Equally sharing a cake between three people – Numberphile

Equally sharing a cake between three people – Numberphile


We’ve got a cake. We’re gonna cut it. Now if there was two of us, as there are, this is quite simple. You just, one person cuts, the other person chooses. All nice and fair, no one’s envious, perfect way to cut the cake. A lot trickier, though, when you have three people. We’ve got three people, Alice, Bob and Charlie. And they’re gonna share this cake. And we want to make sure that it’s as fair as possible. And at the end, nobody is envious of anybody else’s cake. Turns out there’s a very good way to do this that’s actually quite recently discovered, and I’m gonna try and talk you through. So, Alice goes first. And Alice gets to cut the cake into three pieces that she feels are equal, so she would be happy with any of those three pieces. Oh, Gosh. Okay, I’m gonna go there.The thing is I’ve got a get out of jail free card, ’cause it’s a fruitcake, so I can just claim that there’s more fruit in one side. That was what I was aiming for. It’s not bad. Okay. Brady: “I think I can see the dud piece.” Which one’s the dud piece? Brady: “I think that’s a bit smaller.” No. Brady: “Well you have got more there.” It’s got more sparkly bits on it. Yeah. Absolutely. There’s, there’s, I mean, there’s more to a cake than just size. Some people like the sort of crusty bit around the outside, some people like the sugary bits, some people, you know, hate the fruits, there’s, it’s your personal preference that comes into this as well. Okay. So, Alice has cut the cake into three pieces where she wouldn’t mind which of those three pieces she got. Now, it’s Bob’s turn. So, Bob gets to pick which he thinks are his two favorite pieces. Now, you then said that that one was the rubbish piece, so let’s kind of leave that to one side, and assume that Bob goes for one of these two. But Bob, from those two, says which he thinks is the best piece of all. Let’s imagine that Bob thinks this bit’s the best bit. Now what Bob has do, he gets the knife now, and Bob gets to cut a little trimming off of his favorite piece until his favorite piece is now equally as good as his second favorite piece, right? So that little trimming kind of goes to one side, there. So now Bob doesn’t mind between getting this one or this one. He’s fine. He’s happy with either of those. Alice is sort of sitting to one side, desperately waiting for them to hurry up so she can have some cake. Charlie, it’s now Charlie’s go, Charlie gets to pick which piece of cake he wants to eat. He can have whichever he wants. So Charlie might pick one of the pieces that hasn’t been trimmed, or he might pick the piece which has been trimmed. Whatever piece he picks, let’s say he picked this one here, it’s now gonna be Bob’s turn to choose a slice of cake. If Charlie didn’t take the trimmed piece, Bob’s gonna want it, because he’s already said that that’s his top, you know, in his top two, joint top, so Bob’s gonna take that piece, so now Bob’s happy, Charlie’s definitely happy, they got first choice Bob’s definitely happy, ’cause that was his equal top two, and now Alice ends up with this piece, and she’s definitely gonna be happy too, because she cut the cake in the first place, and she cut It in a way that she thought that all three pieces were equally fair. So that’s fine, right? So that’s the majority of the cake sorted. Now, we have to deal with this trimming. Which looks a bit pathetic. Brady: “You can’t just toss that, there? There’s no waste.” No, no, that’s wasteful, Brady. No. Because Charlie went last, on the first main cake cutting section, Charlie now gets to go first, on this bit. So Charlie gets to cut this trimming into three pieces so that he thinks that each piece is equally as good as the others, okay? Now once he’s done that, now Bob gets to pick which of those pieces he wants. Let’s say Bob goes for this one here. Now Bob is definitely happy because he’s got his number one choice out of the trimmings and we already know that he’s happy with his main chunk of cake. Now it’s Alice’s go, and whichever one Alice picks, she’s definitely gonna be happy, because in the very beginning, she thought that all three of the pieces of cake were equally good, she’s ended up with one that hadn’t been trimmed off, and now she’s got an extra bit, so there’s, she’s definitely gonna be happy. And then finally, Charlie ends up with the last little trimming, and he’s definitely gonna be