10 handshakes were exchanged at the end of a party. Assuming that everyone shook hands with everyone else at the party, how many people were at the party?

I saw the above problem posted in set of problems and it caught my interest. My approach was guess and check. I used pictures to help organize my thoughts.

What if there were only 2 people at the party? Naturally, they would only shake hands with each other and only 1 handshake would occur. We can represent this with a picture:

picture 1

For those of you who remember my brainteaser post, this is a graph! The dots represent the people at the party and the line between them represents a handshake.

Ok, let’s try a more interesting party, 3 people. Person 1 and 2 will shake hands, person 1 and 3 will shake hands, and person 2 and 3 will shake hands. Here is a picture:

picture 2

Clearly 3 handshakes take place at this party. The 3 lines represent this fact.

For a party of 4 people the verbiage is going to get complicated. I am going to let the picture do the talking:

picture 3

And 5:

picture 4

Sweet! We found our answer. We need 5 dots to get a total of 10 lines. Or in the language of the problem, we need a party of 5 people to get 10 handshakes. It is amazing to me that the framework of graph theory can have so many applications. This one of the strengths of mathematics; abstract structures can have a multitude of useful applications.


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