hello you there are looking at me matt parker talking to you about these numbers that i'm now
pointing at this is a reasonably famous series of numbers in mathematics and if you've not seen them
before do pause the video and have a go to see if you can work out which one comes next i was given
this on must be my first week of teaching and i spent all day but i cracked it so do pause try it
for yourself people complain i don't leave enough pausing time for these puzzles so i'm going to use
this very brief window to say the video is brought to you by jane street their internships are open
so if you're at the point in your academic career where an internship in maths and finance would be
a good next step details at the end of the video or in the description so uh hello to everyone who
paused welcome back i assume it's months later if you spend all that time and you didn't crack it
you are going to hate this this is the look and say sequence people often try to work out like
the ratio between the numbers and all these things no you look at one number and you say it and that
gives you the next number so the first number is one or more specifically looking at the digits one
one so the next number is one one but now we've got two ones so the next one is two ones but now
there's one two and one one and you keep going each time you look at one and you say it it's the
look and say sequence and while it feels like it's just a ridiculous cheeky trick question there's
actually some surprisingly deep mathematics behind this and it includes one of the rare
examples where the nice seemingly elegant perfect solution to a mass problem was actually a decoy
there is some serious mathematics behind this sequence of numbers no other than john conway
got obsessed with it in the 1980s and what is mathematics if not finding something ridiculous
and then taking it too seriously which is also the motto of this channel so the two questions we want
to ask about this sequence is first of all each sequential term how much longer than the previous
term is it i guess in the limit as it goes on and on and on and secondly what's the ratio between
the different digits like are there more ones are there more sevens what's the ratio between them
in the long run however we are not going to have a look at the traditional look and
say sequence because uh too many digits to put it shortly we're going to swap it out for the binary
look and say sequence the binary version is exactly the same but in binary so it starts
one and then one one but now when you want to write two ones the two is in binary so it's one
zero that's two one and the next line is pretty straightforward 1 1 1 0 1 1 but now you've got
three ones and three in binary is one one and you carry on using exactly the same look and
say principle as before and so we can still ask the question how long is each next term compared
to the previous one as a ratio and in terms of digits we've only got one ratio what is the ratio
between zeros to ones and we can start by just looking at that experimentally as we go through
each one you can count at the number of ones count the number of zeros so the first one that's got
a ratio is one zero one two to one twice as many ones as zeros the next one five to one crazy and
then after a while it starts to settle down into a pattern and once you pass the twenty third term
the pattern looks pretty much like it's just 1.666 so the ratio appears to be one and two-thirds
or is it for a while there everyone was happy to just roll with the conjecture that of course
it's one and two-thirds it's a nice neat ratio that's how maths works but then in 2010 nathaniel
johnson who's an associate math professor at mount allison university in canada thought no
i want to double check i want to prove exactly what that ratio is and so they use some
techniques that john conway had come up with and applied them to the binary look and see sequence
to make the look and say sequence more manageable we want to split it apart into a nice finite set
of blocks that can be recombined to make any of the infinitely many terms in this series however
we can't just use like one and zero as our blocks because we're interested in how the blocks lead
to each other so to explain here here are the 10 blocks right 10 blocks and each one always
goes to a set either one or two other blocks one after the other so if you've got a one by itself
and ignore the fact yes one appears in the other ones there's not like a unique way to split it
apart you just can split any term apart into these chunks the one always goes to one one if you take
it as an individual bit which means that chunk one always goes to chunk two so that's fine now chunk
two one one goes to chunk three and chunk one in that order and you can go through every single one
and write for each block which other blocks or chunks it goes to and you could represent this
as a network like showing which which block goes to which but we want to know is in the
limit once we've done this over and over and over again what kind of nice stable ratio do we settle
into and we're not going to do that using a network we're going to do that using a matrix
and here's the matrix so each column shows you where a certain block goes to so the first column
only has a single one in position number two that's because the first block goes directly
to the second block whereas the second column has ones in position one and three because the second
block goes to blocks one and three so what you can actually do is take any term in this sequence
which is like a series of which blocks you've got and then you multiply that vector of which blocks
you've got in your current term by this matrix and the result of that matrix multiplication
gives you the breakdown of the next term and so the behavior going from one term to the next
is governed by this matrix but how do we understand the long-term behavior of a
matrix if we're multiplying by it over and over wow we need eigenvectors and eigenvalues one of
the most fantastic bits of mathematics that most people have never heard of but it's okay
if you're one of those people it's your lucky day we're gonna have a quick crash course
in how eigenvectors and eigenvalues work in a segment i like to call i can see clearly now
here i have a nice simple two by two matrix zero one two three doesn't get much nicer than that i'm
actually going to turn this into negative two and negative three i'm going to multiply it by
a one by two matrix which is going to be one two nice and simple and straightforward and when you
multiply these matrices together you basically you get this one and you flip it over and you
you do a you know you multiply zero by one which is zero multiply one by two which is two and then
