and hope. The typical exam puzzle leaves 2 cases after the first few clues and one negative clue settles it.

Blocks and gaps

• "X sits immediately between Y and Z" makes a three-seat block, but Y and Z can be on either side. Keep both orders open unless another clue fixes them.
• "Exactly two people sit between A and B" means |A − B| = 3. It does not tell you who is on the left.
• A block of size k in a row of n has n − k + 1 possible starting seats. List them quickly and strike out the ones that clash.

Facing south

If everyone faces south, you can either use the flipped equations or draw the row from east to west so that the person's left becomes your left again. Choose one method and stick to it. Mixing them in the middle of a puzzle is the commonest source of a mirror-image answer.

Negative clues work hard

"D is not a neighbour of A" looks weak, but in a nearly-full row it often removes a whole case. When two seats and two people are left, a single "not at an end" decides both.

Answering the question

Read the question twice. "Second to the right of W" in a north-facing row is seat W + 2. "How many people sit between F and E" is |F − E| − 1, not |F − E|. The number of seats between two people and the number of people between them are the same only when the row is full.

Quick revision

• North-facing: right = +k, left = −k on seat numbers. South-facing: signs flip.
• Absolute clues → attached distances → floating blocks → negative clues.
• Two possible places = two drawn rows; strike one out with a later clue.
• "Immediately between" = a 3-block with the ends swappable.
• People between = |a − b| − 1.