What's special about 277777788888899? - Numberphile

Today we are doing the number 2 777 777 88888 99 So I guess if you want to divide that, two hundred seventy-seven trillion seven hundred seventy-seven billion seven hundred seventy-eight million eight hundred eighty-eight thousand eight hundred ninety-nine. This is a record number when it comes to the persistence of multiplication of a number. So we'll make a smaller one so we can handle it. We would like to give me a small one. I don't know a three or four digit number. Would you like Brady? Oh, I always regret asking you this 5428, okay, so the first thing we do is multiply all the digits.

So 5 and 2 will give us 10, 4 and 8 to give us 32, so it will be 320 and then we continue, we multiply all the digits now, there is a zero, so zero. So you did it in two steps, before arriving. a one-digit number and if we have any one-digit number You just can't multiply the digits anymore and you stop, so the question is: I'm very disappointed. Yes, can I try again? Did you want to try again? Yes, let's try this. It's not called perseverance for nothing. What would he like? See where you went wrong, Brady is a 502 doom spell because he will throw a 0 at the end right away.

So don't make that mistake again, 327. 327. Okay, you're going for prime numbers. look to you? Well. Let's see how he goes. So that's six sevens, which is 42. Someone will correct me if I'm wrong. Four times two will give us eight and then we will plateau at two steps. At home, if you want, try to choose a number. See if you can beat Brady's current record of 2. So how long do you think this one will last? Your big number? Yes, large number. What do you think we are going to achieve with that? Guess,

what

do you think? You've already told me that he has a record.

That's right, but the fact that you're now willing to do it for me makes me think it's not crazy. When have I avoided doing incredibly large calculations? So I'm going to go with something surprisingly smaller. I'm going to take out like 10. I take out my calculator two times seven equals There's one seven two four nine nine Six. ... times six and two... Oh, okay. So now we're at Four times three times eight... What's very exciting about this is that I've never checked it. This is my first time doing it and now I feel a little nervous like that's the Level of Preparation still in Numberphile How much preparation did I put into these videos?

Nice is. What numbers are you currently at? Like, they're kind of interested in this one... two, seven seven times eight 54 5 times 4 is 20 It's 0 1 2 3 4 5 6 7 8 9 10 11 Oh 11 and that's the correct answer. Thank God. It has a persistence of 11, which is the current world record for persistence of multiplying a number. Other numbers equaled the record. But this is the shortest number with the largest persistence currently known, so it is the current champion. Wait, you don't have a limit on the number of digits? No-knock is about managers for everyone and often in these videos I say I went and programmed one thing and then I calculated it and found this.

I discovered that I hadn't programmed it yet because, how is it possible that nowadays they have those little people poorly prepared. I haven't coded it. But we could do it live. I do it? Do it because I always say I code something and I find it. Let's code something. Let's find it. I will receive my laptop. Very funny story last week. My keyboard stopped working and so did the trackpad on my laptop, which is a bit inconvenient. Okay, first things first, we're going to start something that will do this process over and over again, but we want it to stop once it gets to something that's just okay, so this is going to take some time. number and the first thing we're going to say is if the length of the number of digits is the string n, then the nice thing about n is that it's not just a number.

When we worry about the digits, I'll convert it to a string that goes from being like a number represented in binary or any base to just being the base ten digits in a string and if the length of that string is equal to one, then we end up , TRUE? We're at the end of the matter, right? So at that point, print

what

ever n and then be done, so it comes back, I don't know, like "DONE." Alright, that's just going to say it, you know, your job here is done. Well. Otherwise, we need to multiply all the digits. digits are equal.

Let's do it as a list. i for i in the string version of the number, right? Then they are being converted to a string of digits and then each one is removed individually. Oh, but we want them as numbers. So let's convert them back into numbers. How strange, I'm converting them to digits as a string and then taking them out each one. one separately and then convert them back to numbers, right? Which is pretty much, because this is a base 10 thing, one of the sad things about things like this is its specific base, so now we have 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 make the current result equal to one to start and then every time the result is equal to the result Multiplied by that digit and you can make x equal just means make it equal to this multiplied by J, wow , J! Well, I think that's all. and that will give us a new result, but then we have to repeat the process. So this is where we can cheat. And I haven't really tried to code this before, so I don't know if this is going to work. job.

