15 April 2015

Cheryl's Birthday Problem

The Maths world has been abuzz with the Cheryl Birthday Problem.
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 does not know too.
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.
So when in Cheryl's birthday?
There have been two possible responses July 16, and August 17. The originator of the question have indicated that the answer is July 16. This is also the original deduction by James Grime, but his explanation for the difference in interpretation between the two is not convincing. I believe that there are stronger reasons for not agreeing with August 17. My explanation are as follows.

First, it must be clear that Cheryl's birthday puzzle is not for Albert or Bernard, but for you and me. And we should take the entire statement of the problem in helping us solve it. It requires the following steps.

Step 1: Cheryl provides selective information to Albert and Bernard on the month and the day of her birthday respectively. Both of them know that they received part of the information and also know that the other person also received another part of the information.

Step 2: On getting his information of the month of Cheryl's birthday, Albert makes a statement that neither does he know nor does Bernard know Cheryl's birthday. He can make such a statement only if the month of Cheryl's birthday is either July or August, as these do not have any unique days (May 19 and June 18) from the 10 possible birthdays that were initially provided.  

Step 3: Once Bernard comes to know that Albert has made such a statement then he deduces that he now knows Cheryl's birthday. Bernard can make such a statement if the day that he has is 15, 16, or 17.

Step 4: Now, Albert states that he also knows Cheryl's birthday. This can only be possible if the month that he has is July.

Step 5: So, the answer is July 16.

The alternative articulation (for those favouring August 17 as the birthday) begin by stating that Albert had some additional information that the day is neither 18 nor 19.  Now, if Albert had that additional information then the month is not June because if it was June then he would have known that the answer is June 17. Thus, the months must be May, July and August. Now, the only unique number is 17 and the birthday is August 17. I do not agree with this line of argument as they begin with an assumption that Albert had some additional information that the day is neither 18 or 19, which is not based on the problem. It interprets Albert's statement independent of the information that Cheryl provided him, that is, the month of her birthday. Thus, this is not the correct answer. Hence, the correct answer is the earlier deduction, July 16.

1 comment:

  1. Sirjit's argument against thealternative answer of Aug 17 is also not rigid, He suggests that A must have additional information that the day is not 18 or 19. The sequence of events as presented allows A do deduce it cannot be 18 or 19 because B did not said he knows the answer after he was given the date by C. There is, therefore, no additional information given to A except by A's own observation of the events played out and his deduction thereof