Blogtrek

Sitting Next to your Preferred One

I recently came up with an interesting mathematical puzzle today. It is based on some of these conventions and dinner parties that I go to. The problem is this. There is a big banquet at the end of the convention. There is a special someone that you want to sit next to. Perhaps you're in love with this person. Perhaps this person has the key to your next job. Perhaps this person is simply someone you like to sit next to. We will call this person Connie (for "convention").

Most typical banquets feature some 10-20 tables seating 10 each, for a total of 100-200 banqueters. You know that Connie is going to appear at this banquet. So when do you sit down? You don't want to be the first to sit down, as then you have no control over who sits next to you. It's whoever wants to sit next to you, and that might not be Connie. You don't want to be the last person, either. Then there will only one seat left, and that is then yours, and you have no control over who is sitting there either.

The optimal solution is to sit somewhere in the middle, but where? I will measure this by the percentage of people who have seated by the time you choose your seat.

To illustrate this problem, I will assume that everyone is seated in one big circle instead, arbitrarily large. Then you want to get into one of the two seats next to Connie. Suppose you choose time x to sit down. Then the probability is x that Connie is already seated. In that case, the probability that you can sit next to her is 1-x², as x² is the probability that both seats are occupied (for an infinite circle - for a finite set, replacement needs to be taken into account). So this makes a term x(1-x²). In the 1-x chance that Connie is not seated, the probability is near zero that she will sit next to you (unless she is attracted to you, in which case it is certain she will sit next to you, but I am not assuming that is the case; also this assumes an infinite circle or strip of seats). Therefore the probability of getting a seat next to her is x(1-x²). The method here is to differentiate and set equal to zero, and solve for x. When you do that, you get the optimal x to be the square root of 1/3, which is about 0.577. So this mean you will wait until 57% of the people are seated, then you will go in and sit down.

This would make for an interesting problem to work out. I have assumed an infinite strip, but what happens in finite cases? The size of the tables makes a difference; if there are 10 to a table, then Connie can sit anywhere at that table and you can still get to her table. Some seats may be more valuable than others; e.g., next to her rather than across the table from her. And what if you want to sit next to a group of people, and what if the people them cluster into groups or cliques? And once someone works out all those cases, are they prepared to use it at a real banquet?

Stay tuned.