Cheryl’s birthday
In the last week or so, there has been quite a buzz in some circles about a logic problem given to 14 year old students for the Singapore and Asian Schools Math Olympiad (SASMO) for Secondary 3 and 4 students, as follows:
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, to which Albert and Bernard respond by saying:
Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.
Bernard: At first I didn’t know when Cheryl’s birthday is, but now I do know.
Albert: Then I also know when Cheryl’s birthday is.
So when is Cheryl’s birthday?
Answer
The key is to be clear at all times about what Albert and Bernard each know, and what we know as a result of hearing what they say.
Initially:
– Albert knows that the month of Cheryl’s birthday is either May, June, July or August.
– Bernard knows that the day of Cheryl’s birthday is either 14, 15, 16, 17, 18 or 19.
– each knows what the other knows.
- Albert: “I don’t know when Cheryl’s birthday is”.
Given Cheryl’s ten supplied dates, every one of the four months has multiple possible days. So, before Albert told us, both we and Bernard already knew this. - Albert: “I know that Bernard does not know too.”
To say this, Albert must know that May 19 and June 18 are not possible dates for Cheryl’s birthday, otherwise Bernard would know the date of Cheryl’s birthday immediately. So, given that Albert knows only the month of Cheryl’s birthday, the month cannot be May or June.
So now our table of possible dates looks like this…
May 15 | May 16 | May 19 | |||
June 17 | June 18 | ||||
July 14 | July 16 | ||||
August 14 | August 15 | August 17 |
- Bernard: “At first I didn’t know when Cheryl’s birthday is.”
Albert already knew this, and has just told us so. - Bernard: “Now I do know [when Cheryl’s birthday is].”
This means the day of Cheryl’s birthday cannot be 14, otherwise July 14 and August 14 would still be possible dates for Bernard, so he would still not know when Cheryl’s birthday is.
So now our table of possible dates looks like this…
May 15 | May 16 | May 19 | |||
June 17 | June 18 | ||||
July 14 | July 16 | ||||
August 14 | August 15 | August 17 |
- Albert: “Then I also know when Cheryl’s birthday is.”
If Cheryl’s birthday was in August, both August 15 and August 17 would still be possible dates for Albert, so he would still not know when Cheryl’s birthday is. So Cheryl’s birthday is not in August.
So now our table of possible dates looks like this…
May 15 | May 16 | May 19 | |||
June 17 | June 18 | ||||
July 14^{th} | July 16 | ||||
August 14 | August 15 | August 17 |
So Cheryl’s birthday is on July 16.