【Numberphile】10958问题 The 10,958 Problem

I think I have a solution to a very much an unknown problem. It's a problem no one was really looking for a solution for, but I think I found one. There's a thing called the 10,958 problem. To demonstrate what the problem is, I'm going to start with a little number trick, a little something, a little bit of number showmanship, like all good number tricks. It starts with you picking a number. Can you give me any three-digit number? What would you like? Pick a three-digit number. 814. 814, interesting, okay. And can you give me a single digit? What single digit would you like? It can be one of these, it can be a whole new one, whatever you want. Two. Two, okay, right, here's what we're going to do. So, I can now tell you that 814 equals 22 times 2 plus 2 plus 2 to the power of 2 plus 22. I mean, if you don't believe me, we can double-check that. 2 plus 2 plus 2 is 6, I'm prepared to accept that as given. And 6 squared is 36, so we now want 22 times 36 plus 22. 22 times 36 equals 792, add 22 equals 814. There you go. You gave me a three-digit number, and I could write it out as an expression using only the digit you gave me. Do you think that is exciting enough to do a second time? Yes. Alright, what would you like this time? 998. 998, you've gone for a large, but not predictably large number. And what one digit would you like to be used? So we had two last time. One. One, oh my goodness, you want that in... Now, obviously, I could cheat and just write equals 1 plus 1 plus 1 plus... Right, and I could have done that over here, because you happen to pick an even number, and 2, I could have written 2 plus 2. The goal is I'm trying to do it in as few digits written out as possible. So I can actually do 998 as 11 minus 1 will give us 10, to the power of 1 plus 1 plus 1 is just... I've done two pluses there. That is a mega plus. Oh my goodness, that's pretty plus. And so that's 10 cubed is 1000 minus 1 minus 1, and there we are. We've landed on 998. I could go on all day. In fact, I would. But people would cease watching the video quite quickly. And I can confidently tell you that any three-digit number you give me, I will be able to write it out using any digit you give me, because down here I have a paper with all of them in it. So this is single-digit representations of natural numbers. A guy called Tanija, I'm going to pronounce that wrong, he is from Brazil, so I've definitely pronounced that wrong, and it's just tables and tables of three-digit numbers and how you write them out. So this is all three-digit numbers done using the digit 4, and then over here using 6 or 5, anything you want to look up. And so the way I was able to do this is I was cheating. And if you want to cheat, it's really quite straightforward. You put aside a month or two just to memorize this, and then you can cheat with this inside your brain. That's the great thing about the human brain. You can cheat by memorizing your cheat notes, and then you can look at them without anyone being able to tell. It's a real loophole in life. You can cheat by memorizing things. Also known as learning. Also known as learning, yeah. Oh, you learn it, and then you do that, right. And we know that they're all the shortest versions. Has he used some kind of rigorous proof that they're the shortest versions? That is a really good question. I don't know. He claims these are the shortest. I can't remember if he claims definitely the shortest or the shortest he's found. He just says they're minimal possible digits. I don't know how he has confirmed that. So if you trust this guy, then these are the smallest. But they may, I mean, I trust him. Did he stop there? Oh, no. No, no, no, no, no. He went one step further. So let's carry on to, in a different color, sharpie. Let's do any number below 11,111. 7,000. 400. And 15. Okay, so I can tell you that this can be written as. Here's the other one. This one's a bit longer. So this one's like 160-something pages. And I'm just going to look it up. So let's find. I haven't had time to memorize this one. You know, if I had another holiday on the beach, I could sit there and memorize this one too. Oh, you picked a good one. Oh, this one has to use subtraction. Not all of them do. You didn't ask me my digit, my single digit. So you don't get to choose this time because I'm going to use all the digits. So you said 7, 4, 1, 5. 7, 4, 1, 5 is either negative 1 plus 2 times 3, 4, 5, 6 plus 7 times 8 times 9. Or it equals in brackets 9 plus in brackets 8 times 7 plus 6 times 5 in brackets times 43. Close up some brackets. Multiply it by 2. Add 1. Okay, so there is your number written as all the digits in ascending order. So that's 1 to 9 in order. Or all the digits in descending order. That's 9 counting all the way back down to 1. Only using multiplication, addition, division, subtraction, and you can have powers. So only using the simple operations. This guy has also gone through and found every number up to 11,111 ascending and descending using only those operations. This guy's a legend. I love this guy. And he just does it. He just puts them online. You can just find his papers. Yeah, I'd ration them out. You know, I would have a freemium model where you get ascending for free but you pay for descending. Come on. This is premium content. Right, or you know you mess up the middle digits. So now, as you may have realized from the name of this video, there is a problem. And it is the 10,958 problem. If you look up, let's look up 10,958. So it's going to be right at the end. There it is there. So there's 10,958 in descending order. And if you look up the ascending order, it's still not available. It is the only gap in all the numbers up to 11,111. It's the first one you come across. As you go up everything from 1 all the way to here, it's all there. And then you end up with still not available. So it's like the Achilles heel. It's the missing piece, right? It's, you know, it's that little bit. You're like, I've done it all. Except for that little bit. Similar to before, I don't know how comprehensive he has been in assembling this list. So he thinks it can't be done. But he doesn't seem to have completely ruled out that someone else won't do it. Well, I'm never one to leave something like this. At least give it a go, right? There's the spirit. That's my whole philosophy. It may be an impossible task, but you give it a go. And it's just as much fun anyway. It's the Parker Square philosophy. You may be up against an impossible task, but you give it a go. Even though you're probably not going to succeed. I don't want to get you overexcited. It's not a perfect solution. But I think it is. I think it's a perfect solution. But other people may argue otherwise. We get pi. Pi was historically rarely calculated this way. Because it is notoriously inaccurate.