Handshake Problem

Date: 05/07/97 at 12:44:02
From: Bushnell-Prairie City Elem. School
Subject: Handshake problem

Dr. Math,

Our 5th grade math class was learning to solve story problems by 
looking for a pattern and setting up a chart. Most of the problems 
were of the nature: If you had 8 people in a group and each one had 
to shake everyone else's hand one, how many handshakes would take 
place?   

After working through several problems, one boy in the room said "I 
can get it faster without making a chart. If you multiply whatever 
the number is by one less, then divide by 2, you get the right answer 
each time."  

Will this always work?  Why?

Mitchell C. and Mrs. Holland

Date: 05/07/97 at 16:46:32
From: Doctor Anthony
Subject: Re: Handshake problem

Yes, it will always work.  The boy concerned was very observant.  A 
good way to model the situation is to think of an n-sided polygon, 
which has n vertices.

Now consider how many diagonals you can draw between the vertices, and 
also include two lines connecting a particular vertex to the two 
adjacent ones. It is clear that there are (n-1) lines joining any one 
vertex to the other vertices in the polygon. If we consider each of 
the n vertices, each requires (n-1) lines to join to the other 
vertices. There are thus n(n-1) links. But each of these links has 
been produced twice, once from each end, and so the number n(n-1) is 
too large by a factor of 2.

Hence total number of connections joining every vertex to every other 
one is:

   n(n-1)
   ------
     2

This is the result that the boy discovered by observation.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/