The Birthday Party Problem

 

Betty’s friends threw her a birthday party. Everyone (including Betty) met exactly 3 new people at the party. 

When Betty told her mom about the party later, her mom said, “That sounds like fun! How many people were there at the party?” 

“I think there were 14, or maybe 15, including me,” answered Betty.

Question: If Betty was right that there were either 14 or 15 people at the party, then which number is correct? 

For the solution, click "Read More" below.

Solution:

There must have been 14 people at the party. The diagram below shows that it is possible to have 14 people at a party, each meeting exactly 3 people:

In fact, the diagram also helps us see that a solution is possible for any even number of people (greater than 4) because adding another pair of people simply adds another “spoke” to the wheel.

We can also show that an odd number of people in the room would be impossible. The total number of “meetings” is given by the following formula where there are n people in the room, and each shakes 3 hands:

(n x 3) / 2

Notice that the number of meetings is not simply n x 3. We have to divide by 2 because otherwise we are counting each meeting twice – once for each person involved. 

Using the formula: if we had 15 people in the room, we would have 22 ½ meetings, which is impossible. In fact any odd number n would give us a non-integer, which is impossible.