Many members of the club disliked the lack of variety and togetherness at the club. Although the club still had 12 members, some members were threatening to quit because each schedule was so short and there were so few people around each table.

To satisfy their request, the club decided to seat themselves around a big table and create a longer schedule. The twelve members of the club seated themselves in a schedule such that during each block of 55 days, no person was between the same pair of people. How was the schedule constructed?

(Based on The Round Table)

There is an  upper bound to the number of days ("arrangements") that can exist for a given number of people. A simple proof will show that for N people, you can have at most (N-1)(N-2)/2 seatings (exercise for reader). That means for 6 people, you can have at most 10 arrangements and for 12 people, 55.

However, whether there exists such an arrangement is another matter! I've written a program to do an exhaustive search for the full maximal schedule given a number of people (will post source code over the weekend). It successfully finds the 10 arrangement schedule for 6 but completes the exhaustive search for 7 people and 15 arrangements with no solution. I.e. I believe you cannot find 15 arrangements for 7 people.

The search took several minutes to run on my laptop for 7 people but it would probably take millenia to search for 55 on 12 (since this is an exponentially growing search space). However, I'd be well impressed if anyone can find a 15 seating arrangements for the 7 member conversing club

Posted by Glorat on 2004-08-12 05:43:29