I'm going to try to go recursive and then put that new result into the same function. So if it's a long one, it will stop; otherwise it will multiply all the digits and then put them back to the beginning and keep repeating this process, and you know what? Let's make it print the result every time. So we can see them all as it goes and then when it gets here it will stop and say done. Well? That can't go wrong. Let's find out. I really should have checked that in advance. Okay, copy. I'm going to turn on the terminal.

Alright. So what I'm really going to do is the simplest way to run something; literally paste it into the terminal. Let's do the Persistence of 327 which is the second, you said. 42 8 and then 8 forever, right? And then it's done twice and then we leave. Well, I couldn't fix the code to not get the last one twice. I just changed where the check is. So actually what I could have done is have another check here. This way I don't print the result and then print it again before stopping. I could put the check in, but it won't fit. aim.

Well, for a first pass it's fine. And then the first. Let's check the first one to make sure we messed it up. five four two eight It's going to 320 it's going to 0. right? And now, the ultimate test: can you handle 2 six sevens six eights two nines? There we go, that's it and that's much faster! Good? so now it spits out Exactly these all the way and then stops, it doesn't give us a number. Oh, do we want something like... do we want a number of steps at the end? number of steps? Okay, okay Brady will trim this 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 and at instantly we get everything.

We could no longer print every step of the way. It's fun to watch and we get the total number of steps. So, Brady, let's say what would you like? You know, let's put it in while we grind it for a while and see what we get. No, because you can be strategic. Oh, you're right to put a 5. Just put a bunch of fifteen 9s. Fifteen nines. one two three, four, five six seven eight nine 10, 11 12 13 14 15 Which is just nine to the power of 15. No, I put in another 10. Oh, another ten nines. Yes. Well, let's see if it works.

Two steps. Oh, because we got the zero in the answer next time, but what I can do in your terminal is just press up and the previous one is stuck with some more 9 at the end, two steps. Now, what happens if we put eight steps instead of two steps? What about the rope you used on the current record holder? Yes, but put a three in 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 how I love you thinking ready and 2 steps This is harder than it looks! So zeros are like zeros are landmines. Boom, hit a zero and you're out.

This is a minesweeper, but for number searches, it's fine. So this is what I would do. Now I would play with this for a while and it seems that every time you put in a random string of Numbers and boom, just because it was a zero, the next one no longer exists. We need to be more strategic. What happens to all the digits of pi? All of them? I'll get to work on that. three eight one four one five nine two six five I forgot, that's enough - 2 Wow, 11 suddenly seems so much more impressive, no wonder it's the world record.

So does this mean that you could set up a program that would just put each number in one after the other and leave it, you know, leave it for an hour or two? So what we've done now is we've built the basics. checkmark Next, we want to create something to perform the search. Then we could get something to generate random numbers of a certain size. Put them in and email us if you get a good one. The next step would be to be strategic about what numbers we were putting in because you already realized Don't put it in 5, put in 5, it's not going to work, so we could create a search Which one doesn't put in five?

It doesn't include any combination that we know will definitely give us zero, and in fact, we can be even smarter than that because if we look for the smallest number that works for, it doesn't matter what order the digits are in. In fact, all current record holders find the shortest number of digits for different persistence values. So there is 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. In fact, there is no need to search for random versions; you just need that set of digits in that order and then there are other things.

So, for example, this one here we'll use the first one that you put in 5428, that will give you exactly the same result as First of all, you could have Put two four Five eight, which is smaller But actually two times four will give you eight , so you could have put 588 That will give you exactly the same sequence and it is smaller. So if you care about about are the smallest possible values ​​for that subsequent persistence sequence (multiplication persistence). So 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. In fact, you only 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 the sevens eights and nines for the rest so we can reduce our search space dramatically With a little logic, the current search has reached 233 digits, so if you code it, you should start search from 233, it won't be smaller than that. We have already verified it and it is currently. The guess is that you would never be 11. So if people want to try, I mean, I'm always someone who tries.

See if you can write some code and see if it does some smart searching and you know that would be a big step forward. If someone could find a number with a persistence multiplication of 12, would you be the person to make that breakthrough? Certainly, the type of people who solve problems are those who think outside the box, creative thinkers, people who don't follow the herd and Brilliant wants to make you that type of person. Their courses consist of puzzles, like the ones you see on the screen right now. They are carefully designed to make people smarter thinkers.

Not memorizers, not people who just know all the equations and how to pass a test, but problem solvers. People with better connected brains. If you want to learn more about brilliant; what they are doing and how they could help you. Go to shiny.org/

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