The Dinner and Dialogue Club has planned a series of small meetings. Each meeting would consist of two or three members enjoying friendly conversation with each other while eating food from different places all over the world. Each member is scheduled to meet exactly four times. No pair of members will meet twice, but some pairs might not meet at all.

The first thing the club did was schedule and arrange the meetings so that each member knew whom to meet and when. When it came to choosing restaurants, someone suggested that each member eat at two restaurants with eastern food, and two with western food (each restaurant is either one or the other). They liked the idea, but to their dismay, the idea was not possible without rearranging at least some of the meetings.

What possible meeting schedule might cause this to happen? How many members are there in this club, at the least?

(In reply to Least number of members ? by Penny)

Hey, it took me a little while to go through all the numbers and draw myself a diagram like I did for my own solution.  Next, it took me a while to prove to myself that your solution works.  And it does!  Congratulations Penny!  Can I see your program?

Your solution is clearly distinct from mine, which has 14 meetings rather than 13.



Not that my explanation of your solution would be any easier to understand, but I always like to make visuals.  I don't think I could easily draw this one in ascii, so I'll just describe it.

When I drew out a diagram, it looks like a five-pointed star (corresponding to members 1,2,5,7,8).  Members 4 and 6 each intersect adjacent points of the star.  Member 3 is an extended diagonal of the pentagon inside the star, and it intersects 4 and 6 in an assymetrical way.

Posted by Tristan on 2005-11-07 19:19:54