Hi, everyone! So there’s a maths problem that’s gone viral this week, and I’ve got a new blackboard! So I thought– you know… So, the problem–you may have seen it already, it’s called Cheryl’s Birthday Problem, but if you haven’t seen it, well, you know how it works: I’m going to give you the problem, you’ll have time to try and solve it for yourself, if you want, and then we’ll talk about the solutions. So the problem is this: So you’ve got Albert, Bernard, and Cheryl– Or A, B and C if you prefer. Now Cheryl–she sounds like a bit of a nightmare to me– Cheryl says “My birthday is one of these ten dates.” So what have we got here? We’ve got May 15th, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17. So Cheryl says “My birthday is one of these ten dates.” She gives Albert the month of her birthday, she gives Bernard the number of her birthday. And then there’s a following conversation between Albert and Bernard. Albert says, “I don’t know when the birthday is, but I know Bernard doesn’t know, too.” Bernard says, “At first, I didn’t know when the birthday is, but now I know.” And then Albert replies: “Then I know the birthday, too.” When is Cheryl’s birthday? You can use this information to work out when is Cheryl’s birthday. So, I am going to give you time to pause the video, or have a go for yourself, if you haven’t already. And then we’ll start talking about the solution after this flash. So this problem has caused a lot of controversy on the internet this week. There’s been a lot of debate about what the correct solution should be. Now, I can tell you what the expected solution was. The expected solution was July 16, but that might not be the answer you’ve got, and I want to talk about alternative solutions in a minute. But for now, let’s run through what the expected solution was. So, from statement one: Albert knows Bernard doesn’t know. So Albert is holding the month. He can’t have May or June. May and June have unique dates in them. The 18 and 19 are unique numbers. So Albert cannot be holding May or June, or he wouldn’t be able to say that statement. Statement two: Bernard now knows. Bernard has gone through that same logic we’ve just said. He knows the month must be July or August. He’s got the date–he’s got the number. It can’t be 14, because there’s two choices for 14. So it can’t be 14. It must be one of these three: the 15, the 16 or the 17. Statement three: Albert knows the birthday too. If Albert knows the birthday too– he’s gone through the same logic. He’s worked it down to these three answers: July 16th, August 15th, August 17th. He can’t be holding August, because he would have two choices, which doesn’t help him at all! So he must be holding July which means the answer is July 16th. Now, that’s the answer I got when I first saw this posted– it was by Alex Bellos on the Guardian blog. And underneath, there was a lot of comments, there was thousands of comments of people debating the answer. Now, the majority did get July 16th, but it wasn’t an overwhelming consensus. So let’s have a look at some of these alternative answers. It turns out, it might depend on your point of view! So the third most popular answer I saw was August 17. And it all depends on how you choose to interpret This first statement: “I know Bernard doesn’t know.” Now, if you choose to interpret that as fact, then you get this different answer. And let me just run through the process of that for you. So Albert says, “I know Bernard doesn’t know.” So, maybe he’s realized that Bernard hasn’t just jumped in with the answer. So he knows It’s not a unique number. So he can now eliminate May 19 and June 18. Now Bernard says: “Oh! I know the answer!” Well, Bernard has gone through the same thinking, he’s eliminated June 18, May 19 as well. He’s also noticed that Albert hasn’t jumped in with the answer. If Albert was holding June, then June 17 would be unique. So Albert can’t be holding June–he would know the answer. So we can now eliminate June 17 as well. Now, Bernard knows the answer. His number must be unique in the remaining solutions. What have we got? We’ve got two 14’s, so he wouldn’t know. We’ve got two 15’s, so he wouldn’t know. We’ve got two 16’s, and we’ve got two 17’s, But, we’ve just eliminated June 17, so it must be August 17, because Bernard knows that it’s August 17. Meanwhile. Albert–ah!– annoyed that Bernard has beaten him to it, Albert goes through the same process, and gets the same answer, August 17. Now, that is valid, and that all does work but it does depend on your interpretation of this first statement: “I know Bernard doesn’t know.” So what we’ve got here is an example of the difference between a statement, and knowledge of the statement. So the statement here is, “Bernard doesn’t know.” Now, one argument says, we assume this is true, this statement– we go through the logical process, so something, something, something– and–we get the answer, and it was August 17 in that case. The alternative argument is we assume Albert’s knowledge of the statement is true, we go through the process, something, something, something– and we get the alternative solution, which was July 16th. Now, the people who wrote this question originally– it was actually a Maths Olympiad question from Singapore– they have now rejected this alternative solution. They’ve said that this first statement is a statement of knowledge, not a statement of fact. But I can see why people have misinterpreted that first statement. Especially people who aren’t familiar with maths questions like this. I can completely sympathize why they would read that as a statement of fact. Now, the second-most popular solution I saw, I actually have less sympathy for this. The second-most popular solution I saw was June 17th. And let me run through the argument for that. So, from the first statement, “I know Bernard doesn’t know”– so like we’ve done before, we now eliminate the unique dates. So that’s the May 19th and the June 18th. Now Bernard knows the answer. And what people were arguing was Bernard has 17–he knows the answer, so he must have June 17, because it’s all on its own in the month of June. It’s unique in the month of June. It must be June 17–that’s how he knows! Which is–wrong. And I think what they’ve done there is they’ve not quite put themselves in the position of Albert and Bernard. They’re seeing themselves as themselves, like these are statements to you, the reader, and they haven’t put themselves in the position of Albert and Bernard, or the knowledge of Albert and Bernard. And I can see why they’ve done that– so that’s how many books are written– they are written to you, the reader. That’s how many of these maths questions are written– statements are written as information to you, the reader but I think that’s where that falls down, not looking at things from the point of view of Albert and Bernard. So Cheryl’s Birthday Problem is actually an example of something called Public Announcements Logic. It’s where knowledge, or truth, can be determined with new information. And logic really goes to the heart of mathematics, and it is the foundation on which we base all other mathematical results. Another example might be the old mathematician’s joke which you might have heard of before: “Three Logicians Walk Into A Bar.” So three logicians walk into a bar, The barman says, “Does everyone want a drink?” The first logician says, “I don’t know.” The second logician says, “I don’t know.” The third logician says, “Yes.” And I imagine that’s how Albert, Bernard and Cheryl like to spend their evenings. And that’s all from me, so if you have been, thanks for watching.