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The Infamous "Cheryl's Birthday" Puzzle

Who is Cheryl? Why has the world suddenly become obsessed with finding her birthday? And when is it? Keep on reading The Math Dude to learn how to solve this viral math puzzle and determine Cheryl’s birthday.

By
Jason Marshall, PhD,
April 18, 2015
Episode #238

Page 1 of 3

Birthday CupcakeAs you may have noticed, social media has been all abuzz of late over a puzzle from a math olympiad competition for high school students in Singapore. It’s not every day that people around the world get worked-up into hissy fits over a math puzzle, which means this one is really hitting a nerve.

The puzzle goes like this: A woman named Cheryl has two friends, Albert and Bernard, who really want to know her birthday. But she doesn’t want to tell them…at least not directly.

Instead, she decides it’d be fun to write down ten possible dates for both to see (one of which is correct), and then to secretly tell Albert the month and Bernard the numerical day - and then see if they can figure out the right birthday without telling each other what they’ve been told.

Sound strange? Definitely. Completely contrived? Totally. Utterly unrealistic? Without a doubt. Why then has it gotten so much attention? My guess is that it’s because the problem is presented in such a way that it seems unsolvable (you’ll see what I mean in a minute), and this appearance makes it all the more fun (and perhaps a bit frustrating) to find out that it is indeed perfectly solvable.

So, when is Cheryl’s birthday? The world wants to know…so let’s find out!

The Puzzle

Here’s the high school math olympiad problem in all its beautiful glory. Cheryl tells Albert and Bernard that her birthday is one of:

May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17

After Albert is secretly told the correct month and Bernard is secretly told the correct numerical day, they exchange three sentences:

  • Albert: “I don’t know it, but I know that you don’t know it either."
  • Bernard: “Before you said that I didn’t know it, but now I do!"
  • Albert: “If you know it, then I know it!"

At which point they both really do know it. So, when is Cheryl's birthday?

Before we continue on and talk about the answer, I encourage you to stop for a few minutes and try to puzzle it out for yourself. If you’re stuck, read the next section for a hint, but do try it first…it’s fun (really, it is!)

Hint: How to Solve the Puzzle

We need to figure out when Cheryl's birthday is not.

The secret to solving this logic puzzle is to ignore the question posed and instead think about the opposite question. Which means that instead of figuring out when Cheryl’s birthday is, we need to figure out when Cheryl's birthday is not.

As you’ll see, dealing with this opposite question is a lot easier. And once we know when her birthday isn’t, it’ll be easy to go back and figure out when it is.

If you still haven’t thought the puzzle through, now is the time to do so. It’s perfectly fine if you get stuck and can’t quite figure it out, but you should definitely try. If you’re ready to hear my explanation, let’s go...

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