Question

(15) (Bonus 5 pts.) A famous question from the 2015 Singapore and Asian Schools Math Olympiad: Lucky guesses get 2 points, correct answer + evidence of reasoning get full credit Ibert and Bernard are trying to deduce Cheryl's birthday from 10 possible dates: May 15 16 19 June 17 18 July 14 16 August 14 1517 Cheryl tells Albert the month of her birthday, and Bernard the day of her birthday sepa- rately. The following dialog between Albert and Bernard is given below: Albert: I don't know when Cheryl's birthday is, but I know that Bernard doesn't know either. 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 When is Cheryl's birthday?

Answer 1

Let d be the day of her birthday and m be the month of her birthday.

If d=18 or 19, Bernard would know the month as they occur only once in the table. Therefore, d is not 18 or 19.

Albert knew m and was sure he didn't have d=18 or 19, so m cannot be May or June. We now have m=July or August.

After restricting m to July or August, Bernard found out her birthday. This means the date cannot be 14 as it is in both July and August.

After eliminating all those options, Albert found out. This is possible only if the month is July and d=16 because August 15 and 17 are possible options and Albert would not have been able to determine it uniquely.

Answer= July 16