Cheryl's Birthday

Cheryl's Birthday is the unofficial name given to a mathematics brain teaser that 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.^[1] The quiz asked readers to determine the birthday of a girl named Cheryl using a handful of clues given to her friends Albert and Bernard.

Origin

The question was posted on Facebook by Singapore TV presenter Kenneth Kong.^[1] 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,^[2] 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.^[3] The competition was held on 8 April 2015, with 28,000 participants from Singapore, Thailand, Vietnam, China and the UK.^[4] 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".^[5]

The question

The question is number 24 in a list of 25 questions, and reads as follows:^[2]

"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:

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.^[6]

The answer can be deduced by progressively eliminating impossible dates.^[7] This is how Alex Bellos in the UK newspaper The Guardian presented its outcome:^[8]

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 only these numbers appear 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. Since if he had been told August, he would not know which date for certain is the birthday.

The answer, therefore is July 16.

Incorrect solution

After the question went viral, some people suggested August 17 was an alternative answer to the question.^[9] This is rejected by the Singapore and Asian School Math Olympiads as a valid answer.^[9]

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.

The answer of August 17 would be valid if Albert did not deduce that Bernard does not know the birthday in this manner, but instead arrived at that knowledge via some means not mentioned in the problem. As the SASMO organizers have pointed out,^[10] 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.

It would also be the answer if the first statement was made by Cheryl instead:

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.

Sequel

On May 14, 2015, Nanyang Technological University uploaded a second part to the question on Facebook, this time entitled "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?^[11]

Sequel solution

Cheryl first says that she is the oldest of three siblings, and that their ages multiplied makes 144. 144 can be decomposed to factors of prime numbers by the fundamental theorem of arithmetic (144 = 1 x 2 x 2 x 2 x 2 x 3 x 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. On examining all the possible ages, it turns out there are two sets of possible ages that produce the same sum: 9, 4, 4 and 8, 6, 3, which sum to 17, and 12, 4, 3 and 9, 8, 2, which sum to 19. Cheryl mentions that her brothers are the same age, which eliminates the last three possibilities and leaves only 9, 4, 4, i.e. 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. All other possibilities give a unique sum that would have allowed Bernard and Albert to deduce Cheryl's age once Cheryl gave the second clue (that the sum of the ages is the bus number they are on).

See also

• Sum and Product Puzzle

References