A sequence is defined as follows: Start with any positive integer value (𝑛). Each term is found from the previous term as follows: If the value is even, divide it by 2 (𝑛/2). If the value is odd, multiply it by 3 and add 1 (3𝑛+1).

If the first term of the sequence is 30, what would the second term be?

If the second term of the sequence is 30, what would could the first term be?

Fill in the gap: The Collatz conjecture states: no matter the start number (𝑛), the sequence will always reach __________.

What is the first name of Mr Collatz?

In what year did Collatz make his conjecture?

The Collatz conjecture quiz

﻿Hello everyone, it's Mr Miller here, and in this lesson we going to be looking at Gabriel's problem. So first of all I hope that you are all doing well. And just to introduce myself in case you haven't seen any of my videos before on Oak. So my name is Mr Miller. I'm a maths teacher at a secondary school in central London, and I've really been enjoying doing these lessons on Oak, and really looking forward to doing this one, where we going to be looking at Gabriel's problem. And also the next one where we going to be looking at the happy caterpillar problem. So that's to look forward to, but first of all, this lesson, this is Gabriel's problem, so let's see how we are going to do this. So we going to have a grid which you can see on the right, and the numbers one through nine go in the white boxes. You can see that all the white boxes are complete already. That's not going to be the case for all problems, but in this problem all the white boxes are complete. Now each of the blue boxes is the product of the three numbers in that row or column. So for example, if I was looking at this first row here, a product means multiplication. So six times by eight times by three, gives me a 144. And it's the same thing for this second row here. And the third row and all three columns as well. So what I want you to do is, just to warm up, I want you to copy down this grid and complete it. You just need to complete the rest of these blue boxes. So pause the video for two or three minutes to complete the rest of this grid. Okay great, so hope that was nice and straight forward. And you should've got, seven times by one times by five, which is 35. Two times by nine, sorry, times by four, is 72. And in the columns, six times by seven, times by two, gives me 84. And then I've got 72 and finally 60. And there's nothing in this final box there. I can just leave it blank. So anyway that is quite straight forward, and the next problem we're going to have a look at is going to be a little more difficult so, let's have a look. Okay so, again same idea where, remember that in the white boxes we were putting the numbers one to nine. And the blue boxes is the product of the three numbers in that row or column. So here we've got another problem. So I want you to pause the video to copy down this grid and see how far you can get. I'd recommend looking for blue boxes that you can fill out first, and then look to see what other boxes you can fill out. So pause the video now and have a go at completing this grid here. Okay great, so you should've noticed that you can complete this first row and first column nice and easily. So first row, four times by three times by five, is going to give me 60. And the first column, four times by eight times by one, gives me 32. So well done if you found that. And then next you need to think of a slightly different tactic here. Because there's no more blue boxes that we can complete. So we going to need to look at one of the white boxes. So in case you didn't find this out, the box to look for is this one down here. And the reason is, well, I know that one times by something, times by nine gives me 18. So what needs to go in that box? Well if you're thinking a two then really well done. Because one times by two times by nine gives me 18. And then what's next? Well you can think about this middle box here, because now I know that three times by something times by two gives me 42. And I can think about this by thinking, well three times by two gives me six. So what do I need to multiply six by to get 42? Well seven, so there we go. And that just leaves one more white box and one more blue box to fill out. And there's a coupla ways that you can fill out this last white box. The first is by saying, well, eight times by seven is 56. And what is 336 divided by that? That will give you the answer. And the other way is to realise that, as I said the numbers one to nine go in those boxes, and we've only had eight of these numbers, so the last one must go in that box. So we can see that we don't have a six already. So a six must go here and you can also work that out by doing 336 divide by 56. Okay so just one more box to fill out. It's the final blue box which is nice and easy. Five times by six times by nine, gives me 270. And I'm done. Okay, the next one is the independent task. Let's have a look at another problem. Okay, again here is another grid. Same ideas apply. This one is again slightly more difficult. It may be okay if you're stuck to have guess, And see if that works out. So that might be a tactic. But it's really up to you. Pause the video now, copy down this grid, and see how far you can get and see if you can complete it as well. So pause the video now to have a go at this task. Okay great, so let's have a look through. So there's a coupla that we can complete pretty easily. First of all we can complete this white box in the top row here. We know that nine times by three is 27. And then 27 multiplied by six gives me 162. So that is nice and easy. And that then allows me to work out this white box down here. Because six times by two is 12, and 12 times by eight gives me 96. Now, what do I fill