But it is technically to do with pancakes, or at least pancakes are one Nice way to explain what this number is.

We have a pan full of pancakes here on dhe.

The idea with this is there no, all the same size, who have five different sizes of pancake.

And we also have a very inconsiderate chef who makes pancakes all different sizes and then served them up in the wrong order.

Because I'm sure you'll agree.

If you've got a stack of pancakes of all different sizes, the most satisfying way is to have them in decreasing order of science with the smallest one of the top of the biggest one of mom.

So we need to try and get this stack of pancakes back into order.

So I still gotta play.

I use my pancake flipper, stick that on a plate.

The question now is, how do we put these pancakes back into the right order?

There is a rule about how I'm allowed to shuffle these around.

I'm allowed to take the top section of pancakes and turn it over, and that's all I can do.

So, in this case, and what do I want to do?

I want to flip, maybe the whole thing.

Yeah.

I wanna flip the whole thing.

No, in fact, first I'm gonna flip these top two, and the reason for that will become apparent shortly.

So I do that.

So that's one flip, Andi.

I'm not gonna flip the whole thing.

There we go on now.

I'm gonna flip.

I shouldn't have flip the top two.

I'll figure this out eventually.

So flip those three.

That's three flips on.

Now, flip the whole thing again.

Asfour flips.

And if I hadn't put those took top two at the start, I would have had not to do this now.

But never mind, eh?

So that's five flips, which is fine, because I know that any stack of pancakes can be fixed in five or fewer flips.

This one, I suspect, could have been in slightly fewer five.

But in this case, I've used exactly five, which is within the bounds.

So that's fine.

So any sack of five pancakes can be done in five or fewer flips there.

Obviously, some that What?

This is currently in the state where need zero flips, but they're obviously some that could be done in fewer than five on the most you'll need for five pancakes.

Doing it in the most efficient possible way is five.

So the pancake number for five pancakes is there for five.

And this works for any number of pancakes in a stack, so I can show you what happens with three pancakes.

There are six possible ordering CZ for three pancakes so I could have small, medium large three at the bottom again, but instead have to and then want to at the bottom, then three, then one like that, one at the bottom and then two and then 32 at the bottom 13 like that on finally one of the bottom three and then to that gives me all six possible ordering Tze and for each one of these, I can work out how many flips I need to get it back into the sort of standard correct way up.

So in this case, it zero flips, it's already there.

This could be done in one which is just to flip the whole thing.

This one could be done in one flip this one.

If I flip those top two and then flip the whole thing, I'll get back to sorted.

So I'm gonna call that 12 flips this one.

If I flipped the whole thing and then flip those top two around, that'll be two flips on DDE.

In this case, this is the most difficult one, because this can't meet on in fewer than three flips, so I can do it in three.

I would maybe say Flip the whole thing, and then you flip the top two and let the whole thing, but it can't be done in fewer than three.

So for three pancakes, the pancake number is three.

Yes, it does look a little bit partly so.

For three, it's three for four.

It's four for five.

It's five.

After that, it starts to go on, so every so often it goes up by two instead of going up by one.

So it's not quite always just the number of pancakes that you've got, but it is actually something that people try and calculate the trying work out.

What is the pancake number four Given number of pancakes on Dhe.

It's actually an ongoing question because for 20 pancakes we don't know what the pancake number is.

No, no, it's It's an open question.

At Mass.

We don't actually have 20 or above.

We don't actually really know what the answer is to this.

And I guess it's partly because it's just a massive computation.

If you've got 20 pancakes in the stack, the number of possible way she could order 20 pancakes in a stack is massive for any given.

Ordering the number of possible sets of flips you could do is even Maciver, you know, to check all of those and see which one is the most efficient would take a lot of computer time, so no one's managed to crack that yet.

What is it?

90 for 19?

It's 22.

I think something like that s so yeah, we know it's more than 22 on.

In fact, we know it's less than I think 35.

And the reason the person that we have to credit with the fact that we know it's less than 35 is Bill Gates.

So Bill Gates that the guy off of Microsoft when he was a university in 1979 when he was 24 he did a paper.

Unfortunately, paper wasn't called like on the pancake number.

It was called something like algorithms for prefix reversals.

Because that's the technical computer science term for this.

If you're taking the first section and reversing it, that's a prefix reversal.

And his paper.

They proved that there is an upper bound for the pancake number for if you know how many pancakes you've got.

What the A profound is for N pancakes.

So think for n pancakes.

They proved that it's five enforced, five over three, which means that for 20 pancakes it's 105 over three.

That means it's 35.

That is, thanks to Bill Gates.

Since then, they've actually worked out better upper bound for it.

That's slightly nicer.

If they've now shown that anything is less than 18 and over 11 which was found in 2009 which was exactly 40 years later, they managed to improve on that bound.

So there is a way that will always work for any given kind of messed up stack of pancakes.

So the trick is that you take whichever pancake is the biggest pancake.

That's not currently where it's supposed to be on, Put it to the top.

And then I put that one toe where it's meant to be if you keep repeating that so if now safe Pancake three wasn't in the right place.

You do the same thing, you put it to the top, and then each time you you're kind of definitely gonna place at least one pancake.

Possibly Maur.

If you're looking, you will always do it in two and minus three.

So for n pancakes, in the case of five, it would always do it in seven or fewer.

But I know that for five pancakes, pancake number is five, which means there is a way to do all of them in five or fewer.

So that wouldn't be necessarily the most efficient method.

The nice thing about this is they actually originated from, like, a real world problem s.

Oh, there was a mathematician, Jacob Goodman, who was trying to solve a problem in the real world.

And this thing occurred to him is like, Oh, that's an interesting puzzle, you know?

How can I How can I find how many flips it takes to do this, and he wasn't actually using pancakes at the time.

It's really unfortunate.

He wasn't in a kitchen cooking pancakes and discovered this.

He had a stack of towels in the cupboard.

It was really domestic math problem, and he had to try and put these towels in the cupboard.

But he just had a shelf and there wasn't any kind of surface nearby.

And he needed to change the order of the towels so he could just take the top of the stack and flip it over.

And he kind of was like, Oh, actually, that's That's quite a nice problem And he wanted to write it up for a magazine for American Mathematical Monthly, and he was like, How can I reframe this so that it's a nice, accessible thing?

And this idea of using stacks of pancakes with a flipper was his.

His way into this used the pseudonym.

He wrote an article under the name of harried Waiter, which is happy, harried waiter who's carrying the pancakes and so on.

And it was, you know, is a nice little Here's the problem.

People responded to it and wrote back and stuff.

So that was how the thing kind of first originated, but it's since it's become like a real thing and people work on it.

People published papers on it.

So there's a chap called David Cohen, who writes for The Simpsons, who's a mathematician and computer scientists in his spare time.

When he's not writing for the symptoms, I guess he trained as a computer scientist and while he was studying, he also did a paper on this, which was on a thing called the burnt pancake problem.

And in that case you've got pancakes where one side of the pancake is burned and you want to end up with them all in the right order in the stack, but also all the burnt sides down, so that no one can tell that you burn all your pancakes on one side, and that makes it slightly more complex.

And it means that you have to make sure that you do all the the steps in the right order, and you might find that you've you know, you've done it right, but they're all upside down and you need to redo the whole thing.

So that's another thing that he's kind of published a paper on And that's a separate but related problem, I guess.

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