The Handshake Dilemma!

Among the five married couples no one shook more than eight hands. Therefore, if nine people each shake a different number of hands, the numbers must be 0, 1, 2, ..., and 8. The person who shook 8 hands has to be married to the person who shook 0 hands (otherwise that person could have shaken only seven hands.) Similarly, the person who shook seven hands is bound to be married to the person who shook 1 hand. So that the married couples shook hands in pairs 8/0, 7/1, 6/2, 5/3. The only person left who shook hands with 4 is the wife.

There were ten people in total. A person could have shaken hands any number of times between {0...8}, for there were four other couples present. Since there were nine different answers, all of those scenarios did happen; so there is a person who shook hands with no one, a person who shook hands with only one other person, and so on. We can assign every person a number which indicates how many times they shook hands with other people.

Now, the party consisted of people {[0], [1], [2], [3], [4], [5], [6], [7], [8], [X]}, where [X] is the narrator.

The person who shook hands [8] times did so with everyone except for themselves and their partner. Therefore everyone except [8]'s partner has already shaken hands with someone. This leaves it clear that [8]'s partner must be the one who shook hands [0] times.

How about the person who shook hands [7] times? They didn't shake hands with [0], themselves or their partner. They also couldn't have shaken hands with the person who only shook hands once, for [8] already shook their hand. From here we can deduce that [7]'s partner must be [1]. This also means that [7] must have shaken [2]'s hand and [2] cannot have shaken any more hands.

Similarly we continue to observe the person who shook hands [6] times. Once again, they couldn't have shaken [0]'s, [1]'s, [2]'s or their own hand. This leaves [2] as [6]'s partner. The person who shook hands [5] times didn't shake hands with [0], [1], [2], [3] or themselves which leaves [3] as their partner.

The only one left is the person who shook hands [4] times, who by elimination has to be the narrator's wife. (Bonus: The narrator shook hands with exactly [4] people as well.)

I used common sense.

Have you ever been to a party ? In a party you don't just go around shaking hands with people you don't know !

When you know someone, and that person introduces you to someone new, that's when you're supposed to shake hands.

There are 4 other couples and the wife must be knowing one person in each couple and thus shakes hands with the other person of that particular couple.

Hence, 4 handshakes.