When is Cheryl’s Birthday? The Solution

Those who know me, know I love logic puzzles.  I always have a book of them around. I think it’s because I’m not very good at them that I find them fascinating.  So when a Singapore math problem about Cheryl’s birthday went viral, I was hooked.  And this one is a tough one, folks.  My first reaction was how hard is high school in Singapore?  This problem was a Math Olympiad question posed to grade 9 students (age 14-15) in a math competition.  Wow.  On another note, it has made math cool:

Cheryl's Birthday Signapore Math Olympiad Question

Cheryl’s Birthday Singapore Math Olympiad Question

 Problem (Restated)

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

Don’t read this until you’ve given it a fair try.  It’s much more rewarding that way.  Hint to get you started: it’s a process of elimination.

From the first clue:

  • Albert knows the month (May, June, July or August);
  • Bernard knows the day  (14, 15, 16, 17, 18 or 19);
  • Albert knows that Bernard doesn’t know;
    • That means the day number can’t be 18 or 19, eliminating May 19 and June 18, as those are the only occurrences of those numbers;
  • But Albert knows already that Bernard doesn’t know, meaning it can’t be May or June!
    • If Cheryl had told Albert May, then it could have been May 19 – meaning Bernard “might” have known;
    • Likewise if Cheryl had told Albert June, then it could have been June 18 – again meaning Bernard “might” have known;
    • But since Albert is certain Bernard “does not” know, then it can’t be a “might” know, so therefore it can’t be May or June;
  • This leaves July and August for the months and 14, 15, 16, and 17 for the day.

From the second clue:

  • Bernard didn’t know, but he knows now;
    • This implies that Albert’s statement gave him enough information to determine Cheryl’s birthday;
  • Albert narrowed it down to July or August;
  • Day number 14 repeats, so can be eliminated;
  • That leaves July 16, August 15, and August 17.

From the third and final clue:

  • Albert confirms that he now knows the birthday;
    • If it was in August, he wouldn’t know as there are two dates (15 & 17)
    • So it must be in July, as there is only one date (16)
  • Therefore, Cheryl’s birthday must be July 16

Ahh, Cheryl, what a troublemaker.  And what a great question.

That’s it.

k

 

Karim Sultan
My vision is for digital innovation across the divide, by securing and simplifying the Internet and extending its reach to every citizen on Earth and beyond.

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