# Finding number of handshakes

1. Dec 24, 2012

### SithsNGiggles

1. The problem statement, all variables and given/known data
A couple invites $n$ couples to a party. Upon arriving, some people shake hands with each other and some do not, but nobody shakes hands with one's own spouse or with oneself. After all the guests have arrived, the hostess asks each of her guests as well has her husband how many individuals the person shook hands with. Amazingly, she comes up with $2n + 1$ different numbers. The problem now is this: with how many people did the hostess shake hands, and with how many people did the host shake hands?

(Suggestion: Work this out first for $n = 3$ and then $n = 4$, and then find a general pattern that works for an arbitrary positive integer $n$. You will need to prove that it does indeed work.)

2. Relevant equations

3. The attempt at a solution
Not much was done on my part; I don't know how to approach this. The suggestion as to "work it out for $n=...$" is over my head. So far, I wrote that there are $n+1$ total couples (including the host), so there are $2(n+1) = 2n+2$ individuals. I don't know where to go from here.

Any ideas? As always, much appreciated.

2. Dec 24, 2012