# 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:

## 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.

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