I have been very passionate in Mathematics very early on, and eager to present the beauty
and my love of Maths to others. That is why on the screen right now you can see a website
that I designed when I was thirteen. It is definitely not well-maintained and I have
lost the files, so I cannot update the site any more. You are more than welcome to click
the link in the description to see my work.
I learned about Catalan numbers in a combinatorics course outside school when I was thirteen,
and I didn’t think too much of it. However, one year later, I came across this kind of
square maze where you need to go from the bottom left to the top right. Suddenly, I
thought of the question: given a particular size, how many mazes could there be? This
is an utterly vague question, because first of all,
What does a maze even mean? And what did I mean when I said a given size? So, I narrowed
down the question into thinking only about the solution path. Then this path lies on
a particular grid, and so we can talk about the size of the grid; but of course there
are problems with the simplification, which I completely ignored at the time: for one
thing, one maze could have different solution paths, and the other point is that different
mazes could have the same solution path. But I didn’t care, as counting solution paths
is way less ambitious than counting mazes.
I told my math teacher in school about this problem that I had been working on for fun.
He told me that I should be looking for something called self-avoiding walks, which are paths
on the grids that don’t visit any vertex more than once. This is perfect for my project
because the solution path should have this exact property: if the so-called solution
path has a vertex that is visited twice, we can cancel this portion and obtain a shorter
solution path, and in a maze, nobody would say this is a solution path, right? And something
like this is totally legitimate because it is possible that there are walls to block
you from going with the shortest route.
When I went back home and scrolled down the Mathworld page, I found that there are formulas
known only for small grids, and this, for some reason, excites me. I could have made
an original discovery! Of course, I was being arrogant here, because even proper mathematicians
couldn’t work out those formula, why should I have any reason to believe so much in myself?
To emphasise the difficulty of the problem at stake, think about the number of self-avoiding
walks on this simple grid from the bottom left to the top right. Pause the video and
work out the answer. Are you ready? The answer is 12 - these 12. You probably got this right.
But what if I just increase the side length of the grid by 1? Pause the video again to
work out the answer. The answer is 184. That escalated quickly. In fact, there is an OEIS
page dedicated to this particular problem, recording the nth number in the sequence as
the number of self-avoiding paths in an nxn grid. And here, you can see that the logarithmic
plot of the sequence behaves quadratically, so the growth of the sequence is more than
exponential. I quickly gave up on the idea of making an original discovery because I
drilled on that for too long and still haven’t had a clue how to tackle that problem, because
even for this case, I couldn’t count properly to give an answer of 184! I turned to something
that I knew of - the Catalan numbers.
I didn’t remember exactly how I drew parallels between the very ambitious problem of the
maze, and the Catalan numbers, but I think I might have thought of this. Catalan numbers
can also be formulated in a one-dimensional way. Assume a robot is initially on the origin
walking along a non-negative number line. It can either walk to the left or to the right
but it cannot cross the 0 mark. If after 2n steps, it goes back to the origin, the number
of ways the robot could do this is the Catalan number. This problem has some sort of constraint
in the sense that the number of right steps must be at least the number of left steps.
In the maze problem, we can also say that the number of right steps must be at least
the number of left steps, and also the number of up steps must be at least the number of
down steps. Of course, these are not the only constraints for the self-avoiding walk problem,
because these two conditions don’t exclude the self-avoiding part of the counting, but
that’s close enough to make me think there might be a connection between the two problems.
The problem is, we can solve the Catalan number problem using a 2-dimensional grid, which
we have seen before in the video series; so we need to use a 4-dimensional grid to solve
the self-avoiding walk problem even if we ignore the self-avoiding constraint. That
was definitely out of my element, but I didn’t give up yet. What I tried to do was to study
the Catalan number problem a little bit further, then try to think of a way to generalize this
to the supposedly 4-dimensional problem that I am eventually after.
So just to be clear about the timeline of the narrative, first of all I thought about
the maze problem - how many mazes are there given some size; then I simplified it to counting
the number of solution paths, which happened to be self-avoiding walks, which many mathematicians
studied; but then this is still too ambitious a task, so I went back to studying the Catalan
numbers further instead. So I generalized the Catalan number like what you saw in the
previous video. That generalization part was done when I was fourteen. These were done
all by myself, and I felt very proud of it, and called it a day.
One year later, there was something called a “mathematical modelling mini-thesis competition”,
and my math teacher encouraged me to join. I totally ignored the mathematical modelling
bit, because I thought, I had been thinking about this whole thing for a long time, this
was about time I finished it. So I actually did even more generalizations that I have
described towards the end of my last video, as well as two more cases. Again, these were
all done by myself. But then two unfortunate things happened.
Firstly, I suddenly realized the phrase “mathematical modelling” and I needed to come up with
a real-life situation for all of these to fit. That’s why I thought of this bank deposit
or loan scenario. Now, this can’t be called mathematical modelling because nobody in the
real world will count how many ways there are to solve a problem. But this is still
better than having no quote unquote real-life situation in it, right?
The second unfortunate thing is that towards the end of my project, I figured something
called Catalan triangles and trapezoids, which were exactly what my generalizations had been
about. So I couldn’t exactly call those my original discovery. The best that I can
say is I discovered them independently. So at the last minute, I decided to just cite
those sources about the Catalan triangles and trapezoid.
I didn’t get a medal or anything, because that was not even remotely a modelling problem.
I literally just force a made-up scenario into my pure math problem. I got honorable
mention in the end. Naturally, I would really want to see what other contestants of the
competition wrote that got the medals. I really just wanted to know how a modelling should
have been done, because throughout this project, I had not learned anything about mathematical
modelling. So I saw the title of the mini-thesis that got bronze, and it says, “the mystery
of five-pointed stars”. I didn’t know what the content of that article was about,
but I was pretty sure that it was not more real-life than my mini-thesis. It did not
even hide the fact that they were not doing a modelling! Ok, I am not salty, I am not
salty, I am not salty.
But in any case, this was the first proper project or research that I did, so it really
meant something to me even if I didn’t get any concrete thing (like prizes) out of it.
I hope that this video gives you a bit of context behind this video series. I don’t
know if you guys like these type of not-so-mathematical videos once in a while, because this type
of video is fun for me to make, not sure if it is the case for you as well. Tell me in
the comments whether you like it or not and I would tune the frequency of these types
of videos.
If you enjoyed this video, give it a thumbs up and subscribe with notifications on. See
you in the next video! Bye!