There were 12 people at the Christmas party.

Quick Scoop: What’s Going On?

This is the classic handshake problem :
Everyone shakes hands with everyone else exactly once , and we know the total number of handshakes is 66.

To find the number of people, we use this idea:

  • Each pair of people produces one handshake.
  • The number of ways to choose 2 people from nnn people is n(n−1)2\frac{n(n-1)}{2}2n(n−1)​.

So we solve:

n(n−1)2=66\frac{n(n-1)}{2}=662n(n−1)​=66

Multiply both sides by 2:

n(n−1)=132n(n-1)=132n(n−1)=132

Now find integers that satisfy this:

n2−n−132=0n^2-n-132=0n2−n−132=0

Factoring:

(n−12)(n+11)=0(n-12)(n+11)=0(n−12)(n+11)=0

So:

  • n=12n=12n=12 or n=−11n=-11n=−11.
  • A negative number of people makes no sense, so we take n =12n=12n=12.

Mini Story Version

Imagine you walk into this Christmas party and decide to count handshakes:

  • When there are 2 people, there’s just 1 handshake.
  • With 3 people, there are 3 handshakes.
  • As the group grows, the number of unique pairs (and thus handshakes) grows much faster.

By the time you reach 12 people, the number of unique pairs—and therefore handshakes—is exactly:

12×112=66\frac{12\times 11}{2}=66212×11​=66

So a party with 12 guests perfectly matches the 66-handshake count.

Key Formula Highlight

If you want to reuse this for any similar riddle:

Number of handshakes when everyone meets everyone once:

Handshakes=n(n−1)2\text{Handshakes}=\frac{n(n-1)}{2}Handshakes=2n(n−1)​

where nnn is the number of people.

TL;DR: 66 handshakes means 12 people were present.

Information gathered from public forums or data available on the internet and portrayed here.