out next? Well it's not actually super straight forward to work out what I do next. So there's a number of ways you could've thought about this. But I'm just going to talk you through one of these ways which I think makes sense. Now, if you have a look at this second row here. You have got two missing white boxes. But have a think what two different numbers must go in those boxes. Well, you know that it's something times by two, gives me 14. So you know that these two boxes here have to multiply together to give me seven. Because seven times by two is 14. And how do you get seven by multiplying together two numbers. Well because seven is prime and it must be a seven and a one. So you must have a seven and a one, or a one and a seven in these two boxes. Now you could've tried out both of those combinations, and if you tried a seven in this box here, well you would've found out that it wouldn't work, because nine times by seven, looking at this first column here. Nine times by seven is 63, but I want to get 45, so that is not going to work. So the seven can't go in this box. It must go in this one. And one must go in this one, and that turns out to work and you can complete the rest of it really easily. So a five goes there, a four goes there, and finally, we've got three times by seven which is 21, times by four, so this is an 84. So again, work getting a little bit trickier, but well done if you managed to work that out. Well done if you did it with some clues, and for the final task, the explore task, it's going to get even trickier again. So let's have a look. Okay so for this explore task, again we have got the same problem. But this time you have noticed that we have all of the numbers in the blue boxes, but we need to find out what goes in the white boxes. And remember that in the white boxes we have the numbers one through to nine. So this is the trickiest one you've seen so far. So I've given you a clue in the speech bubble to help you get started. You know where the five must go, if you think about which two numbers five is a factor of. So anyway, pause the video now to copy down this grid. See if you can work out the first step, where the five goes. And even feel free to workout where all of the numbers go if you are feeling confident. Great so I hope that you managed to work out at least where the five went, and you know that the two numbers in the blue boxes that five is a factor of, 40 and 60 off course. So therefore where does the five go? Well the five has to go in this box here. Because you know that when you multiplying together different numbers, you then have to multiply a five by something to get 40 and a five by something to get 60, because five is a factor of both of those numbers. Now, you can actually do the same thing with some of the other numbers. So the number seven and nine will be useful ones to find out. So pause the video now if you haven't worked it out already, to find out where the seven and the nine must go using the same idea. Okay great, so you should've found that the seven must go here, the nine must go here, and this information actually allows you to work out the other numbers as well, because you can work that six goes here and you can work out the number in the top left as well. That's got to be a two. And then the rest of the numbers are quite straight forward. You could next look at this second column here, you know that the numbers three and one go in the two missing boxes, and you know by looking at the numbers at the end of the first two rows that the three goes here, the one goes here, and therefore the missing numbers are going to be eight there and four there. So that is that problem. And if you managed to do this, really really well done. And if you're thinking about why is this kind of important for maths, well really it's about this idea of factors and prime factors and prime factorization which you would've had a look at already, probably. So yeah, this task really shows that the factors of a number is really important and it's a nice way of seeing that in this task here. Anyway, that is it for this lesson. Apart from on the next slide, I'm going to show you some more problems that you could do if you find this interesting. So here are some additional problems. So again, here are some additional problems. The source here is nrich math, which have got some really nice problems, so that is where I got this from. So again, similar ideas here, but you looking at more different numbers and more setups of the grids here. Anyways that is it. So pause the video here if you want to have a go at these problems. And on this slide here, you can share your work if you have your parent or parents permission on twitter using the hashtag, sorry, the handle @OakNational and the #LearnwithOak, if you would like to show your completed grids. Anyway that is it for today's lesson. Hope you enjoyed it. And next time we going to be look at the happy caterpillar problem which I promised you is very exciting. So looking forward to that. Thanks so much for watching. Well done, have a great day and see you next time.

Each blue box is the product of the 3 numbers in that row or column. All the questions refer to this picture below. First, what is the value in box A?

What is the value in box B?

Which of the following could be a value for C? 

Gabriel's problem:

In this lesson, you will learn about a famous maths problem called Gabriel's problem.

Maths

Y8 Maths

Key Stage 3

Year 8

Gabriel's problem

Famous maths problems

UNIT

Goldbach's conjecture

The Collatz conjecture

The Happy Caterpillar Problem

Fill the gap: Every even integer greater than two can be written as the sum of two __________ integers.

Lesson overview:Gabriel's problem

Core Content

Famous maths problems: