Seating Arrangement

What a seating puzzle is

You get a group of people and a list of clues about where they sit. Your job is to rebuild the one seating that fits every clue, then answer questions such as "who sits second to the right of D?" or "how many people sit between P and T?". There is no trick of calculation here. The whole skill is reading each clue correctly and drawing it the same way every time.

Left and right depend on the direction you face

This is the source of most wrong answers.

• Facing north (the default in a row): your left is the west end of the row, your right is the east end. Draw the row left to right on paper and the person's left is simply your left.
• Facing south: everything flips. The person's left is the east end, their right is the west end. If you draw the row as usual, a south-facing person's "right" is to your left on paper.
• Rule of thumb: if the question says "facing south" or "facing away from the viewer", draw a small down-arrow on every seat and read left and right from that arrow, not from your own hands.

Circles

• Facing the centre: left = clockwise, right = anticlockwise.
• Facing outward: left = anticlockwise, right = clockwise.
• A circle has no ends and no fixed first seat. Place the first named person anywhere; every other seat is fixed relative to that one.
• "Opposite" exists only when the number of seats is even. With n seats, the person opposite is n ÷ 2 places away in either direction.

Counting words

• "Second to the right of X" means one person sits between X and that seat.
• "k-th to the left" means k steps, so there are k − 1 people in between.
• "Immediately between X and Y" means X and Y are the two neighbours of the middle person.
• "Between X and Y" without "immediately" means somewhere strictly inside the stretch from X to Y.
• Position from the ends: in a row of n people, the person p-th from the left is (n − p + 1)-th from the right.
• People between two seats at positions a and b = |a − b| − 1.

Worked example

Five people sit in a row facing north. T sits in the middle. U sits second to the right of T. V sits to the immediate left of T.

• Seats 1 to 5 from left: T = 3.
• U = 3 + 2 = 5 (right end). V = 2.
• Two seats remain: 1 and 4. If the question adds "W is not at an end", then W = 4 and X = 1.

Now turn the same row to face south. T is still in seat 3. U, "second to the right" of a south-facing T, is two seats to the west, so U = 1. V, "immediately left" of T, is at seat 4. One word changed and the whole answer changed.

The three-step method

1. Fix the frame. Draw the seats first: a line of n boxes, or a circle with n marks. Write the facing direction on the frame.
2. Place the sure clues first. Ends, the middle seat, "k-th from the left end". These need no guessing.
3. Attach the chains. "A is left of B, B is second right of C" — hang each new person on someone already placed. Use negative clues ("not at an end", "not a neighbour of D") to kill the extra cases at the end.

Quick revision

• North-facing row: person's left = your left. South-facing row: person's left = your right.
• Circle facing centre: left = clockwise. Facing outward: left = anticlockwise.
• k-th to the right means k − 1 people in between.
• p-th from left in a row of n = (n − p + 1)-th from right; people between seats a and b = |a − b| − 1.
• Draw the frame, place fixed clues, hang chains, then use the "not" clues.