you add them together so that's gonna equal the one at the top here is gonna be uh two and then
the one at the bottom here is gonna be one times negative two two times negative three it's gonna
be negative eight so that goes down there okay and that's that's matrix multiplication and
if you think of these as vectors this matrix has taken this vector and turned it into that
vector which has a new length and indeed a new direction but what if i made the one slight change
to that being negative one at the top well okay let's redo the multiplication
this time it equals well that's just still well this top one is still two hasn't changed
here's that vector and the bottom one now is now uh well that's gonna be positive two
times this so it's gonna be negative 4 this time okay no major change but then if you have
a good look at this you're like wait a minute this vector is that vector just multiplied
by negative 2. so actually i could turn this back into the original vector negative 1
2 but put a negative 2 out the front and so for this specific matrix or vector if you
multiply it by this matrix you get the same vector just scaled and that in a nutshell
is eigenvectors and eigenvalues this is an eigenvector of this matrix because if you
multiply it by the matrix you get the same vector but with a value at the front the eigenvalue not
even the only option so instead of using negative 1 2 as our vector there we could put in the vector
negative 1 1 and it turns out if you multiply that by this matrix you end up with exactly
the same vector negative 1 1 but the eigenvalue is negative 1. so each time you multiply it by
this matrix all it's doing is flipping the sign of that vector crazy stuff and this is useful
because the process of constantly multiplying by the same matrix and getting out the same thing but
scaled means that the ratios of the values within these vectors aren't changing and that's exactly
what we want we want in the limit when the ratios between the different chunks cease to change so
what we have to do is find the eigenvector and eigenvalues for the big matrix we had
and then that tells us once you hit this steady state and we're going to look for the biggest
eigenvalue version of the many eigenvectors that's the ultimate steady state that our
series will settle into and nathaniel knew about this because their actual research
is looking at uh quantum information theory we're looking at different quantum states and you use
the same idea of eigenvectors and eigenvalues when you're doing quantum physics calculations
and nathaniel went ah i'm going to use them for this which is much more important i specifically
put these negative values down here because if you just do 0 1 2 3 you can get eigenvalues and
eigenvectors but they're way more complicated they're not nice neat integers you get like
square roots of 17 everywhere i hope now you can uh matrices how eigenvalues and eigenvectors work
although to be honest three blue one brown will have a much better video with a much better nice
visual explanation of this so go check that out um the downside is uh their theme song smells good
we just need the eigenvector and eigenvalue for our binary look and say matrix but how do we
find those well in nathaniel's blog post which i'll link to below they explain how they did it
they say using maple it is simple to derive this value they just put it in the software
package maple and i just spat the answer out and the eigenvalue is four six five five seven
one and probably more digits and that's it so in the limit it settles down into a nice rhythm
where each term is 46.6 five five seven one percent longer than the previous one
and the eigenvector looks like this interpreting this eigenvector shows us
that the first two chunks are irrelevant in the long run the chunk with just a one or two ones
don't have to worry about them and then we have the ratio between the other ones and so that b
squared means that the third and fourth chunks are b squared times as likely to appear or as
frequent as the chunks with the one next to them and we've got more complicated terms involving b
b is just an expression involving a and a is an expression involving the square root of 93.
the eigenvector is all about the square root of 93 when it comes to this matrix and this is just
a nice neat way to collapse all those terms down so that the eigenvector is a bit easier to look at
and understand but in theory you should substitute all those things in to get the original ridiculous
vector now to get the limiting ratio between ones and zeros we take the limit ratios we have between
all the different chunks and we multiply each of those likelihoods of that chunk being there
by the ratio of ones and zeros in that chunk and then we add them all up and we do not get
five divided by three we get this so on one hand mathematicians had one and two thirds
on the other hand they had this ridiculous expression and they're like oh which one
which ones are going to be in fact what you're looking at now allow me to say is
the thumbnail good thumbnail thanks for watching the video and it turns out this ridiculous one
is the correct answer which to be honest if you do work it through comes out to 1.66572
and some stuff very close and a good reminder that just because it looks like something
is the answer in mathematics you don't know for certain until you do the maths
thank you for looking at me say to you thank you for looking at this video it's been fun
and thanks to nathaniel johnson for doing the maths in the first place and thank you
to jane street who support my channel they saw very complex mathematical problems in
the financial world which means they just want more mathematicians that they can potentially
hire which is why they support my channel and why they run a very generous internship
program this is in all three of their offices the london new york and hong kong officers do this
and i've visited all of them oh i'm doing hong kong again soon the other ones i think i've
specifically seen the interns but no promises i am the least appealing a part of the intern
process more importantly you'll learn about all things like uh quantitative trading and software
engineering and quantity of research and business development which i just read off a list you've
seen what i do for a living it's this anyway j street uh they're incredible what they do and
if you want to do an internship i highly recommend going to janestreet.com join j street with hyphens
in it and if you want uh slash internships or something the link is on the screen do check
it out they're great and maybe i'll see you at one of the offices there you are thank you jane street
i can see clearly now the rain is an equivalent scaler
i can see all vectors that correspond
gonna launch
it's matrices be a bright high bright sunshine