# Cheryl's Birthday

Cheryl's Birthday is a logic puzzle, specifically a knowledge puzzle.[1][2] The objective is to determine the birthday of a girl named Cheryl using a handful of clues given to her friends Albert and Bernard. It was asked in the Singapore and Asian Schools Math Olympiad, and was posted online on 10 April 2015 by Singapore TV presenter Kenneth Kong. It went viral in a matter of days.[3]

## Origin

The question was posted on Facebook by Singapore TV presenter Kenneth Kong.[3] The posting drew thousands of comments, including several humorous ones—many aimed at Cheryl who apparently didn't want to disclose her birthday straight away. Kong posted it out of his debate with his wife, and he incorrectly thought it to be part of a mathematics question for a primary school examination, aimed at 10- to 11-year-old students,[4] although it was actually part of the 2015 Singapore and Asian Schools Math Olympiad (SASMO) meant for 14-year-old students, a fact later acknowledged by Kong.[5] The competition was held on 8 April 2015, with 28,000 participants from Singapore, Thailand, Vietnam, China and the UK.[6] According to SASMO's organisers, the quiz was aimed at the top 40% of the contestants and aimed to "sift out the better students". SASMO's executive director told the BBC that "there was a place for some kind of logical and analytical thinking in the workplace and in our daily lives".[7]

## The question

The question is number 24 in a list of 25 questions, and reads as follows:[4]

"Albert and Bernard just become [sic] friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates:

 May June July August 15 16 19 17 18 14 16 14 15 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. [sic]
Bernard: At first I don't [sic] 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?"

## Solution

The answer to the question is July 16.[8]

The answer can be deduced by progressively eliminating impossible dates.[9] This is how Alex Bellos in the UK newspaper The Guardian presented its outcome:[10]

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

All Albert knows is the month, and every month has more than one possible date, so of course he doesn’t know when her birthday is. The first part of the sentence is redundant.

The only way that Bernard could know the date with a single number, however, would be if Cheryl had told him 18 or 19, since of the ten date options these are the only numbers that appear just once, as May 19 and June 18.

For Albert to know that Bernard does not know, Albert must therefore have been told July or August, since this rules out Bernard being told 18 or 19.

Line 2) Bernard: At first I don’t know when Cheryl’s birthday is, but now I know.

Bernard has deduced that Albert has either August or July. If he knows the full date, he must have been told 15, 16 or 17, since if he had been told 14 he would be none the wiser about whether the month was August or July. Each of 15, 16 and 17 only refers to one specific month, but 14 could be either month.

Line 3) Albert: Then I also know when Cheryl’s birthday is.

Albert has therefore deduced that the possible dates are July 16, Aug 15 and Aug 17. For him to now know, he must have been told July. If he had been told August, he would not know which date for certain is the birthday.

Therefore, the answer is July 16.

### Incorrect solution

After the question went viral, some people suggested August 17 as an alternative answer to the question.[11] This was rejected by the Singapore and Asian School Math Olympiads as an invalid answer.[11]

The solutions which arrive at this answer ignore that the latter part of:

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

conveys information to Bernard about how Albert was able to deduce this. Bernard would only have known the birthday if the date was unique, 18 or 19. Albert therefore is able to deduce that "Bernard doesn't know" because he heard a month that does not contain those dates (July or August). Realizing this, Bernard can rule out May and June, which allows him to arrive at a unique birthday even if he is given the dates 15 or 16, not just 17.

As the SASMO organizers have pointed out,[12] August 17 would be the solution if the sequence of statements instead began with Bernard saying that he did not know Cheryl's birthday:

Bernard: I don't know when Cheryl's birthday is.

Albert: I still don't know when Cheryl's birthday is.
Bernard: At first I didn't know when Cheryl's birthday is, but I know now.
Albert: Then I also know when Cheryl's birthday is.

Cheryl: Bernard doesn't know when my birthday is.

Albert: I still don't know when Cheryl's birthday is.
Bernard: At first I didn't know when Cheryl's birthday is, but I know now.
Albert: Then I also know when Cheryl's birthday is.

Note: The final statements by Albert in the two alternative examples only completes a dialogue, they are not needed by the reader to determine Cheryl's birthday as August 17.

