Self locating strings are to do with decimal places of irrational numbers so
we need a number with an infinitely long decimal expansion that is irrational of
course so we're basically gonna do PI
you probably guess we're going with pi so these are the first few digits of the
decimal expansion of pi and 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
the first number so 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 first one 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 going to hope that there should be
some more so if the second number there had been a two that will work as well
whether it's a four or so exactly so if this is a this would need to be a two
this would need to be a three four five six seven so none of these work yet and
the other thing is once we of course get to double digit or triple digit
positions if we were for example considering the twelfth number of the
decimal expansion we would need the 12th position to be a 1 and in next to it the
2 so you actually would need a 1 in the 12th position and a 2 in the 13th
position if you think about that a little bit you're probably that sort of
most common near the beginning when you just have one number that needs to be in
one fixed position and then when you get to two digit 3 digit 4 digit you have
several requirements that need to happen in a row so your probability gets
smaller so for position 1 million you would need position 1 million to be a 1
and then the next six positions after that to all be 0 exactly so that's very
very unlikely let's be honest right compared to just having let's say you
know a 7 at position 7 or something if the numbers are truly random which is
generally what we assumed in the expansion of pi they each one appears
approximately equal probability if that is indeed true then you know
you're gonna see or expect to see more of the lower down self occasion digits
compared to the higher up ones further along we've got a first one to a good
start so what we now need to do is I need to write out more digits of pi so
I'm gonna have to load this up I'll be honest that that's about as good as it
gets for me as I'm doing pi what people want for seven decimal places you know
you don't really need more than seven decimal places 100 there we go the first
100 digits of pi that was painful we've got our first self locating string here
at position 1 and what we can now do is continue to look through these numbers
and we're not even doing this I can already tell you there aren't any more
the next ones are really really far away so after one right at the very first
digit we get ourselves a casing string it's super exciting we think they're
gonna be really common they're not the next one is at sixteen thousand four
hundred and seventy so I'm definitely not writing out 16,000 digits of pi back
that's that's a job for the viewer and then after that we then jump up to forty
four thousand eight hundred and ninety nine then we go up to the big one at 79
million eight hundred and seventy three thousand eight hundred and eighty four
so that's our first one in the very first position then we go to our second
one at sixteen thousand the third one forty four thousand and then the fourth
self locating string at 79 million (Tom would you guess that there are an
infinite number of these) I'm inclined to say yes with with the idea
being that pi is an infinite string of decimals and 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 then you've got this extra condition that not only
does the number have to appear 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 all right
there's another way to look at this there is indeed so and this is the
reason why I did write the first 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 as zero
so if we call this now the zeroth spot and then this one the first and the
second and of course we continue in the same way we now have a slightly
different index we're going to 0 1 2 3 4 5 6 bingo
so we've got our first one now at the 6th position indexing from 0 and then if
we carry on 7 8 9 10 11 12 13 14 15 16 17 18 19 20 so there's a 2 in position
20 but then there's a 6 so it doesn't work 21 22 23 24 25 26 27 we have a 2
and then we've got a 7 so we've actually found our second self locating string in
positions 27 or 28 we've got 6 we've got 27 and then we jump up to a number kind
of near sixteen thousand thirteen thousand five hundred and ninety eight
is our next third self locating string so it's not too far away from the 16,000
on index well then we go up to forty three thousand six hundred and eleven
again maybe it's a coincidence maybe it's not but it's quite similar to what
we saw before and then finally this one is is less close but at least has the
same number of digits we get 24 million six hundred forty three thousand five
hundred and ten and that is the final that's going to be what five so we
actually have one more self locating string in the first 100 million digits
of pi when doing it indexing zero so as I said there isn't a convention between
index 0 or 1 but we're gonna now stick with index 1 from now on just just to be
clear and that perhaps more interesting question about these it's super cool
finding them and you should totally go away and try find your own but the other
interesting thing you can do is ask the question of do these things loop back
around on themselves so what I mean by that is if I pick
