10/3/08

The Bises Problem

So when I idly posed the following math problem in my last post, I wasn't really thinking about it, but it's actually a little bit interesting. 

The Question: If a party of nine people is breaking up, in which there are five girls, four guys, and no relations, how many kisses will be exchanged?

The Assumptions: Any pairing involving a girl (that is, girl-girl or girl-guy) will result in two kisses (one on each cheek). Since none of the guys are related, any pairing of two guys will result in no kisses.

To warm up, let's consider the Handshake Problem: If a roomful of n people all shake hands with one another, how many handshakes will be exchanged? For simplicity's sake, let's say n in this case is 10. That means each person in the room shakes hands with 9 other people, so you might be tempted to multiply 10 by 9 and arrive at 90 handshakes. Actually, though, you've double-counted each handshake, since A shakes hands with B and B shakes hands with A, but that's only one handshake total. So you divide by 2 and arrive at 45 handshakes. Or, more generally, in a room with n people, you will have [n*(n - 1)]/2 hanshakes.

My problem is a little more complicated, since the guys don't faire les bises among themselves and instead of one handshake, we have 2 bises. If you look at the picture (sorry, it's not the most beautiful graph ever, since I drew it by hand and photographed it with my webcam, but it will have to do), where G = girl, B = boy, and each line = 2 bises, all the girls are connected to everyone else but the boys are only connected to the girls. For the moment, let's pretend kisses are like handshakes, i.e. one between two people. All you have to do is calculate the number of kisses for a normal group of 9 and then subtract the number that aren't being exchanged by the group of 4 boys, in other words:

(9*8)/2 - (4*3)/2 = 36 - 6 = 30

Now, remembering that each exchange of bises actually involves 2 kisses, we multiply by 2 to arrive at 60 total kisses in the above scenario. In other words, Elizabeth wins!

(Is it glaringly apparent how much I miss math?)