## Sequel

On May 14, 2015, Nanyang Technological University uploaded a second part to the question on Facebook, this time titled "Cheryl's Age". It reads as follows:

Albert and Bernard now want to know how old Cheryl is.
Cheryl: I have two younger brothers. The product of all our ages (i.e. my age and the ages of my two brothers) is 144, assuming that we use whole numbers for our ages.
Albert: We still don't know your age. What other hints can you give us?
Cheryl: The sum of all our ages is the bus number of this bus that we are on.
Bernard: Of course we know the bus number, but we still don't know your age.
Cheryl: Oh, I forgot to tell you that my brothers have the same age.
Albert and Bernard: Oh, now we know your age.

So what is Cheryl's age?[13]

### Sequel solution

Cheryl first says that she is the oldest of three siblings, and that their ages multiplied makes 144. 144 can be decomposed into prime number factors by the fundamental theorem of arithmetic (144 = 2 × 2 × 2 × 2 × 3 × 3), and all possible ages for Cheryl and her two brothers examined (for example, 16, 9, 1, or 8, 6, 3, and so on). The sums of the ages can then be computed. Because Bernard (who knows the bus number) cannot determine Cheryl's age despite having been told this sum, it must be a sum that is not unique among the possible solutions. On examining all the possible ages, it turns out there are two pairs of sets of possible ages that produce the same sum as each other: 9, 4, 4 and 8, 6, 3, which sum to 17, and 12, 4, 3 and 9, 8, 2, which sum to 19. Cheryl then says that her brothers are the same age, which eliminates the last three possibilities and leaves only 9, 4, 4, so we can deduce that Cheryl is 9 years old and her brothers are 4 years old, and the bus the three of them are on has the number 17.

### Second sequel: Denise's Revenge

On May 25, 2015, mathematician Alex Bellos published the second part of the puzzle, called "Denise's Revenge", in his The Guardian's column "Alex Bellos's Monday Puzzle"[14]. This sequel was written by Dr. Joseph Yeo Boon Wooi, the original Cheryl's Birthday puzzle author. In the article, Bellos explains that in the sequel, a new character joins the team. The puzzle states:

Albert, Bernard and Cheryl became friends with Denise, and they wanted to know when her birthday is. Denise gave them a list of 20 possible dates.

17 Feb 2001, 16 Mar 2002, 13 Jan 2003, 19 Jan 2004
13 Mar 2001, 15 Apr 2002, 16 Feb 2003, 18 Feb 2004
13 Apr 2001, 14 May 2002, 14 Mar 2003, 19 May 2004
15 May 2001, 12 Jun 2002, 11 Apr 2003, 14 Jul 2004
17 Jun 2001, 16 Aug 2002, 16 Jul 2003, 18 Aug 2004

Denise then told Albert, Bernard and Cheryl separately the month, the day and the year of her birthday respectively. The following conversation ensues:

Albert: I don’t know when Denise’s birthday is, but I know that Bernard does not know.
Bernard: I still don’t know when Denise’s birthday is, but I know that Cheryl still does not know.
Cheryl: I still don’t know when Denise’s birthday is, but I know that Albert still does not know.
Albert: Now I know when Denise’s birthday is.
Bernard: Now I know too.
Cheryl: Me too.

So, when is Denise’s birthday?

### Solution to Denise's Revenge

The next day, Alex Bellos published the solution to Denise's Revenge, which is solved in the same way than Cheryl's Birthday: by successive eliminations[15].

Recapitulating:

```Albert: I don’t know when Denise’s birthday is, but I know that Bernard does not know.
```

Albert, having only the month, obviously doesn't know. However, for him to know that Bernard doesn't either, it means that he knows the date's day number is not unique, which means it's not 11 or 12 (the only dates with a unique day). Again, for Albert to know that, we can conclude that Albert knows that the month is not April nor June. This eliminates 11 Apr 2003, 12 Jun 2002, 13 Apr 2001, 15 Apr 2002 and 17 Jun 2001.

```Bernard: I still don’t know when Denise’s birthday is, but I know that Cheryl still does not know.
```

This statement gives us two clues. First, Bernard says, AFTER making the same deduction as Albert, "I still don’t know when Denise’s birthday is...". Bernard has been given a day, so if he followed Albert's reasoning but still haven't got the right answer, it means that the correct date is NOT one of the ones which now have unique days; i.e. 15 may 2001 and 17 feb 2001.

