Solution for Cheryl's birthday

It is 16 July! Below is the explanation of this very pleasing problem.

Cheryl has told A and B that her birthday is one of these ten dates:

May 15, 16, 19

June 17, 18

July 14, 16

August 14, 15, 17

She has told A the month and B the number of the day, and they both know this.

A says, 'I do not know C's birthday.

This tells us nothing, because whatever month he had been told A would not be able yet to deduce the birthday.

But B would know C's birthday if he had been told that the day was, say, 19: because 19 occurs in the list of options only in May. Likewise B would know the birthday if had been told that the day was 18: because this occurs only in June.

So, then A says:

'... but I know for sure that B does not know either.'

A would not be able to say this, if the month he had been told was May or June, because for each of these two months there is the possibility that B actually knows the birthday already.

From this B (and we) can deduce that the month of C's birthday is neither May nor June.

B replies: 

'At first I did not know C's birthday, but I do now.'

This means that the number of the day (which B knows) must occur in either July or August but NOT in both these months. For example, if B knew the day was 17 then he would know it had to be August 17. Likewise, B would know the birthday if the day was 16 (July) or 15 (July).

A has deduced all this as well! He knows that the day is 15, 16 or 17. But A also knows the month. And he then replies:

'Now I know as well!

Now A could only know the birthday if the month (which he knows) has only one of these three options in the list: 15, 16 or 17. That month is July.

This is how we (and B presumably now) know that Cheryl's birthday must be 16 July.

Cheryl's birthday

I'll give my solution to this logical reasoning problem in my next post, so anyone reading this has the chance to solve it themselves first. This is the problem from a Singapore mathematics test that has apparently 'gone viral'. It was constructed by Dr Joseph Yeo Boon Wool, a mathematics professor at the Singapore National Institute of Education.

So here is the actual problem.

Albert and Bernard want to know the birthday of their new friend Cheryl. She tells them that it is one of the following ten options:

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

She whispers in A's ear the month of her birthday and tells B that she has done this.
She then whispers in B's ear the day of her birthday and tells A that she has done this.

Then A says, 'I do not know C's birthday; but I know for sure that B does not know either.'
B replies: 'At first I did not know C's birthday, but I do now.'
A replies: 'Now I know as well!'

This is an excellent example of a logical reasoning problem that involves making deductions from what people say about what they know or do not know. These puzzles always assume that all the people involved have high powers of deductive reasoning, so you can assume that if something can be deduced they will deduce it!

Here is another example, much easier than finding Cheryl's birthday!

A, B and C are told that in a bag there are 3 red hats and 2 blue hats. B is blindfold, so is unable to see anything. One hat is put on each person's head and they have to work out what colour hat they are wearing. A and C can see the hats on the other two, but none of them can see their own hat.
A says: I do not know what colour my hat is.
C says: Nor do I.
B says: Then I am wearing a red hat.
How did B work that out?

Solutions in my next post.