happy, too, because he cut the trimmings in a way that he thought that they were all equal. So there you go. That is envy-free, totally fair and totally delicious cake. There is actually an n-person version of this algorithm. The only thing is, the number of cuts that it takes to get this sort of envy-free solution gets a bit bonkers. Brady: “Yeah, so what do we start getting trimmings of trimmings?” Trimmings of trimmings, and trimmings of trimmings of trimmings trimmings. So in general, the number of cuts that It takes to divide a cake between n people fairly is n to the n to the n to the n to the n to the n. Which is kind of bonkers. You are never aware of how much noise your body makes until you are sat in front of an unbelievably sensitive microphone with, you know, headphones sort of looping yourself back, you know, into your ears. What do you mean, like a grumbly tummy? Grumbly tummy, my God, you never notice how grumbly your tummy is. When you’re in an audio booth, it’s like, it’s like roaring in your ears continually. If you don’t listen to audiobooks, it is something you should try, and you should go to audible.com slash numberphile because audible have the best range of audio books out there. And if you sign up for their 30-day trial, using that URL on the screen, you can download a free book. You can start off with a free one, and I’m gonna recommend well, this is a physical copy, but I’m gonna recommend the audiobook of this one, which is called, The Mathematics of Love, Patterns, Proofs and the Search for the Ultimate Equation. Do you like that? a little bit of curly writing going on there. The Mathematics of Love and it’s, oh! It’s by you! Oh, my God! And the reason I’m talking about it now, is because, not only did Hannah write this book, but she recorded the audiobook version of it. And I want to quickly talk to her about that, because I’ve never met someone before who’s done this. The one that you feel really self-conscious about is, it’s really kind of a bit gross, actually, but like, the sort of the sound of, that your mouth makes. Wet noise? Yeah, exactly. So, it’s sort of almost like clicking. Just the way that your saliva moves around in your mouth. So, there are a few tricks that they have to try and minimize it. One of them is to make you eat green apples. You’re kind of sat in this audio booth eating green apples, trying to reduce the clicking sound. Oh, and you’re also not allowed to have milk or chocolate before you do It. You can’t have milk or chocolate before you record, and you have green apples with you. Yeah, milk and chocolate makes you a bit too mucousy. So disgusting. So when I listen to. Have you ever listened back to your audiobook? Like, have you got it on your…? Absolutely. It’s like someone pouring honey in your ears. Well. That sounds interesting. I’m joking. So if you would like to have honey poured in your ears by Hannah, then can I recommend you go to audible.com slash numberphile 30 day trial on audible. You can, you can download any book as your free one, but I’m gonna recommend the honey in your ears that is The Mathematics of Love by Hannah Fry. We haven’t actually said what the book’s about, because I’ve been so fascinated by – It’s obvious. It’s obvious what it’s about. It’s all about, like, how mathematics applies to things like dating and relationships and stuff like that. Yeah. Yeah. Exactly. How to use maths to make your own love life a little bit more efficient. And it is, it is quite a good book. I have read it. And I have listened to it. Honey? … dunno. Maybe, maybe. More mucousy milk. That’s right.

9 thoughts on “Equally sharing a cake between three people – Numberphile

  1. Try https://www.audible.com/numberphile and check out The Mathematics of Love by Hannah Fry

    Envy free cake cutting (the paper mentioned): https://arxiv.org/abs/1604.03655 — A clarifying note from Hannah: n^n^n^n^n^n is the maximum number of cuts in the n person case.

  2. It is easier to cut it into four, then recursively cut one of the resulting pieces into until the sizes are negligible.

  3. You can do it with fewer cuts if Bon and Charlie divide the trimmed part between themselves as if they split a cake between two parties.
    It is also more fair, as the Alex shouldn't really get additional piece.

  4. Do mathematicians have to come up with some practical application when publishing these things, or can they just publish anything?

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