2

777 777 88888 99

So I guess if you want to split that up, two hundred seventy seven trillion seven hundred and seventy seven

Billion seven hundred and seventy eight million eight hundred eighty eight thousand eight hundred and ninety nine.

This is a record-breaking number when it comes to the multiplication persistence of a number.

so we'll do a smaller one just to get a

handle on it. We'd like to give me a small. I don't know like three or four digit number. Would you like Brady?

Oh, I always regret asking you this

5428, alright, so the first thing we do is we multiply all the digits together.

So 5 and 2 is gonna give us 10, 4 and 8 to give us 32, so it's gonna be 320

and then we continue we multiply all the digits together now, there's a zero so zero

So you made it two steps,

before we hit a one digit number and if we have any one digit number

You just you can't multiply the digits anymore and you stop so the question is I'm so disappointed

Yeah, can I have another try did you want another go? Yeah, let's try this. It's not called persistence for nothing. What would you like?

See where you went wrong brady is a 502 spells doom because that's gonna chuck a 0 on the end straight away

So don't make that mistake again

327? 327. Nice, you're going for primes. look at you? Okay. Let's see how that goes

So that's six sevens which is 42. Someone will correct me if I'm wrong

Four times two is gonna give us eight and then we're stuck

Two steps. At home if you want try and pick a number. See if you can beat Brady's current record of 2

So, how long do you reckon this one goes for? Your big number? Yeah big number

What do you what do you think we're gonna get out of that one. have a guess, what do you reckon?

You've already told me it's a record holder. That's right

But because you're willing to do it for me now makes me think it's not crazy big.

When have I shied away from a crazy-big calculations?

So I'm gonna go for something surprisingly smaller. I'm gonna go for like

10

Getting my calculator out

two times seven equals

There's one seven

two four

nine nine

Six. ... times six times two.... Oh, okay. So now we're down to

Four times three times eight.... What's very exciting about this is I've never actually checked this

This is the first time I've actually and now I'm feeling a little nervous

like that's the level of preparation yet on numberphile how much prep I put into these videos pretty is like

What numbers are you into currently? I like are kind of interested in this one... two, seven seven times eight

54 5 times 4 is

20

Is 0

1 2 3 4 5 6 7 8 9 10 11

Oh

11 and that's the correct answer. Thank goodness. It has a persistence of 11, which is the current

world record for multiplication persistence of a number other numbers equaled the record

But this is the shortest number with the biggest

currently known

Persistence and so that is the current champion.

Hang on so you're with no limit on the number of digits?

No-knock is about managers everyone and often in these videos

I say I went away and I programmed a thing and then I calculate it and I found this I found that I have not

Programmed this yet because as you may have those little underprepared today, I haven't coded it up

But we could should we do it live do me do it?

Do it cuz I always say I code something and I find them. Let's code something. Let's find it

I'll get my laptop so funny story last week my

Keyboard stopped working and the trackpad on my laptop, which is a bit awkward. Okay

So first things first, we're gonna set something going which is going to do this process over and over and over again

But we want it to stop once it gets to a thing which is only one

okay, so this is gonna take some number and the first thing we're gonna say is if the

the length of the number of digits to the string

n so the great thing about n is, it's not only a number

When we care about digits

I'm going to turn it into a strings

that goes from being like a number represented in binary or whatever base to just being

The base ten digits in a string and if the length of that string equals one

Then we're done right? We're at the very end of the thing, right?

So that's at that point print whatever n is and then finish so return I don't know like "DONE"

All right, that's just gonna tell it you know, your job here is done. Okay. Otherwise we need to

multiply together all the digits.

digits

equals

Let's do it as a list. i for i in

The string version of the number, right?

So they're turning into a string of digits

and then it's taking out each one individually

Oh, but we want them as as numbers

So let's turn them

back into numbers

So bizarrely I'm turning it into digits as a string

then taking out each one separately then turning them back into numbers, right?

Which is kind of, because this is a base-10 thing one of the sad things about stuff like this is its base

Specific so now we've got all the digits

for

j I'm just using j is the placeholder in

Digits so now we want to multiply them all together. Okay, you know, let's have

Current result

Result equals one to start with and then each time

result equals result

Multiplied by that digit and you can do x equals just means make it equal to this times

J whoops

J. Okay, I think that's it

and that's gonna give us a new result, but then we've got to repeat the process

So this is where we can cheat

And I haven't genuinely haven't tried coding this before so I don't know if this is going to work

I'm going to try and get recursive and then put that new result into the same function.

So, if it's one long, it'll stop otherwise

It'll multiply all the digits together and then put it back into the beginning and it will

Keep repeating through this process and you know what? Let's make it print the result each time

So we get, we get to see them all as it as it loops through and then when it hits here

It'll stop and say done. Okay?