Then he goes on to say "...but I know that Cheryl still does not know." The only way for Cheryl to know the answer would be if she only had one remaining date in the year she has been given. The only year that has one remaining date is 2001 (13 Mar 2001), so if Bernard knows that Cheryl doesn't know the answer, it means that the day (which is the only information Bernard knows) is not 13. That eliminates 13 Mar 2001 and 13 Jan 2003.

```Cheryl: I still don’t know when Denise’s birthday is, but I know that Albert still does not know.
```

It is not possible that Cheryl knows the right date for Denise's birthday, since every year still open has many possible dates, so this statement is redundant. However, for her to know that Albert does not know, she would need to know that the correct date does not have a unique month among the remaining alternatives. The only possible date with a unique month is 19 Jan 2004; therefore, the only way for Cheryl to know that Albert does not know yet would be if Cheryl knew that the year is not 2004. This reasoning eliminates 19 Jan 2004, 18 Feb 2004, 19 May 2004, 14 Jul 2004 and 18 Aug 2004.

```Albert: Now I know when Denise’s birthday is.
```

Up to this point, following the first half of the puzzle, the only possible dates still remaining would be: 16 Mar 2002, 14 May 2002, 16 Aug 2002, 16 Feb 2003, 14 Mar 2003 and 16 Jul 2003. Now, if Albert, who only knows the month, says that he already knows Denise's birthday's right date, it would mean that the right date has a unique month. This eliminates both dates in March, 16 Mar 2002 and 14 Mar 2003.

```Bernard: Now I know too.
```

Again, for Bernard (who knows only the day) to know, it would mean that the right date has a unique day. This eliminates all remaining dates with repeated days, i.e. 16 Aug 2002, 16 Feb 2003 and 16 Jul 2003.

```Cheryl: Me too.
```

Following the reasoning up to this point, it is obvious that the only remaining possible date, which should be the right answer, is 14 May 2002. Therefore, this statement is redundant, and it's here only to finish the conversation.

## References

1. ^ van Ditmarsch, Hans; Hartley, Michael Ian; Kooi, Barteld; Welton, Jonathan; Yeo, Joseph B.W. (2017). "Cheryl's Birthday". arXiv:1708.02654.
2. ^
3. ^ a b "When is Cheryl's birthday? Singapore math question for kids stumps internet". CBC News. 14 April 2015. Retrieved 15 April 2015.
4. ^ a b Bellos, Alex (13 April 2015). "Can you solve the maths question for Singapore schoolkids that went viral?". The Guardian. Retrieved 15 April 2015.
5. ^ Withnall, Adam (13 April 2015). "Singapore maths question: How to solve the problem that has stumped the world". The Independent. Retrieved 15 April 2015.
6. ^ Carter, Claire; Gray, Richard; Harding, Eleanor (15 April 2015). "'When is Cheryl's birthday?' The maths problem set for teenagers that has baffled the world". Mail Online. Retrieved 15 April 2015.
7. ^ BBC News, 14 April 2015
8. ^ Dua, Ruchi (15 April 2015). "Cheryl's birthday: Singapore school maths problem stumps the Internet l". India Today. Retrieved 15 April 2015.
9. ^ Chang, Kenneth (14 April 2015). "How to Figure Out Cheryl's Birthday". The New York Times. Retrieved 15 April 2015.
10. ^ Bellos, Alex (13 April 2015). "How to solve Albert, Bernard and Cheryl's birthday maths problem". The Guardian. Retrieved 15 April 2015.
11. ^ a b Grime, James (15 April 2015). "Why the Cheryl birthday problem turned into the maths version of #TheDress". The Guardian. Retrieved 15 April 2015.
12. ^ Henry Ong, Executive Director, Singapore and Asian School Math Olympiads, SASMO's Reply to Queries Why Cheryl's birthday is not Aug 17
13. ^ Lee, Min Kok (15 May 2015). "'Cheryl's birthday' poser is back with Part 2: Cheryl's age". The Straits Times. Retrieved 29 December 2015.
14. ^ Bellos, Alex (25 May 2015). "Cheryl's birthday puzzle part two, Denise's revenge - can you solve it?". The Guardian. Retrieved 23 September 2019.
15. ^ Bellos, Alex (26 May 2015). "How to solve it! Cheryl's birthday puzzle part two: Denise's revenge". The Guardian. Retrieved 23 September 2019.