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)

(In reply to get started.....a hint by Ady TZIDON)

From the cyclic set for three:

ABC

it is easy to get to the set of 3 for four by inserting a D at each possible place:

ABCD ABDC ADBC

To these, it's easy enough to add another three to make the set of 6 for five people:

ABCDE AEBDC ADBCE

ABECD ABDEC ADEBC

by just duplicating the original set and placing E between different pairs in each.

But with six people, we are to add only 4 additional rings.  Each time an F is placed between pairs of letters, two arcs of three people are broken, that must then be placed in the four additional rings.  A determination has to be made as to how to keep track of where these can go.  As the number get bigger, and the number of additional rings gets smaller in comparison, it gets trickier.  To make a program, one would have to know either a definite algorithm for placing the new person, or a way to keep track of the needed arcs so that they could be pieced together in the new rings. As the number of such rings grows, there comes the combination problem of the choices of putting which arcs with which other arcs in making them up

Posted by Charlie on 2004-03-29 21:57:58