Handshake ProblemDate: 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/ |
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