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  • it is cool, balanced Turnberry, and what you do is you write your number in base three and then your place all the tools by minus ones.

  • So let's do it.

  • Zero is fixed.

  • One it's fixed to in base three just says to Andi, The rulers replace all the tools by minus ones, so this is now minus 13 Internally, it's one not so it doesn't change.

  • Four is 11 intern ary, and it doesn't change.

  • Still, four five is one to which becomes one comma minus one.

  • Now one common minus one means 13 minus one.

  • So it's 26 is 20 which becomes minus 10 which is minus three.

  • Let me remind you were writing numbers and base three.

  • So it's minus one comma zero sub three, which means minus three plus zero zero on the ones in the one's position that suits my history.

  • Seven is to one, which becomes minus +11 So that's minus three plus one, which is minus two and so on.

  • And if you do this, you get all the insurgents both positive and negative.

  • Exactly ones.

  • It's a way, if in a concise way of the numerator ng all the all the insurgents.

  • And if you look at the graph, it looks like this It looks like Star Wars.

  • A scene from Star Wow, Isn't that great?

  • Yeah, Yeah, Look at them.

  • They're all those star destroyers here here, And that graph will eventually as expands out floor every single interview, negative and positive here little they're just keeps coming.

  • Mysterious.

  • So now the sequences you take, n you write it in base 10 and you subtract the product of the non zero digits.

  • One product is one Subtract we get 02030 Very boring up 290 because the number of minus the product of its digits is zero.

  • If there's only one digit Okay, 10 10 has one non seven digit one.

  • Its product so subtracted We get 9 11 has the product is one.

  • So we get 10.

  • 12.

  • Product was to only get 10 13.

  • Look at the product of the non zero digits and subtract it from the number 10.

  • 14 is 10 again and well, let's jump forward by the time we get up to 25.

  • Products of digits is 10.

  • So we subtract 10 from 25 when we get 15 so it's drastically changed.

  • 26.

  • Subtract 12 from 26 we get 14 and so on.

  • The graph looks like this.

  • I call it mysterious because to a gardener this looks very like wisteria purple flowers you see in the trees.

  • Nice.

  • It's like the group gets bigger, but it never droops further than well.

  • You're not subtracting very much because you're looking at the product of the digits, and that's much less than what you started with since the digits rule nine or less.

  • So once you start getting interesting numbers like, you know, 4692.

  • Yeah, four times, six times nine times two is whatever it is, but it's not very big, so it's not.

  • It doesn't drew very far from what you started with something like some pretty big groups there.

  • Well, yes, they had a lot of big digits like that.

  • Yeah, mysterious.

  • So this is the forest fire and rule is that we put down the Mexican graphically earliest sequence, meaning you always pick the smallest numbers.

  • If you have a choice, go small.

  • I'll just do it.

  • That's easiest way we start off when we have a choice.

  • Pick the smallest positive number.

  • So so far we don't have anything to worry about, so I can pick a one.

  • It's gotta be positive.

  • Okay, I can put another one because it's not a big deal.

  • The rule says you may not have three terms that air in an arithmetic progression, meaning the difference between here and here is the same as the difference between here and here.

  • So a of I plus Jay Miner save I must not equal a of I Plus two J minus ev I plus J.

  • We've got two ones here.

  • I would like to put a one here because that's the smallest legal move.

  • But then the difference between then these three terms if we did that, we'd have a one and a one and a one, and they will be equally spaced.

  • That's not allowed.

  • OK, there's no loud.

  • They jump I zero each time equally spaced numbers with a constant difference between them.

  • It's not allowed, so we go to the next time.

  • There's nothing wrong with putting a two there.

  • What about the next one?

  • Well, I could put a one there because there's no conflicts.

  • We've put one there.

  • Now.

  • What about here?

  • Could I put a one there?

  • Yes, I could.

  • I can.

  • And therefore I must, because there are no three ones that are evenly spaced.

  • Yes.

  • Let's go on.

  • You'll see now.

  • So the sequence begins.

  • 11211 Now, what can I put here?

  • Should always ask.

  • Can I put a one?

  • Can you put a one?

  • You start off.

  • Can I put it one?

  • Can I put it to We can't put a one because we'd have 111 And that would be a violation of the rule.

  • Constant increment of 03 equally space terms.

  • Not all that can afford it.

  • So I think we can do it too.

  • So we've got 112112 All right.

  • What's the next term now?

  • Could it be a one?

  • Well, I think it's gonna be a problem if we have one.

  • Because 1 to 4 to seven, we would have a one jump of three on a one again jump of three.

  • That's not allowed.

  • That would be an arithmetic progression.

  • 111 at three terms which share equally spaced.

  • That can't be a one.

  • Could it be a too?

  • Well, Yes, I are.

  • You sure?

  • It looks like it could be a two.

  • Yeah.

  • Okay, too.

  • Nothing wrong.

  • Okay.

  • What about the next term?

  • Could it be a one where it can't be a one by the same rule?

  • Because we have 258 That would be a 111 No, for Bolton.

  • Forbidden.

  • No, lad.

  • So comfy.

  • One could be a two.

  • Well, no.

  • Because then we'd have to to to Okay.

  • Could it be a three?

  • Well, if it was a three now, here we see something a little bit more subtle.

  • If this was a three, we would have 123 constant difference of one at three equally space points.

  • No, can't be a three, but it could be a four.

  • And it keeps going.

  • That's the sequence.

  • It's well defined.

  • It's unique.

  • And it's crazy.

  • It looks like this Here is a graph of it.

  • It's actually bean filtered out a little bit.

  • The picture is very busy.

  • So one of my friends Shan Greg, remove some of the the blackness from it.

  • So it looks like this.

  • This is 100,000 terms and you notice this This pattern of smoke which is repeating it has a fractal structure and we do not have any explanation for that.

  • You think?

  • Wow, this is such an elegant definition.

  • What's going on?

  • No one has analyzed this.

  • What do you mean by no one's analyzed it?

  • What do you want them to find out about it?

  • Like what would What would be?

  • Well, you could say.

  • For instance, what we have here is something like the Easter Island heads that get bigger and bigger and bigger.

  • And they're spaced at such and such an interval.

  • There is some regularity.

  • What is it?

  • When is the next blob gonna be?

  • How biggest the blob?

  • What's going on?

  • So when we first started plotting sequences in the O.

  • E I s or converting them to music, I made two artificial sequences to display this The music when is for at Lisa converted to sounds that sequence a 123456 If you're interested to display the graph, I made this sequence and you say worth unearth What sequences?

  • This?

  • Well, it's very simple.

  • If you turn it around, it is a famous profile head of a woman in profile from the Cleveland Museum.

  • Why'd you pick that one?

  • Clean line.

  • And it's also a beautiful drawing.

  • And if you found any properties to the No, No.

  • Great.

  • You found out you were right.

  • All right, then.

  • This is brilliant.

  • Have sponsored today's video.

  • I'm gonna go back into this geometry fundamentals.

  • Course, I've been doing it before doing a rock.

  • See what I'm up to here?

  • 27 unit use a packed together to create magic.

  • A Rubik's Cube.

  • If all eight off the corner cubes of removed, how much will the surface area of the figure increase?

  • So I'm removing three.

  • I'm exposing three.

  • I think it will stay the same way, you know.

  • Yes.

  • Ah, yes.

  • Nice.

  • Shall we have a look at what's next?

  • How many of the following nets can fold up into this square pyramid showing above?

  • Look, people, if you want to check out this course or all the other ones on brilliant, have a look at this.

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