I've been reading some Martin Gardner books, all sorts of gems in there, here's one I haven't practiced very much. It just needs a die. I want you to pick a number on the die, don't
tell me what it is. So I can't actually see anything at all which means that you can feel free to sort of,
I don't know, show the the video watchers- (Brady: Let's go for that number.)
- A whole number right? You haven't got a picked pi or something? (No it's a- it's an integer.)
- Okay (And it's on a die.)
- Good. You should be able to see three numbers on the die.
(Yes I can) Can you see your number?
- (No) Interesting. Can you see your number now?
- (Yes) That's interesting. Can you see your number now?
- (Yes) (Is it the one on top?)
I think so. (Yep, you got it.) [Laughter] Well I'm excited. [Laughter] (How did you do that? Can you can) (you teach it so people can do it to) (their parents?)
- I can do. Do you have any idea- because I've never tried this trick on anyone before and, first of all, with all magic tricks there's a trick right? But
when you know the trick the thing's lame.
I've just done a lame thing. Is it at all interesting that I could know with- blindfolded what your number was? (Yeah!)
- Okay, can we try it one more time? And I think you might start to spot what's going on, like all magic tricks you repeat it and you're like wait a second. So so you you decide a number again.
- (Let's go) (for - uh - let's go for that number.)
I'm trying to point a corner towards you so- it's interesting on a D6, you can only ever see three numbers. Can you see your number now?
- (No) Interesting. So let's try the
question again there, can you see your number now? (Yes)
- That's interesting, what about, what about now?
- (No) That's very intriguing, why would you choose that number? (Yes!) Why would you choose a 1? I mean that's what lost you in that pig game When I first saw this I was like, ah it's like what they call a binary search. A binary search is where you're hunting for something that
you don't know and you try and eliminate half the possibilities in one go. And that's kind of the key to this; I'm asking three questions. And that should be enough information, it's just that doing it with a blindfold somehow makes it feel surprising. I'm still surprised that I get it right- Audrey I know you're impressed too- (Audrey!) (Audrey - out!)
- Do you want to roll a die? - No? Okay.
- (Out. How do we-) (how do we do how do you do it?)
- So a binary search - get rid of half the options with each question. And a moment's thought reveals that I've only got six options. So first of all I could ask six questions; the trick would be a bit rubbish - is it 1? No, is it 2? I guarantee you- I think I did those in three questions and if you halve something three times actually you've reduced from eight to one; so three halvings should be plenty of questions to get this. So that's the that's the base principle; like I'm just asking enough questions. So I think I can talk you through this; the first thing is is one of the six numbers I ask a question which is, can you see your number? And if you say yes what do I know?
- (Narrowed it to three.) Yeah and I'm blindfolded but I can still feel there's three faces there there and there that I know you can see. If you say yes I know it's those three, if you say no I know it's one of those three. So I've halved it. And then the next question is, if I ask you then- let's say you say no on that and I spin it round, if I then ask you can you see your number? It's a stupid question right because I know you can, so I need to hide one of these. And all I did was spin that forward and now I know that two
of them are ones that might be your number. And if I say can you see your number and you say no, I knew that these two were possibles and the one on the bottom is also possible. And you can't see it so the one on the bottom is your number. But if you said yes then it's one of these two so I just need to hide one of those two and now that one's the only possible one that you can see, or it's the one that you can't see. So if you say no it's that one, if you say yes there's that one.
- (Let's) (use the number 1 again.)
- Okay so you've chosen 1 but I don't know that; I'm blindfolded, I can't see anything. But I know that if I spin the die so that you can see three then I ask a question, yes or no,
can you see your number? And in this case you would say yes, so I know it's one of these three. If you'd said no I would have known it's one of those three. In this case it's not so I've already got half the solutions out of the way. I now want to ask again; if I didn't move it I wouldn't be getting any new information so I'm going to hide one of these three, gonna hide that one. So now I know that these two faces are possibles and the one on the bottom. So can you see your number?
- (Yes..) So I know, even though I'm blindfolded,
that it's one of these two, so let's hide one of them. Can you see your number?
- (No) It's the one I just hid
which is on the bottom and that's it. And because of
this halving business I'm guaranteed to get down to only one left in at most three turns. In fact it might go even quicker. (Let's do it now if the
one wasn't visible) (in the first go.)
- Okay, and I'm also going to rig it slightly to show how it could go even quicker but- can you see your number?
