Seating Arrangements

A seating arrangement problem gives a set of clues about where several people sit — relative to a row's ends, relative to each other ("immediately to the left of," "second from the right," "between two named people") — and asks for the complete seating order, or for one person's exact position.

Linear arrangements. For people seated in a single row, number the seats from left to right (1, 2, 3, and so on). Fix the clue that pins down a definite position first (an end of the row, or "third from the left"), then work through the remaining clues one at a time, filling in a position whenever a clue makes it certain. If a clue is genuinely ambiguous — meaning more than one final arrangement fits ALL the given clues — the honest answer is that the arrangement cannot be uniquely determined, not a guess.

Circular arrangements. For people seated around a round table, there is an extra subtlety: whether the people are described as facing the CENTRE of the table or facing OUTWARD/AWAY from it changes which physical direction counts as a person's own "left" and "right." The standard convention used in this chapter is: when people are seated facing the centre, moving to a person's IMMEDIATE RIGHT means moving one seat in the CLOCKWISE direction (as the arrangement would be viewed from above); moving to a person's immediate left means moving one seat ANTICLOCKWISE. (If the people instead face outward/away from the centre, this rule reverses.) The safest method is to list the seats in clockwise order first, place people according to the clues one at a time, and only then read off any "immediate left/right" relationship from the completed clockwise list. …