Hi everyone. So I've decided to start a series of videos on Olympiad questions.
So that might be questions from the International Mathematical Olympiad,
the Australian Mathematical Olympiad, possibly even the Putnam competition
which isn't really an Olympiad but same sort of level of difficulty, and just
whatever other Olympiads I can find from around the world. So if you're not
familiar with my channel, I already have series of videos on very hard maths
questions for the younger students around 13 years old and also quite hard
questions for the older students around 17 years old so if you're looking for
something a bit easier than Olympiads you might want to check out those
playlists. So Olympiad questions are really really hard and they're hard even
for me. Now I'm saying that as someone who has a degree in physics, first class
honours; I was top of my year in maths again and again. I've had a lot of extra-
curricular problem-solving practice, so even for me they're really really hard.
So I just wanted to emphasize growth mindset here because I feel like even
though they're really hard for me even now I'm gonna get better and better at
them and you're probably in the same boat. Like when you look through these
Olympiad questions you might feel completely demoralized and just defeated
because they are really hard but the more you do them, the more you try to
understand them and read through the solutions and work on them, the better
and better you'll get. Now the question that I've chosen to
work on today is one that I actually could do; I found one that was easy enough.
So this is from the 2017 International Mathematical Olympiad and I'll put a
link below to where you can download that paper. So you might want to
pause the video here, read through the question and see if you can figure out
how to prove it. Alright now the first thing that I would
do with a question like this is just try to sort of draw up a particular
rectangle and play around with it just to make
sure that I understand the question completely. Because you know some of this
might be a bit confusing this "distances... all odd or even" - what does that mean? So we
get a much better idea if we draw a picture. So the first thing I did was
draw up something like this. I made it a 7 by 15 rectangle. Often it's good to
start with something small, but I felt like a smaller rectangle would be like...
smaller rectangles would fill it up very quickly so I wanted to make something
bigger. So we've got a rectangle R with odd integer side lengths and then we
want to divide it into small rectangles. Now I think that the first time that I
read this I thought that it would be sort of divided like you draw a line
down there and a line across there and then you'd have like 4 rectangles
in there and like it had to be like that. And then I reread the question and
realized, well it could be something a bit more interesting, like we don't have
to have straight lines going all the way across. So we might have something that
looks maybe like this. So now we've divided it into small rectangles with
integer side lengths. "Prove that there is at least one among the small rectangles
whose distances from the four..." Okay so what does that mean? "Distances from the
four sides". Let's just pick one of these rectangles like this one for instance. So
"the four sides", so that's, you know, these four sides of R... its distance from
this side... well it's touching that side so that distance is zero. Its distance from
this side is like, you know, the number of squares between them, so that's 5.
Distance to this side is just 4, and distance to that side is 4 as well.
So that's what it means about the distances from the four sides. Now
we want them to be either all odd or all even, so we've got even even even odd so
that's not one of the small rectangles that it's talking about here. So maybe we
can work out which one it means. Let's look at this one.
So its distance from these two sides is both 0 and then the distance to this
side is 4, and the distance to that side is 8, I think... yeah.
So that one has the distances all even, so that's one of the rectangles
that it's talking about. So this sort of process of like drawing stuff up and
trying a particular rectangle is really useful to help you understand the
question. You're going to want to like reread the question a few times
because it's easy to sort of think it says one thing when it actually says
another thing. So this is very useful. Now I played around with this for a while,
and I noticed that what we need is a rectangle with an odd width and height,
because if it's even then you've got like an odd number of squares left above
and below it. Do you know what I mean? So you can't split an odd number up into even and
even or odd and odd. I hope that explanation made sense to you. If it
doesn't then just play around with it for a while by yourself and, you know,
draw things out, see what happens and you'll start to understand. So I
thought that maybe a good first step might be trying to prove that there's
always a rectangle with odd width and height like it's usually a good idea to
invent an easier problem for yourself to try to tackle first
before you tackle the full problem, because proving that there's a rectangle
with an odd width and height doesn't necessarily ensure that it's gonna have
these distances all odd and even but it might be a good idea. So I think that was
my first idea. I didn't really get anywhere with that, to be honest. So that
didn't turn out to be useful. But then I started to see some similarities to a
question that I read about a long time ago called the "mutilated chessboard
problem" God that sounds awful, but the question goes like this: you've got a
standard chessboard and you cut off the two opposite corners so they're gone and
then the question is: can we tile that chessboard with dominoes? So I know kids
these days don't really play with dominoes so a domino is just a little rectangular
thing like that; it's just like two squares stuck together basically. So, you