- (No.) So under the cover of my hands, if I'm blindfolded I'd spin this all the way around and that's my mental check, right it's one of those three numbers; and I would hide one of them. Now I don't know which one to hide but I've just got a sort of consistent way of rolling it so I'm convinced I know where it is. And now I say, can you see your number?
- (No) And I know it must be the one I've just hidden then so which
point I just clean up, walk away, in two turns and that's nicer. Although as a performer for me I'm like it just feels really lame, I've just like just asked a couple of questions. But I still think the effect is quite nice and it's a nice demonstration of searching efficiently. And it also means that if eight is what doubling three times gives you - so halving three times gets you back to one - I've got more than enough information, I should be able to do this on a D8. (Can you do it with the D8?)
- Mathematics says I can; practice means I'm a bit nervous but we can have a try. This is an octahedron and it does have eight sides-
- (But I can see four at a time now!) Yeah that's true, so we've got- and I think if I aim a corner towards you you can only see those four - can you see those four? Okay, the most difficult thing with me in my blindfold is like orienting the dice so you can actually see the four, because if I put it like that I don't know what you can see. (Well do you want to do this one) (unblindfolded?)
- No we'll do it blindfolded (All right, should I choose a number?) Choose a number; don't tell me whatever you do.
- (All right,) (let's go for that number.)
- So you and the camera can see four numbers? Can you see your number?
- (No) You can't see your number? So I'm kind of hiding what I'm doing, you might want to speculate about what I'm doing. Let's go with- can you see your number?
- (Uh no I can't.) You can't see your number? This is the bit that I'm just super feeling my way about. Can you see- Oh dear - can you see your number now? (No!)
- Interesting Your number is on top.
- (On top there?) (Do you want the good
news or the bad news?) I want both. (You got it!)
- Yes! But I knew I was going to get it all along because I'm supremely confident in mathematics and my tactile blind handling of a D8. (Let's do the five again.)
- So you go for five so can you see your number, yes? But all I know is it's one of these four. My next job is to halve the amount of information so I need to hide two of these faces; so I did that in this case by just rolling over like that. I know that two of the ones you could see are now missing but two of them are visible. So can you see your number? (Yes)
- So I know it's one of these two. I'm kind of feeling with my first two fingers each time thinking it's one of those two. If you'd said no I knew it'd be one of those two that I first showed you, but I've got to keep track of where they are
with my blindfold. But since he said yes I know it's one of these two, so I just got to hide one of those two. So let's just roll it
onto there, so I know that one of them is on the bottom, that helps me know where it is, and one of them is visible and I know it's the front one. I say can you see your number and you say no, in which case I know it's the one I just hid so I flip it over and flourish or something. And sort of start sweating about whether I've actually got it.
- (Nice!) But it's it's a nice example of like searching efficiently if you've got a lot of things, trying to halve it each time is one of the most efficient; and that's partly why binary as a number system is like used in computers because it's kind of an efficient way of thinking about breaking up numbers. And that's just a nice demonstration of a binary search. The fact that you can do it on a D6 - Martin Gardner talked about doing on a D6 and I was like, that means it must be possible on a D8. (Can you do a higher D now?)
- There's some options right? There's a D10 - that could be done in four questions. D12, the dodecahedron icosahedron, that could be done in four; and that could be done in five. But it's hard, I mean I'm finding the octahedron hard enough. So I'd need a lot more practice but it could be done and the thing is I start to doubt whether the impressiveness- like four questions feels like quite a lot. The fact that the D6 sometimes works out in two is what gives that little kicker like boom boom, got it. Brush up on your dice mathematics as part of this Casino Probability course on Brilliant, today's episode sponsor. Now it doesn't matter if you get things right, or occasionally wrong, you're always learning with Brilliant's fantastic quizzes and puzzles and their deep dive courses. I do really enjoy doing these, interactive, gets my brain turning, yeah. Premium subscribers get access to everything on Brilliant but there's also plenty of stuff to see for free, so why not go check them out see what you make of it? If you do want to become a premium subscriber, Numberphile viewers are going to get 20% off by going to brilliant.org/Numberphile, that's brilliant.org/Numberphile there on the screen. And if you're already a member, why not consider gifting one of those premium subscriptions to the student or the lifelong learner in your life? ...Well of course we know what it means in practice, we mean if we throw it, you know, it should land on each of the 30 faces a 30th of the time. But let me talk about that for a second. When I was a graduate student we had a guy who was a retired executive who wanted to test the laws of chance- And well this means more often than that that's the end of the story. So the totem pole continues like this, this one beats this one and this one beat this one. So I think- this is almost for the simple game you're not playing
against someone else, aim for 20 is a good piece of advice.
