Hasan and Lauren attended a dinner party with 4 other couples. Since some people already knew some of the other guests, every person at the dinner party shook hands with every person they had not met before.

Lauren noticed that everyone else (excluding Lauren herself) ended up with a different number of handshakes!

Can you figure out how many people Hasan shook hands with?

#### Solution

Yes, Hasan shook hands with 4 people.

There were 9 people other than Lauren, and each person had to have met one other person already (since each person was part of a couple). So if everyone except Lauren had a different number of handshakes, that means there were people that shook hands 0, 1, 2, 3, 4, 5, 6, 7, and 8 times – let’s call them Person 0 through Person 8 based on how many people they shook hands with.

- Person 8 shook hands with every person except their partner. That means Person 0 is their partner, since everyone else must have shaken hands with at least one person (Person 8).
- Person 7 shook hands with every person except their partner and Person 0. That means Person 1 is their partner, since everyone else must have shaken hands with at least two people (Person 8 and Person 7).
- Continuing this logic, we know Person 6 and Person 2 must be a couple, Person 5 and Person 3 must be a couple, and Person 4 and another Person 4 must be a couple.
- But since Lauren noticed everyone else had a different number of handshakes, Lauren must be one of the Person 4, and so Hasan must be the other Person 4.