So we need a number with an infinitely long decimal expansion.

That is irrational, of course.

So we're basically deep.

I put you probably guessed we're going with pie.

So these are the first few digits off the decimal expansion of pie on DDE.

What we want to do for these self locating strings is to label each position after the decimal point.

So we're going to start with the first number.

This is the first position, the second position, the third.

And then the question you ask yourself to find the self locating string is you want the number to match its position in the decimal expansion.

So here we've got our 1st 1 straight away.

The very first digit number one is in the first place, So that would be our first self locating string.

And then what we want to do is continue along through the numbers and see if we can find any others.

Of course, the expansion is infinite, so we're gonna hope that there should be some more.

So if the second number there had been a to that well worked as well what?

It's a four.

So exactly so if this is that this would need to be a two, this would need to be a 34567 So none of these work yet on the other thing, is once we, of course, get to double digit or triple digit positions.

If we were, for example, considering the 12 number of the decimal expansion, we would need the 12th position to be a one and next to it the two.

So you actually would need a one in the 12 position in a two in the 13th position.

If you think about that a little bit, you're probably They're sort of most common near the beginning, when you just have one number that needs to be in one fixed position.

And when you get to two digit three digit four digit, you have several requirements that need to happen in a row, so your probability gets smaller.

So for position one million you would in a position one million to be a one on, then the next six positions.

After that Toby zero.

Exactly.

So that's very, very unlikely.

Let's be honest compared to just having let's say you know a seven position seven or something.

If the numbers are truly random, which is generally what we assume in the expansion of pie they each want it appears approximately equal probability.

If that is indeed true, then you know you're going to see or expect to see Maur of the lower down, self locating digits compared to the higher up one's further along, we've got the 1st 1 to a good start.

So what we now need to do is I need to write out more digits of pi s.

So I'm gonna have to let us up.

I'll be honest.

That's about as good as it gets for me.

Says I'm going by what people want for seven decimal places.

You know you don't really need all the seven decimal places way.

The 1st 100 digits of pi was painful.

We've got our first self locating string here at position one on what we can now do is continue to look through these numbers on without even doing this.

I already tell you there aren't any more.

The next ones are really, really far away.

So after one right to the very first digit, we get ourselves like 18 string.

It's super exciting.

We think they're gonna be really common.

They're not.

The next one is at 16,400 and 70.

So I'm definitely not writing out 16,000 digits of Pi.

That that's that's a job for the viewer.

And then after that, we then jump up to 44,899.

Then we go up to the big one at 79,873,884.

So that's our 1st 1 in the very first position.

Then we go to our 2nd 1 at 16,000 the 3rd 1 44,000 and then the fourth self locating string at 79 million.

Tom, would you guess that there are an infinite number of ways I'm I'm inclined to say yes with the idea being that pie is an infinite string of decimals on dhe.

You.

You might expect that because it's infinite.

You would hope that every string of numbers should be in there at some point, but and you got this extra condition that not only does the number have to appear, it is to appear in a certain place So that's why these are quite rare.

It's a difficult question for sure.

It's a good question.

There's another way to look at this.

There is indeed so and this is the reason why I did write out the 1st 100 So I'm gonna take a different color because you may have wondered why I labeled the first position here the one as as index one.

Because you could also, of course, completely, you know, it could be completely valid to label.

This is zero.

So if we call this now the zeroth spot, then this one the first on the second.

And of course, we continue in the same way.

We now have a slightly different index.

We're going to 0123456 Bingo.

So we've got our 1st 1 now at the sixth position.

Indexing from zero.

And then, if we carry on 789 10 11 12 13 14 15 16 17 18 1920.

So there's a two in position 20.

But then there's a six, so it doesn't work.

2122 23 24 25 26 27.

We have a two and then we've got a seven.

So we've actually found a second self locating string in positions 27 28.

We've got six, we've got 27 then we jump up to a number kind of near 16,000.

13,000 598 is our next third self locating string, so it's not too far away from the 16,000 on index wall.

Then we go up to 43,611 again, baby.

It's like coincidence.

Maybe it's not, but it's quite similar to what we saw before.

And then Finally, this one is less close, but at least has the same number of digits we get 24,643,510 on that is the final.

That's going to be, what, five.

