Optimum Seating Arrangements and Tuscan Squares

Abstract

It may be desired to seat \(n\) people along a row (as at a lunch counter), or \(n+1\) people around a circular table, in \(n\) consecutive rounds of seating, so that each person \(x\) has every other person \(y\) on their right exactly once, and on their left exactly once, in one of the seatings. Alternatively, it may be desired to seat \(2n\) people along a row, or \(2n + 1\) people around a circular table, in only \(n\) consecutive rounds, so that each person \(x\) is adjacent to every other person \(y\) (either on the right or the left) exactly once. We show that these problems are solved using the rows of Tuscan squares to specify the successive rounds of seatings.