It’s Follow-Up Friday: Birthday Puzzle edition!

Welcome to Follow-Up Friday!

By this time, you know the drill. Follow-Up Friday is a chance for us to revisit the subjects of previous posts and bring the PuzzleNation audience up to speed on all things puzzly.

And today, I’d like to return to the subject of birthday brain teasers!

Working on the Cheryl’s Birthday brain teaser a few days ago reminded me of another birthday-fueled puzzle that’s been around forever.

How many people do you need to enter a room before the probability of any 2 or more people sharing a birthday (day and month only, not year) is greater than 50%?

Assume for the sake of the puzzle that birthdays in the population at large are equally spread over a 365 day year.

Now, given that there are 365 days in the year, you’d assume the number of people necessary to get that probability of a shared birthday above 50% would be more than half of 365, or 183 people.

But it turns out that, statistically speaking, you don’t need anywhere near that many people.

Let’s break it down. Person A has a birthday. Person B has a birthday. There’s only one possible pairing, A-B. Person C has a birthday, but creates three possible birthday pairings: A-B, A-C, and B-C.

Person D could have a different birthday, but the introduction of Person D begins escalating the number of POSSIBLE shared birthdays. With these four people, we have SIX possible pairings: A-B, A-C, A-D, B-C, B-D, and C-D.

Our fifth person, Person E, allows for TEN possible pairings: A-B, A-C, A-D, A-E, B-C, B-D, B-E, C-D, C-E, and D-E. The probability of a shared birthday is increasing much faster with each new person.

As it turns out, it only takes 23 people to give us a 51% probability of a shared birthday.

And that would certainly save on catering.

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Are you smarter than a Singaporean student?

We love brain teasers here at PuzzleNation Blog. Whether we’re dealing with curious parking spaces, men in hats, the crew of the Enterprise playing games, or the seesaw-based conundrum that so baffled Captain Holt on Brooklyn Nine-Nine, we thoroughly enjoy tackling these often diabolical and curious logic problems.

And one has been making the rounds on Facebook, Twitter, and other social media platforms recently. This one comes from a Singaporean classroom, and has made headlines all over the Internet.

Hmmm, doesn’t seem like a lot of information, does it?

So, we have ten possible dates.

Let’s put them in a chart to organize them as best as we can.

Now, let’s analyze each statement in order, since the progression is the key to solving this brain teaser.

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

Since Albert is told the month, and there are multiple options for each month, there is no way he could know. At first. But he does know Bernard doesn’t know either. How?

Deduction. If Cheryl told Bernard 18 or 19 (the only days that appear once), Bernard WOULD know Cheryl’s birthday. So Albert can eliminate those two options.

And for Albert to KNOW that, Cheryl cannot have told him May or June, since those were the only months with days that appear once.

A lot of information in a single sentence. Let’s move on to the next sentence.

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

Bernard is on the same track as Albert. He’s eliminated May and June. And he says he knows Cheryl’s birthday. If you look at our chart now, there are three singlet dates (15 for August 15, 16 for July 16, and 17 for August 17). If he was told 14, he wouldn’t know if it were July or August, so we can eliminate those.

From ten possible days, we’re down to three. And Albert’s final sentence finishes the job.

Albert says: Then I also know when Cheryl’s birthday is.

Since Albert is only told the month, it has to be July, because there are two possible dates left in August.

Therefore, in impressively brisk fashion, both Albert and Bernard have deduced that Cheryl’s birthday is July 16. And so have we!

We’ve also deduced that Cheryl is sort of a pain in the ass. But I suspect that wasn’t much of a brain teaser.

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