starting point for example 169 that suppose it's my favorite number well
actually let's suppose it's the favorite number of Dan Sikorsky who is the guy
found this out so this is Dan's favorite number and what we do is we then look
for where where does the number 169 appear in the decimal expansion of PI
and we should be able to spot it there it is 169 is down here and because I did
this earlier this is going to come in at position 40 starting here so the number
169 so we've got 48th 41st 42nd appears in position 40 so now you say we started
with 169 and that is sort of taken us to position 40 in the expansion so then you
say well now where does 40 appear in the expansion of pi so we're going to try
and create some kind of pathway through this infinite decimal expansion I
mentioned 40 appears all over the place exactly so we're gonna look for the
first occurrence of 40 that's sort of this the lowest down one that's the idea
so 40 is going to appear at the 70th position so the fours are 70 and the
zeros are 71 so we started at 169 40th position we looked for 40 that's taking
us to the 70th position so now we look for 70 that's the idea so 70 is coming
in right now on the bottom and this is why I wrote out a hundred digits so 70
is down here in position 96 is the starting number so we've gone 169 244 72
96 and now we're going to look for 96 and we're going to carry on and we're
going to see where we end up so 96 is going to take us miles away to 180 then
we go this is a pretty long route we then go through 3664 then we look for
that number which appears in the 24,000 717 position then we go to 15,000 492
then we go 84,000 198 so I think this is the highest point of our path that
descend into the madness of the decimal expansion of pi so that then goes back
down to 65 4 8 9 3 7 2 5 there is a point to this I promise then we get to
16 9 7 441 702 3 7 8 8 5 7 5 7 1958 14 six oh nine six two eight nine two forty
four seven four five and then we go to nine three eight five and finally we get
one six nine again so we loop back round to 169 which appears in the fortieth
position so the mathematical question here is given any random starting number
that we just choose have any number of digits do we always get a loop
that's the 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 can 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 loop back
around and we have no clue as far as I'm aware this is just a sort of puzzling
question and 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
sixteen thousand four hundred and seventy you don't move because one is in
position 1 so you get to one and then you just loop back around or you get to
16 thousand 470 and loop back around so they're like dead ends or they're
basically dead ends so that's our the self locating strings stop these loops
from happening and I do have an example of one loop which which terminates at
one if we start at 211 so a random starting number this appears actually
back in here at position here two one and then one down here so annoyingly
over two lines but we've got a number here 211 and this is at position 93 this
one here the 93rd position so we go 211 to 93 where does 93 appear 93 but
there's an earlier one hey there's good so you always go to the
earliest this is the the it's just a rule you it's an arbitrary rule and you
could change the rules but for purpose of our loops we're going to the first
appearance of that number so you're completely correct but this is going to
be the one and this is at position fourteen and now where does fourteen
appear in PI I think you know this one Brady may be right at the beginning so
fourteen is in position one so we've gone 211 293 to
14 and then we've gone to one and then we just loop back around forever and
we're stuck at one so one our self locating string here is the dead end
you either have this big loop that goes or you know some size of loop where you
just keep going round around 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 and there's again far as I'm aware no one has found such a
path where you just keep going forever and ever and ever as far as I'm aware
there's no point to this this is I said I found this on the PI search page it's
recreational maths but then it does lead to quite a deep question
the patterns of numbers within the expansion of pi so whilst even if we
were able to answer that question it probably wouldn't tell us much about PI
itself but the the methods and the techniques you're going to use to do
such a proof are going to probably be brand new and lead to all kinds of other
awesome stuff it's like the classic case of you know trying to climb a particular
peak there's not really that much at the peak but it's everything all the other
Peaks you scale to get to that one it's all your the new stuff you discover then
has all these other amazing uses and loads of really big results in maths
come from recreation or playing around with numbers this kind of thing this is
the first five million digits of pi printed on two thousand pieces of paper
and if you would like your very own piece of pie I'm gonna be signing these
and sending them out to numberphile viewers to find out how to get one have
a look in the video description or go to patreon.com/crashcourse member so we say
it's not just great in the one it's much greater than one so sort of a thousand
and above turbulence and your equations so we've got rid now