[00:01] you. I want you to think of any number between 1 and 100. Just think of it in All right, picture it in your head. Try and transmit it to me. Yeah? Yep, doing it. Yeah, you do it. You do something weird with your
[00:13] I'm pushing it towards you. I'm pushing the number out of my brain into your >> [laughter] >> Like some sort of rectal or something. do is I want you to take I want you to look at this list. There's a lot of
[00:27] >> I want you to think of your think of your number again and I want you to tick every row that it appears in. Okay. You're going to have to hold the is expensive piece of kit. How do I know I'm pointing in the right direction?
[00:41] >> Every one my one's in? >> Yeah. Okay. And make sure you um >> Otherwise it'll ruin it. And and try to transmit it to me. Your arm's in the >> I can't. Uh
[00:55] >> No, I'm not. No, okay. Yeah. I've got to impress people with this. I think I I'm right or if I'm wrong. Okay. >> turn that back. Okay. All right. Okay, now to keep keep transmitting that
[01:08] >> [laughter] >> All right. Yep. >> All right. Yep. Let [snorts] me think about it.
[01:24] a seven in it or a two in it. Is it 72? [laughter] All right, there we go. So, you impressed? Mm. What do you mean mm? How many numbers there are there? Yeah,
[01:39] okay. Yeah. You should be impressed. Yeah. All maybe you'll be impressed by the maths of it. Okay. Right. Okay, now let's have at these numbers in this little this little thing. Yeah. All right, the first
[01:53] number in each row, every single one of them is Fibonacci. 1 2 3 5 8 13 21 34 55 numbers? I don't know, yeah. >> Yeah, so that's that sequence where you
[02:09] and then you keep adding to the the the two before. So, if you wanted to get generate the Fibonacci sequence, you've got you take Fn, which is the nth, you add it to Fn + 1 and that will give you Fn + 2, right?
[02:23] this again. >> Yeah, 1 1 2 3 5 8. Exactly. So, these One is Fibonacci two two is Fibonacci three. This is the Fibonacci sequence,
[02:36] right? And all I did was to get that answer, that 70 What was it? 72? I took these numbers Yeah. >> and added them together and it gave you your answer and it always works. Right. >> Okay, but why is it working?
[02:49] It was >> Okay. 1 3 1 Yeah, 3, so that's 4. Uh 13, so that's then that's I'm giving you 17 and then 55. So, that's 72. Always
[03:03] Fibonacci. There's something called the Zeckendorf decomposition. So, what do I number. It can be written in terms of Fibonacci numbers. So, let me give you take number 27, for example. Okay, I can write the number 27 as 21 + 5 + 1,
[03:21] right? I can certainly do that. Yeah. And these are all Fibonacci numbers. which I could write 27 in terms of Fibonacci numbers. I could write 13 + 8 Fibonacci numbers. I could write 13 + 8 + 5 + 1. I could do 21 + 3 + 2 + 1,
[03:37] ways I can do it, but there's something special about this first one. Now, this first one, what you'll notice is that none of the Fibonacci numbers that appear in it are adjacent to each other in the Fibonacci sequence.
[03:51] >> Yeah, exactly. So, if I look at this one to this this decomposition I've written here, 13 and 8 are next to each other in the Fibonacci sequence. And in this one, for example, are next to each other in the Fibonacci sequence. This
[04:06] decomposition this this particular choice which is unique for 27 is called the Zeckendorf decomposition and it's a decomposition in terms of Fibonacci numbers in which there's no two adjacent ones. And can every number be made in
[04:21] >> Yes. Yes, it can. And this gives rise to base Fibonacci. I can probably squeeze this in there now. You've got I've got another piece of paper here. another piece of paper here. >> [music]
[04:34] So, you know, base base two which is binary, base 10 which we're used to there's also something called base Fibonacci which is based on the Zeckendorf decomposition. So, let me just write down the the few of the
[04:47] could carry on, of course, going that way. If I want to write 27, how could I write it in base Fibonacci? Well, I know it's got what If I do the Zeckendorf decomposition, it's got one of 21s, it hasn't got any of these, hasn't got any
[05:00] of these, eight. It's got one of five. It's got no threes, no twos and it has got a one. So, this is how you would write 27 in base Fibonacci. you hack it off the total and see what >> Exactly. And actually, Brady, you've
[05:14] really really important about the Zeckendorf decomposition and why it called a greedy algorithm. So, what's what's what's So, what do I mean by this? And this this can explain why you never get two adjacent ones. So, so
[05:26] never get two adjacent ones. So, so let's give an >> [laughter] >> So, right, we like sweets, right? Now, so this this this Let's say there's 100 sweets in this sweet shop. Then I go in
[05:41] first, right? And I'm greedy. I'm going to buy as many sweets as I can, but it has to be a Fibonacci number. Go in, there's 100 sweets. I'm going to buy as many as I can. I need to find the largest Fibonacci number, okay, that's
[05:54] below 100. Okay, that is 89. right? Now, you follow me in and you're a bit annoyed cuz there's not as many left now. And you go in. There's 11 left now, right? And you go in and you buy as
[06:06] many as you can and it's got to be a Fibonacci number, greedy algorithm. So, how many can you get? You can get eight. Okay? And we can keep going like that. Now, there was no way we could have ever had an adjacent number of of um of
[06:20] Fibonacci numbers. Why? Let's take an arbitrary amount of any number of sweets arbitrary amount of any number of sweets in there. I go in first. I buy as many as I can and it's a Fibonacci number. Let me call it Fn + 1, okay? I buy this
[06:32] Then you go in, okay? And there's less now, right? But you buy the biggest the adjacent Fibonacci number. Let's say you do that. Let's imagine that's right? And together that means we've got this many, which is
[06:47] >> Another Fibonacci number, but I took as many as I could possibly take, right? taken. >> That's how many I would have taken. So, protecting this greedy algorithm proves it's impossible to get the to pick up
[07:01] And that's really the basis for the Zeckendorf decomposition and why you get these unique representations in terms of Fibonacci numbers for any number. And this is the core of this magic trick as well. So, how were these lists created
[07:17] to work? What's the process? >> So, every number is if if if it's got if >> So, every number is if if if it's got if it's if it's got an entry one in here, basically the idea. So, let's look at it. Let's look at 27, right? So, so so
[07:32] claim 27's going to appear in the one row. Yes, it does. It's going to appear It doesn't. Or the three row. No, it doesn't. It will appear in the five row, can see it's going to pick it's going to pick out the Zeckendorf decomposition.
[07:46] This is exactly how it's working. It'll pick it in it'll be in the 20 21 21 row there as well, which So, it literally picks out each one. And that's how it with the with the Zeckendorf decomposition. I just think the trick's
[07:59] got What is this? [laughter] It's mathematical magic. It's not that magical if you have to carry around this list with you and and require that much work and information from me. It detracts from the trick, but
[08:12] Well, I I think you're too fussy about your presentation on your before? I think I did it on my daughters and they weren't impressed, either. I don't know. I think it's good.
[08:30] know if it's infinite at the moment. What I do know is this this this next one in the sequence is at least as big as 2015. >> [laughter] >> It is You're right. You're right, Brady.
[08:43] You're right. It's way bigger. It's actually way bigger.