That can't go wrong

Let's find out. I really should have checked this in advance.

Okay copy. I'm just gonna fire up terminal

Okay. So what I'm actually gonna do this is the laziest way to run something; literally paste it into terminal.

Let's do the

Persistence of 327 which is the second one, you said.

42

8 and then 8 forever, right? And so it's done twice and then going we're out

Okay, I couldn't fix the code to not get the the last one twice. I just changed where the check is

So actually what I could have done is have another check in here

So I don't print the result and then print it again before stopping

I could put the check in but I'm not gonna

It fit for purpose. Alright for a first pass it's fine. And then the very first one

Let's just check the first one just to make sure I've messed this up five

four two eight

It goes to 320 goes to 0. right? And now the ultimate test can it handle 2 six sevens six eights two nines

There we go, right so and that's that's so much quicker! Right? so it's now spat out

Exactly these all the way down and then it stops doesn't give us a number

Ah, do we want like a, do we want a number of steps at the end?

number of steps?

Okay, okay Brady will cut this out and put it on the second channel

2 one two, three, four five six oops 1 2 3 4 5 6 1 2

There, total steps 11, okay, right

So now we can put a number in

and instantly we get everything.

we could no longer

print every single step along the way. It's kind of fun to see and we get the total number steps

So Brady, let's put on what would you like? You know, let's

Put put as we just mash it for a while and see what we get. No, cuz you can be strategic

Ah, you're right to put a 5

Just put in a bunch put in

Fifteen 9's.

Fifteen 9's. one two three, four, five six seven eight nine 10, 11 12 13 14 15

Which is just nine to the 15.

No, I put in another 10.

Oh another ten nines. Yep

Well, let's just see if that one works

Two steps. Oh

because we got the zero in the answer next time but what I can do in your in terminal just push up you get the

Previous one stuck a few more 9s on the end

two steps now what if we um,

Put in eight instead

two steps

What about, what about the string you used like at the current record holder? Yeah, but put a three on the front

I love it. Okay, so 3

2 1 2 3 4 5 6 1 2 3 4 5 6 9 9 ah brilliant

I love like I love you thinking ready and 2 steps

This is harder than it looks!

So it's zeros are like a

Zeros are the land mines

Boom, hit a zero, you're out. This is minesweeper, but for number searches, okay

So this is what I would do I would now play with this for a while and it seems to be whenever you put in

some random string of

Numbers and boom, right because it was a zero the next one it's gone

We need to be more strategic

What about all the digits of pi.

All of them?

I will Get cracking on that. three eight one four one five nine two

six five I forgot, that's that's enough -

2

Wow, 11 seems a lot more impressive all of a sudden, no wonder it's the world record.

so does this mean that you could set up a program that would just

Put every number in one after another and just leave it, you know, leave it for an hour or two

so what we've done now is we've built the basics of checking

Next, we want to build something to do the search.

so we could just get something to generate random numbers of a certain size

Shove them in and send us an email if it gets a good one. The next step would be to be

Strategic about what numbers we were putting in because already you realized

Don't put it in a 5 you put in a 5 it's not gonna work so we could create a search

Which doesn't put in fives?

Doesn't put in any combination we know will definitely give us

Zero and actually we can get even smarter than that because if we're looking for the smallest number that it works for

It doesn't matter what order the digits are in

So in fact all the current record holders the shortest number of digits for different persistence values

So there's a record holder for 10 and for 9 and so on

They're always, the digits are always in ascending order

Because you want the smallest number with those digits. So in fact, you don't have to search for

Shuffled around versions; you just need that set of digits in that order and then there's other things

So for example this one here

we'll use your first one you put in 5428, that's going to give you exactly the same result as

First of all, you could have put in two four

Five eight, which is smaller

But actually two times four is going to give you eight,

so you could actually have just put in 588

That's going to give you exactly the same sequence as that and it's smaller

So if all you care about are the smallest possible values

for that sequence of persistence afterwards (multiplication persistence)

Then you never want to have a 2 and a 4

You never want to have two threes because I could be a 9 you never want a 5 at all

In fact, you only ever end up with a few small numbers at the beginning

Never more than one 2 never more than one 3

Never more than one 4 I think and then all sevens eights and nines

for the rest of it so we could reduce our search space

dramatically with a little bit of logic the current search has gone as far as

233 digits so if you do code it up you've got to start searching from

233 it's not gonna be smaller than that. We've already checked and it's currently

The conjecture is you would never be 11

So if people want to have a go, I mean, I'm always one to give it a go

See if you can write some code see if it does a clever search and you know

It would be a major breakthrough.

If someone could find a number with a multiplication of persistance of 12

Will you be the person to make that major breakthrough?

Certainly the sorts of people who cracked tough nuts

are those who think outside the box, creative thinkers, people who don't follow the flock and

Brilliant wants to make you that type of person.