Lesson video

Hi, I'm Miss Kidd-Rossiter and I'm going to be taking today's lesson on the Collatz conjecture.

This is an unsolved puzzle in maths that's been stumping mathematicians for generations.

You might find it helpful for this lesson to have some A3 or big paper.

If you don't have that, A4 will be fine.

Make sure you've also got something to write with.

Try and get yourself in a nice, quiet place where you're free from distractions so that you can fully concentrate.

If you need to pause the video now to get any of that sorted, then please do.

If not, let's get going.

So then I'd like you to try this: Yasmin and Zaki are writing number sequences.

To get the next number, they apply these rules to their sequence.

Now the rules depend on the parity of the number.

So parity might be a new word for you.

That just means whether the number is odd or even.

So if your number is even, you just divide it by two.

If your number is odd, then you multiply it by three and add one.

So you're going to have a go at this, choose a number between one and ten to start your sequence, continue the sequence, and what do you notice? I think you'll know when to stop and choose a different number and try again.

So try this with three or four different numbers and see what you notice.

When you're ready to go through it, resume the video.

Excellent work.

I'm actually not going to talk about this because it's going to form the connect part of the lesson.

So Collatz, he was born in 1910 and he died in 1990, so not that long ago.

He defined a sequence as follows, and this is similar or it's actually the same as what you did in the try this.

So start with any positive integer value.

So I just limited you to the ones between one and ten, but you could start with any positive integer.

And you find each term from the previous term as follows.

If the value is even, you divide it by two.

And if the value is odd, you multiply it by three and then add one.

So Collatz, in 1937, which was just a couple of years after he'd finished studying, came up with the collapse conjecture, which has got lots of different names in maths now.

So, some people call it the three X plus one problem.

Some people call it a hailstone sequence, but I'm going to go with the Collatz conjecture today and the Collatz conjecture states that no matter the start number, so whatever number you choose, it doesn't matter so long as it's a positive integer value, Mr. Collatz says that the sequence will always reach one.

Did your sequence reach one? What happened when it reached one? Let's have a look at some together.

So, I'm going to start my sequence with the number three.

Three is odd, so I multiply it by three and add one and I get ten, Ten is even so I half it.

And I get five.

Five is odd, so multiply it by three and add one.

And I get 16.

16 is a special kind of number and it's even.

I get eight because I half it.

Half eight gives me four, half four gives me two, and then half two gives me one.

So I have got to one when I started with three.

What's the first number between one and 10 that is not on our sequence here? So, we've got one, two, three, four, five.

We haven't got six.

So if we started with six, Ooh, that's even so I'd half it.

And that gives me three.

So I can just join that onto my sequence here.

What's the next number then? Seven is not in my sequence yet.

So seven is odd.

So I would multiply it by three and add one to get 22.

22 is even so I half it and get 11.

11 is odd.

So multiply that by three and add one to get 34.

34 is even so I half it and get 17.

17 is odd.

So I multiply it by three and add one and get 52.

52 is even, so I half it and get 26.

26 is even so I half it and get 13.

13 is odd.

So I multiply it by three and add one, which is 40, 40 is even so I half it and get 20.

20 is even so I half it and I get 10.

So that means I can add this in now to my sequence here.

So now I've got all the numbers from one to seven.

I've also got eight.

I haven't got nine yet.

So let's add nine in.

So, nine multiplied by three, add one gives me 28.

28 is even, so I half it, it gives me 14.

14 is even so I half it, and that gives me seven.

So I can add it in here.

So now you can see that my sequence has got all the terms in it from one to ten.

This is like what we call a tree.

So lots of mathematicians have drawn, really beautiful ones of these really big ones that go up to huge numbers.

Now they haven't managed to prove that this is true for every positive integer yet.

They believe that it's true, but they've not yet proven it.

So you're now going to apply your learning to the independent tasks.

So pause the video here, navigate to the independent tasks.

And when you're ready to discuss it, come back and we'll talk about it together.

Pause now.

How did you get on with that? Did you have some fun drawing out those trees, which number between one and ten gives the longest sequence? Well, we already saw that in the connect, didn't we, that nine made the longest sequence.

Which number between one and 20 gives the longest sequence? Oh this was a bit of a mean question because there are actually two answers.

So both 18 and 19 give equally long sequences.

Are there any shortcuts you could take with the conjecture? Well, I actually was thinking about this and when you think about three plus one, if you start with an odd number and do 3 plus one, it gives you what? Exactly, an even number.

So what you could do to do two steps in one is to do three plus one divided by two.

Because here you would divide by two, wouldn't you? And then you would get another odd or even, we're not sure.

But you could do odd.

So you could just combine these two steps.

Now, this one I really liked because you have to think of it mathematically here, and it shows you how multiple numbers can link into the same thing in these trees.

So for example, here, we know that 32 is even so we know that half of that must be 16 and that links in with the rest of our chain.

What other number could give me 16? So I've done the divide by two, one here.

So something let's call it X multiplied by three, then add one gives me 16.

Let's just make that look a little bit more mathematical three X plus one equals 16.

So we can solve this now by taking away one from both sides.

So we get three X equals 15 and then dividing by three on both sides to get the X equals five.

So we know that this one here must be five and so on, and we have to apply this process over and over and over again with our mathematical thinking to fill in this tree.

So here are your answers.

Hopefully you've got all those.

Well done if you did.

Let's look at the explore task now, then.

Carla is thinking about the Collatz conjecture.

She says that some sequences will only contain even numbers.

Do you agree with her? And can you find some examples that either show or don't show this to be true? Pause the video now and have a go at this task.

How did you do, what did you think? Hopefully you thought, Oh, mess, what you're talking about? All the sequences we've looked at so far have got one in them and one is an odd number.

So if we take one out of the equation, can we think about any sequences that have all even turns apart from one? Excellent, anything where it's two to the power of something is your start number will give you all even terms apart from the last one, which is one.

So for example, if we did two to the power of four, which we know is 16, then we go 16, eight, four, two, straight to one.

If we did two to the power five, then we would just add 32 onto there, if we do two to the power of six, we'd add on 64 onto there.

If we did two to the power of seven, we'd add on 128 and so on.

So if we've got two to the power of anything, it will have all even terms apart from one.

Hope you've enjoyed this lesson on the Collatz conjecture, I've really enjoyed teaching it to you.

If you'd like to share your work with Oak National, then please ask your parent or carer to share your work on Twitter.

Again, at Oak National and hashtag Learn with Oak.

Don't forget to go and take the end of lesson quiz so that you can show me what you've learned and hopefully I'll see you again soon.

Bye.