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.
As 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…
What Does Albert Know?
Cheryl told Albert the month of her birth. Which means he knows which of May, June, July, or August is correct. Knowing just this month, is there any way Albert could know Cheryl’s actual birthday? The answer is no—since she listed more than one possible day for each of these months, he can’t yet know. Which is exactly what he says to Bernard.
But he also says that Bernard can’t know Cheryl’s birthday at this point either. How can he be sure? Because he knows that Bernard has been told the numerical day of Cheryl’s birth. And he understands that if the month which he knows to be correct were to contain a numerical day that appeared only once amongst all 10 possibilities, and if that day was Cheryl’s birthday, then Bernard would know not just the day, but the month, too.
Since Albert says Bernard can’t know, we can conclude that Cheryl wasn’t born in any of the months containing numerical days that appear only once. Looking over her list of possible days, this means she was not born in May or June, since the numerical days 18 and 19 appear uniquely in those months.
So all 5 days in May and June can be crossed of our list of possibilities, leaving us with 2 days in July and 3 days in August to choose from.
What Does Bernard Know?
As soon as Bernard is told that he can’t know Cheryl’s birthday, he says that he knows it.
As soon as Bernard is told that he can’t know Cheryl’s birthday, he says that he knows it. Sounds crazy, right? But Bernard’s reasoning is actually pretty straight forward.
It begins with the fact that, like us, he hears Albert’s declaration that neither of them can know Cheryl’s birthday. When Bernard hears this, he is able to make the same deduction that we made to figure out that Cheryl must have been born in either July or August.
Knowing only these months, how can Bernard claim that he knows Cheryl’s birthday? Remember, Bernard knows the actual numerical day. So it must be that this numerical day he knows occurs only once in Cheryl’s list of possible dates in July and August.
That means that July 14 and August 14 cannot be Cheryl’s birthday, since those 2 dates share the same numerical day – and Bernard has ruled that out. Which means we’re left with only July 16, August 15, or August 17 as possibilities. So which is it?
When Is Cheryl’s Birthday?
At this point, we’re almost done. We’ve narrowed down Cheryl’s original list of 10 possible days to 3, and Bernard knows which of these is correct. As soon as Bernard proclaims that he knows Cheryl’s birthday, Albert says that he knows it as well. How?
Remember, Cheryl told Albert the correct month at the outset. So he isn’t stuck wondering if it’s either July or August—he knows which month it is. If he knows that the correct month is August, he wouldn’t be able to say that he knows her birthday, since there would be 2 possibilities to choose from (and he wouldn’t know which is correct.)
But after using Bernard’s declaration to cross July 14 off the list of possibilities, Albert knows that July has only one remaining date. And that date must be Cheryl’s birthday.
And thus, you, me, Albert, and everybody else can say that Cheryl’s birthday is July 16!
Wrap Up
Wrap UpOkay, that’s all the viral math goodness we have time for today.
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Birthday cupcake image courtesy of Shutterstock.
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About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.