So we actually have one Maur self locating string in the 1st 100 million digits of pi when doing your indexing zero.

So, as I said, there isn't a convention between index 01 but we're gonna now stick with index warm from now on, just just to be clear on DDE that perhaps more interesting question about these, it's super cool finding them, and you should totally go and try to find your own.

But the other interesting thing you can do is ask the question off.

Do these things loop back around on themselves?

So what I mean by that is if I pick a starting point, for example, 169 that suppose it's my favorite number.

Well, actually, let's suppose it's the favorite number.

Off.

Down.

Sikorski, Who is the guy?

Found this out.

So this is Dan's favorite number on DDE.

What we do is we Then look forward.

Where does the number 169 appear in the decimal expansion of Pie Onda?

We should be able to spot it.

There it is.

169 is down here on because I did this earlier.

This is gonna come in at position 40 starting here.

So the number 169 so got 40th.

41st 42nd appears in position 40.

So now you say we started with 169 and that was sort of taken us to position 40 in the expansion.

So then you say Well, now where does 40 a pair in the expansion of pie?

We're gonna try and create some kind of pathway through this infinite decimal expansion.

Imagine, 40 appears all over the place.

Exactly.

So we look for the first occurrence of 40.

That's sort of the lowest down.

Well, that's the idea.

So 40 is going to appear at theseventy ith position.

So the force of 70 on the zeros at 71.

So we started a 16 9/40 position.

We looked for 40.

That's taken us to the 70th position.

So now we look 70.

That's the idea.

So 70 is coming in right now on the bottom.

And this is why I wrote out 100 digits.

So 70 is down here in position 96 is the starting number.

So we've gone 169 to 40 for 70 96 now we're going to look for 96 we're going to carry on and we're gonna see where we end up, So 96 is going to take us miles away to 180 then we D'oh!

This is a pretty long route where they go through 3664.

Then we look for that number which appears in the 24,717 position.

Then we go to 15,492.

Then we g 0 84,198 So I think this is the highest point of our path that we descend into the madness of the decimal expansion of pie.

That then goes back down to 65 for 8937 to 5.

There is a point to this, I promise.

Then we get to 16.

97 4 41 70237885757 1958 14 6096 to 89 to 44 745 And then we go to 9385 Finally, we get 169 again.

So we loop back around to 169 which appears in the 40th position.

So the mathematical question here is given any random starting number that we just choose of any number of digits.

Do we always get a look, huh?

That's the mathematical, so it seems like a bit of fun, but there's like a really deep mathematical question into this.

Do you always get this loop that's going to happen, given any starting number.

Will you always find a loop?

The loop could be of any length.

It could be very short.

It could be quite long.

It could.

But you know, as long as it's not infinite length, then you would look back around on dhe.

We have no clue.

As far as I'm aware, this is just a puzzling question.

But there is one thing that we can say, which is that not every number will have an infinite loop, because this is where we come back to the self locating strings and my new favorite thing at the moment.

Because if you end up at a self locating string, so if you somehow come across one or 16,470 you don't move because one is in position one.

So you get to one, and then you just loop back around or you get to 16,470 look back around there like they're basically dead ends.

So that's where the self locating strings stop these loops from happening on.

I do have an example of one loop, which which terminates at one.

If we start at 211 so random starting number.

This appears actually back in here at position here to one and then one down here so annoyingly over two lives.

But we've got off number here.

211 on This is a position 93.

This one here is a 93rd position.

So we go 211 to 93.

Where does 93 appear?

93 million.

Okay, there's good.

So you always go to the earliest.

This is the It is just a rule.

It's an arbitrary rule, and you could change the rules.

But for the purpose of our loops, we're going to the first appearance of that number.

So So you're completely correct, but this is gonna be the one.

And this is a position 14.

And now where does 14 a pair in pie?

I think you know this one.

Brady may be right at the rickety, so 14 is in position one.

So we've gone 211 to 93 to 14 and then we've gone toe one, and then we just look back around forever and we stuck at one.

So one ourself locating string here is the dead end.

You either have this big loop, the goes or, you know, some size of loop where you just keep going round round in circles, you either reach your dead end, as you called it, the self locating string, which just circles back to itself forever.

Or can we find a path again?