Cheryl’s Birthday Problem



Albert, Cheryl and Bernard (wikimedia.org)

Albert and Bernard meet up with their friend Cheryl. Albert politely asks Cheryl: “When is your Birthday please Cheryl?” Cheryl doesn’t give them a straight answer. She says: “I’m not going to tell you, but I’ll give you some clues.” She then writes down a list of possible dates:

May 15, May 16, May 19

June 17, June 18

July 14, July 16

August 14, August 15, August 17

“My birthday is one of these.” She says.

Then Cheryl whispers to Albert, the month, and only the month.

To Bernard, Cheryl whispers the day, and only the day.

“Can you figure it out now?”, she asks Albert.

Albert says: “I don’t know when your birthday is, but I know that Bernard doesn’t know either.”

Bernard says: “I didn’t know to begin with, but I do now.”

Then Albert says: “Well, now I know too!”

When is Cheryl’s Birthday?

The Solution

To go about finding Cheryl’s birthday, it’s best to first put these possible dates into a table, and also state what Albert and Bernard know concisely:-

May   15 16     19
June       17 18  
July 14   16      
August 14 15   17    

Albert knows the Month
Bernard knows the day

First the Table

There are four months: May, June, July and August. There are only six days, 14, 15, 16, 17, 18 and 19. Now, 14, 15, 16 and 17 appear twice and 18 and 19 only appear once.
If Bernard had been told 18, then he would know straight away, before anyone says anything, that it is 18 May. Because 18 only occurs once.
Similarly, if Bernard had been told 19, then he would know straight away, before anyone says anything, that it is 19 June. Because 19 only occurs once.

Now the Statements

Albert says:

“I don’t know when your birthday is, but I know Bernard doesn’t know, either.”

If Albert had been told May or June then he is aware that Bernard may have been told 18 or 19. So he couldn’t know, for sure, that Bernard didn’t know either.

From Albert’s statement then, we know that it must be July or August.

Bernard then says:

“I didn’t know originally, but now I do.”

Bernard is also aware that if he, Bernard, had been told 18 or 19, then he would know the birthday date straight away. For Albert to say that he knows that Bernard doesn’t know either, then, from this, Albert figures out that Albert had not been told May or June.

For Bernard to then know, after Albert’s statement, then it can’t be July 14th or August 14th. Because Bernard only knows the day, not the month. So Bernard still wouldn’t be able to decide between July or August.

We are now only left with three possibilities. August 15th, July 16th and August 17th. If Cheryl said any of these three days to Bernard then he would know the month and hence know the birthday. However, we, the reader, still wouldn’t know though. We readers need the final statement.

Albert replies:

“Well, now I know too!”

Albert only knows the month. If the day had been the 14th then he knows that Bernard still wouldn’t know. If Albert had been told August then he still wouldn’t know, after Bernard’s Statement, whether it was the 15th or the 17th. He could be sure that Bernard knew, but he, Albert, couldn’t know. So the only way that Albert could also know is if he was told July, because there is only one July day left, the 16th .

The answer is July 16th

Took me ages to figure this out. I have a background in Maths and Problem Solving but I do find these Verbal Reasoning problems difficult.
The exchange between Albert and Bernard is shown to be at a conversational pace, but you and I have to sit and scratch our heads for a while to figure it out.
I wonder if either of the boys actually gave Cheryl anything at all for her birthday after this!
There is only one solution.

Of course, this is a famous problem and has been covered elsewhere, even Wikipedia and The Guardian. I just wanted to see if I could make the solution a little clearer for people.

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