## Wednesday, April 15, 2015

### When is Cheryl's birthday?

Question ((Singapore and Asian Schools Math Olympiad test 2015))

Albert and Bernard just become 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 don't 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 with analysis:

First, we draw a mapping diagram and map the given months to their corresponding date as shown in the diagram above:

Next, we need to decipher the meaning and hence the implication from the dialogues between Albert and Bernard.

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

Implication 1:

Albert said he knows Bernard does not know the answer, because he realized from the rule of probability that the possibility Bernard could hazard the correct month based on the date he knew is practically nil.

Wow, this is very informative, and hence we can cross out the possible dates that associate with the month of May and June.

If you are a novice to this kind of logical implication and probability related problem, you might not see why this should make any sense to you. I will try my very best to explain, hopefully, you can make the head or tail of it after you read the explanation below.

Let us prove it by the method of contradiction and bearing in mind that Albert said he knew Bernad did not know the answer.

Now, we let May be the birthday month of Cheryl and Cheryl told Albert so. Albert would then think to himself, what if 19th is the date that Cheryl has told Bernard? In that case, Bernard would point out the birthday date of Cheryl on the spot, since 19th is the only date being mapped once in the mapping diagram above. So, Albert wouldn't tell as in the dialog that he knows Bernard does not know the birthday date.

These two statements:

Albert wouldn't tell as in the dialog that he knows Bernard does not know the birthday date

leads to a contradiction and therefore May could not be the possible candidate for the month of the birthday of Cheryl.

This applies too to June, since 18th is the another date that being mapped once in the mapping diagram and it is being matched to June.

This observation narrows down the possible candidates to:

Our next step is to study the dialogs between Bernard and Albert, as they said:

Bernard: At first I don't know when Cheryl's birthday, but I know now.

Albert: Then I also know when Cheryl's birthday is.

Implication:

Bernad said he now knows the answer because he knows the date and the date is now being mapped once in the diagram and he can now safely point out the month that only corresponds to the date he knows at hand.

This implies we can rule out 14th as the birthday date of Cheryl, since if 14th is the answer, then Bernard has to make a wild guess whether July 14th or August 14th is the correct answer instead of knowing for certain the birthday of Cheryl.

Let's say Cheryl told Bernard 15th is the actual birthday date, then Bernard would point out August 15th is the birthday of Cheryl since that is the only possible candidate left. But then after listening to Bernard that he claimed he now knows the answer, Albert said he knows the answer too. This can't be true, since if August is the birthday month, then Albert would have to guess if August 14th, August 15th or August 17th is the answer, and he could not possibly point out the correct answer.

If 16th is what Cheryl has told Bernard, then Bernard can safely conclude that July 16th is the birthday of Cheryl, since that is the only answer, and if July is the month that Cheryl has told Albert, when Albert heard Bernard who said he now knows the answer, he can safely single out 16th is what the date that Cheryl has told Bernard.

And yeah! July 16th is the birthday of Cheryl!