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Cheryl’s Birthday

The answer follows logically. (Source: BBC/Kenneth Kong)

The answer follows logically. (Source: BBC/Kenneth Kong)

Mathematics is an entrancing subject, covering cold logic to modelling hot lava. Logical deductions and analytical reasoning form a fundamental part of mathematics, and of wider analysis, whether in marketing, business strategy or operations.
Published in the Singapore and Asian Schools Math Olympiads (Sasmo) for 15 year-olds, the problem of Cheryl’s birthday was wrongly reported to be for Singaporean school-children. The problem is as follows:

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?

Solving the Problem

It is easier to visualise this problem by looking at the grid of all possible dates. For clarity, Albert knows the month of Cheryl’s birthday, whilst Bernard knows the day.

cheryls-birthday-part-i

What we can observe is that May 19 and June 18 are peculiar, as they are the only days with a single month associated with them. If Albert was told that the month was either May or June, then Albert could not deduce that “I know that Bernard does not know too”. This is because Bernard could have been told 18 or 19 as the day, which means he would know Cheryl’s birthday: June 18 or May 19 respectively.

If the month is either May or June, then we reach a contradiction. Due to this contradiction, our assumption that the month is either May or June must be incorrect. This leaves only the possibility that the birthday is in either July or August. Now the possible list of dates is: July 14, July 16, August 14, August 15, August 17.

cheryls-birthday-part-ii

This is the point where Bernard says: “At first I don’t know when Cheryl’s birthday is, but I know now.” This means he must have a day that, in the remaining list, only has a single month associated with it. If Bernard was told 14, then he wouldn’t know if the birthday was either July 14 or August 14. That removes the possibility that Bernard was told 14 was the day. Hence, the reduced lists of dates is: July 16, August 15, August 17.

cheryls-birthday-part-iii

Now, Albert says: “Then I also know when Cheryl’s birthday is.” If the birthday was in August, then Albert has a choice of two dates: August 15 or August 17. This would mean he could not know when Cheryl’s birthday is, which contradicts his statement. Consequently, Albert was told the month of July.

Therefore, Cheryl’s birthday is July 16.

cheryls-birthday-part-iv

Mathematics is not simply arithmetic: logical reasoning underlies debates, from the boardroom to the legislature.

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This entry was posted on April 16, 2015 by in Other Interests and tagged .
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