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Albert and Bernard

By David, 22 May 2015

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates

May 15            May 16            May 19

June 17            June 18

July 14             July 16

August 14        August 15        August 17

Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too.
Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.
Albert: Then I also know when Cheryl’s birthday is.

So when is Cheryl’s birthday?”

 

For the past month, this brainteaser has puzzled the world. It has appeared in newspapers, TV shows and online. It’s one of the trickiest problems from a very tricky test, the Singapore and Asian Schools Math Olympiad. So how do you solve it, and what does it have to do with maths?

At first glance, it is hard to see how to start solving this puzzle. Albert knows the month and Bernard knows the day, but it may seem like Albert and Bernard are just saying “I don’t know” until they magically do know.

This puzzle works by the process of elimination. Starting with lots of possible solutions, you eliminate some solutions at each step of thinking through the puzzle. Many people use this technique when solving multiple choice quizzes – instead of working out the right answer, just eliminate all the wrong ones.

Have a go at solving this puzzle yourself, and then take a look at our answer below.

 

 

Image: CSIRO/ Carl Davies CC/BY

Image: CSIRO/ Carl Davies CC/BY

 

Cheryl gave Albert and Bernard several possible dates, and only one is the right answer. She told Albert the month, and Bernard the day.

To get started, one option is to put the dates into a table.

May 15, 16, 19; June 17, 18; July 14, 16; August 14, 15, 17.

Most of the columns have two entries. But two of the columns, 18 and 19, only have one entry. These seem like a good place to start with the problem. Imagine Cheryl’s birthday was on 19 May – the conversation might go like:

 

Bernard (who knows it is on the 19): I can work out when Cheryl’s birthday is!

Albert (who knows it is in May): the only special date in May is the 19, so that must be Cheryl’s birthday!

 

If Cheryl’s birthday were on 19 May or 18 June, Bernard would be able to work out the answer straight away. We can be sure the answer is not one of these dates, so we have coloured it red in the table to remind us that these are not the right answers.

May 15, 16, 19x; June 17, 18x; July 14, 16; August 14, 15, 17.

Albert’s first comment is confusing at first, but contains a lot of information.

 

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too.

 

Albert knows the month and not the day. If Albert knows for sure that Bernard doesn’t know the answer, it must be because that month does not contain the days 19 or 18. Remember the last step? If Cheryl’s birthday were on 19 May or 18 June, Bernard would be able to work out the answer straight away. If Albert had been told May or June, he couldn’t be 100% sure that Bernard doesn’t know the answer.

Seeing as Albert knows Bernard doesn’t know, Albert must have been told the month is July or August.

May 15x, 16x, 19x; June 17x, 18x; July 14, 16; August 14, 15, 17.

So far we have excluded the days 18 and 19, and the months May and June. Bernard can also work this out from Albert’s statement. He says:

 

Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.

 

If Bernard was told 15 or 17, Bernard would be able to work out the birthday was in August (the only month left with those days). And if he was told 16, he could work out it was in July. So those are all possible answers. But if Bernard was told 14, he would not be able to work it out.

Seeing as Bernard can work out the answer, we can exclude day 14.

May 15x, 16x, 19x; June 17x, 18x; July 14x, 16; August 14x, 15, 17.

The final statement comes from Albert:

 

Albert: Then I also know when Cheryl’s birthday is.

 

If the birthday was in August, Albert would still be in the dark. He wouldn’t know if Cheryl’s birthday was on day 15 or 17. We can exclude those two dates, leaving us with the correct answer. The birthday must be in July, on the 16.

May 15x, 16x, 19x; June 17x, 18x; July 14x, 16; August 14x, 15x, 17x.

What’s happening?

Using a process of elimination helped solve this problem. In a process of elimination, you use logical thinking to exclude possible answers until you are left with the correct answer. It’s a useful way of thinking, though it takes some practice. Doctors sometimes use a process of elimination to diagnose a disease.

 

More information

A different party brainteaser

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