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 re: Partial seating arrangement generator by Hugo)

Hugo,

I think your concentric circles are a good idea and I want to work with it to see if I can come up with something.  I'm positive that you need at least three wheels.  The way I see it, you want to keep two numbers next to each other as long as possible.  Minimizing the movement for each session will increase the total possible moves without re-occurance. 

You have to move at least 4 people for each meeting.

One of the first things I tried was to move all of the odd numbers clockwise, which is works out quite similar to your concentric circles.  Then I tried to find patterned sub-permutations for each position.  I consistantly found that I would eventually have a seating where 2 numbers out of the 12 were between two numbers they were previously between(I hope that makes sense, because it really sounded wierd saying it).  The question is, why didn't all 4 end up between two numbers they were previously between (Damn it! I did it again).  This led me to believe that my method was flawed. 

My conclusion:  Moving 6 people at a time eliminated too many future possibilities.

Then I started moving only 4 people at a time using different patterns.  18 days was the best I got before running into an arrangement where only 2 not 4 were between two numbers they were previously between (It's like an addiction; I can't stop doing it).  At that point I would stop.

I decided early on that I'm looking  for a pattern, not random patterns that took me to 55 days.  If any pattern led me to a point where 2 and not 4(I'm not gonna say it), then there must be a better pattern.  I think this is logical. 

Then I considered that 4 is the minimum I can move.  6 is more than I want to move. What about 5?  Then I came up with the star idea. 

5 is one of my ideal numbers.  5*11=55 right? That's what I've really been looking for all along.  A pattern of 5 and 11.  (I don't know why other than my own stupidity)

What I want to do now is take your concentric circles idea and see if I can come up with some way to make 5 circles, with yeah you guessed it, 11 unique sub-permutations.

It will be a few days before I can work on it.  My job is shift work, and I'm on my 4 twelve hour night shift right now.  It's hard to think when you haven't slept.  (The proof:  check my postings.  I've said some pretty dumb things.)

What kills me is that Gamer originally posted this with a difficulty of 3. 

And now for something completely different:  "A moose once bit my sister." I bet you know where that came from.

Posted by Bruce Brantley on 2004-09-20 08:46:09