A few weeks back, it was a color-changing dress that blew out the neural circuits of the Internet. Though it may not have quite the mass appeal, this week it is a math problem that is making bushels of brains hurt.

It started with a posting on Facebook, by Kenneth Kong, a television host in Singapore.

From there, people around the world have been trying to figure out Cheryl’s birthday, or at least wondering why she couldn’t just save everyone a lot of trouble and be more direct with Albert and Bernard.

The original Riddle goes like this;

Albert and Bernard just met Cheryl. “When’s your birthday?”

Albert asked Cheryl.

Cheryl thought a second and said, “I’m not going to tell you, but I’ll give you some clues.”

She wrote down a list of 10 dates:

May 15 — May 16 — May 19

June 17 — June 18

July 14 — July 16

August 14 — August 15 — August 17“

My birthday is one of these,” she said.Then Cheryl whispered in Albert’s ear the month — and only the month — of her birthday. To Bernard, she whispered the day, and only the day.

“Can you b it out now?” she asked Albert.Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either.Bernard: I didn’t know originally, but now I do.Albert: Well, now I know, too!

When is Cheryl’s birthday?

The wording of the problem is terrible, so here is a clearer version, which makes some of the assumptions more obvious but which does not change any of the underlying logic of the problem:

Albert and Bernard just met Cheryl. “When’s your birthday?” Albert asked Cheryl.

Cheryl thought a second and said, “I’m not going to tell you, but I’ll give you some clues.” She wrote down a list of 10 dates:

May 15, May 16, May 19

June 17, June 18

July 14, July 16

August 14, August 15, August 17

“My birthday is one of these,” she said.

Then Cheryl whispered in Albert’s ear the month — and only the month — of her birthday. To Bernard, she whispered the day, and only the day.

“Can you figure it out now?” she asked Albert.

Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either.

Bernard: I didn’t know originally, but now I do.

Albert: Well, now I know, too!

When is Cheryl’s birthday?

The problem actually came from a math olympiad test for math-savvy high school-age students.

Answer:

let’s examine what Albert and Bernard say. Albert goes first:

I don’t know when your birthday is, but I know Bernard doesn’t know, either.

The first half of the sentence is obvious — Albert only knows the month, but not the day — but the second half is the first critical clue.

The initial reaction is, how could Bernard know? Cheryl only whispered the day, so how could he have more information than Albert? But if Cheryl had whispered “19,” then Bernard would indeed know the exact date — May 19 — because there is only one date with 19 in it. Similarly, if Cheryl had told Bernard, “18,” then Bernard would know Cheryl’s birthday was June 18.

Thus, for this statement by Albert to be true means that Cheryl did not say to Albert, “May” or “June.” (Again, for logic puzzles, the possibility that Albert is lying or confused is off the table.) Then Bernard replies:

I didn’t know originally, but now I do.

So from Albert’s statement, Bernard now also knows that Cheryl’s birthday is not in May or June, eliminating half of the possibilities, leaving July 14, July 16, Aug. 14, Aug. 15 and Aug. 17. But Bernard now knows. If Cheryl had told him “14,” he would not know, because there would still be two possibilities: July 14 and Aug. 14. Thus we know the day is not the 14th.

Now there are only three possibilities left: July 16, Aug. 15 and Aug. 17. Albert again:

Well, now I know too!

The same logical process again: For Albert to know, the month has to be July, because if Cheryl had told him, “August,” then he would still have two possibilities: Aug. 15 and Aug. 17.

The answer is July 16.

Alternate explanation:

Remember, Albert is told either May, June, July or August.

Bernard is told either 14, 15, 16, 17, 18 or 19

Let’s go through it line by line.

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too.

All Albert knows is the month, and every month has more than one possible date, so of course he doesn’t know when her birthday is. The first part of the sentence is redundant.

The only way that Bernard could know the date with a single number, however, would be if Cheryl had told him 18 or 19, since of the ten date options only these numbers appear once, as May 19 and June 18.

For Albert to know that Bernard does not know, Albert must therefore have been told July or August, since this rules out Bernard being told 18 or 19.

Line 2) Bernard: At first I don’t know when Cheryl’s birthday is, but now I know.

Bernard has deduced that Albert has either August or July. If he knows the full date, he must have been told 15, 16 or 17, since if he had been told 14 he would be none the wiser about whether the month was August or July. Each of 15, 16 and 17 only refers to one specific month, but 14 could be either month.

Line 3) Albert: Then I also know when Cheryl’s birthday is.

Albert has therefore deduced that the possible dates are July 16, Aug 15 and Aug 17. For him to now know, he must have been told July. Since if he had been told August, he would not know which date for certain is the birthday.

The answer, therefore is July 16.

Singapore Birthday Puzzle: Visual Explanation

Cheryl’s Birthday Explained