Cheryl’s Birthday Math Puzzle Explanation

Recently we posted this problem to our social media accounts:

In this Blog Post I will explain how to determine Cheryl’s birthday. This problem appeared on a Math Olympiad question for high school students in Singapore. At first, it may seem impossible to find a solution. However, after carefully considering the dialogue in the problem one can conclude an accurate solution.

 

To begin, lets examine each line individually.

• Albert says, “I don’t know the birthday, but I know Bernard doesn’t know either.”

The first part of this statement is obvious. Of course Albert does not know the birthday, but the second part is a clue. Albert is sure Bernard cannot know the date. If Bernard was told the date of the 18th or 19th, he would automatically know Cheryl’s birthday because those date are only listed once. How is Albert sure Bernard did not hear the date of 18th or 19th? It must be because Albert knows the birthday is not May or June (The month of May and June list the two dates that do not have a duplicate date). Meaning Albert was told either July or August.

We can eliminate: May 15, May 16, May 19, June 17, June 18

The first statement reduces our list of possible birth dates in half, to 5 new possible options:

 

July 14 July 16

August 14 August 15 August 17

 

• Bernard then says, “I didn’t know at first, but now I do know.”

Upon hearing Albert’s statement, Bernard figured out the month must be July or August. He then also said now he could figure out the birthday. If he had been told 14, the month could still possibly be July or August. So he must have been told 15, 16, or 17.

We can eliminate: July 14, August 14

The second statement reduces our list from 5 possible birth dates, to 3 new possible options:

 

July 16

August 15 August 17

 

• Albert then says, “Now I know Cheryl’s Birthday.”

If Albert was told August, he still would not be sure. So he must have been told July.

We can eliminate: August 15, August 17

The third statement reduces our options from 3, to the final date.

Therefore, Cheryl said July to Albert and 16 to Bernard.

 

July 16

 

Cheryl’s birthday is July 16!