know, we might stick a domino on those two like that and stick another one there
like that and so on. So you might like to pause the video here and see if you can
work out the answer to that question. But okay
so what happens is you'll find that you can't tile the mutilated chessboard
with dominoes because each domino covers one black and one white square,
because it covers two adjacent squares, and this chessboard that's been cut up
has more black squares than white squares, so that's not gonna work; we're
gonna be left with two black squares at the end which aren't gonna be adjacent
so we can't stick a domino on them. Okay so that's that question. I don't remember
where I read about that; I know that Martin Gardner covered it in one of his
books and I'll put a link to that below but I don't think that that's where I
read about it because I seem to remember the author talking about how much harder
the question would have been if it was just presented as like you've got
an 8x8 grid and you cut off two opposite corners like if they haven't mentioned
chessboards at all, you would sort of have to come up with this idea of
coloring it in like a chessboard and then that makes it a much harder
question because you have to come up with this very creative idea of coloring
it in like that. So that sort of stuck with me; I don't know how many years ago
I read about this. If anyone can think of a book that mentions that and brings
it up as like "what if it wasn't presented as a chessboard?" then let me
know in the comments, because I don't remember what book it was that I read that
in. Alright so then what I did was colour in my rectangle like this, like a chessboard,
so let's play around with that idea. So you can try out various, you know,
rectangles. We still want like ones with odd width and height like we found
before, so we might pick one like that.
So that one has a distance there of 1, distance there of 4... 3 again, I mean
odd again, and 6 there so even like that one, but that one's got mixed, so those two
are even and those two are odd, so that's not what we want. Let's try another one.
Let's do something like that. So that one's got a distance there of 1,
distance there of 1, distance there is... 9,
and that distance is 3, so that one's got all odd, so that satisfies our
condition there. So if you play around a bit you'll see that what you want is a
rectangle with all four corners black, like this one has its four corners white.
So why might that be? So going back to the idea of the mutilated chessboard
problem, where the problem there was that that there were too many black squares
to cover with dominoes, so... well if we look at this big rectangle you'll find
that it has more black squares than white squares. Now this rectangle here with
the four white corners, that has more white than black whereas this one has
more black than white. If you had a rectangle with mixed corners, so some
of them black and some of them white then you get an equal number of white and
black squares within it. So you can kind of see the outline of the proof here, that there
must be at least one rectangle with four black corners because that's the only
way that you're going to get more black squares than white squares in total, and
a rectangle with four black corners is going to satisfy this condition. So
that's basically the proof; that's the idea of the proof. So I kind of I wrote
out something... I write this out. You can read through that if you want; it's just
putting into words what I just said. What I just said was in words, but you know
what I mean. Okay, I wasn't too happy with this sentence here though; I just sort of
glossed over the idea that a rectangle with all for black corners would... would
satisfy this condition; would have either all odd or all even distances. So I feel
like this is the kind of thing where if I was actually doing the
olympiad I would probably write all that out and then sort of resolve to
come back and flesh out that idea in more detail if I had time at the end.
Now, to be honest I couldn't actually think of a good way to word that and
explain why a rectangle with four black corners is gonna satisfy the condition.
But I read in the solution... they explain it pretty well here. They're
just using green instead of black and yellow instead of white. So what they've
said is that because this is green and this is green, that means that that
distance and that distance have the same parity. Now parity just means whether
something is even or odd. So they haven't explained why these both being
green mean that these two numbers have to have the same parity; they just sort
of held it as self-evident, but... so they go on to say, like, okay so x and u have
to have the same parity. Now because that's green and that's green, that means
that that has to have the same parity as that and so on around the rectangle. So
all the numbers have to have the same parity. So that was a pretty good
explanation. Now I don't think I would have been able to solve this question if
I hadn't heard of the mutilated chessboard question before. So one of the
really important things for being good at mathematical problem-solving is just
having seen lots of different questions before and being able to apply solutions
to old questions to new questions. So one really good
source of questions like this is books by Martin Gardner, also Ian Stewart is
another author who writes this sort of recreational mathematics books and
I'll put a link below to where you can get those sorts of books. It also
reminded me of something that Paul Lockhart said in his article, A
Mathematician's Lament. He was talking about the area of
a triangle, so the way that he presents it is that you've got a box and a
triangle that fits inside it like that, and you want to know, what is the area of
the triangle? Like how much space of the box does it take up? So you get this
brilliant idea to cut it like that, and then you can see that that is half of
that and this is half of that, therefore the triangle area must be half
of the box area. So he is making a point that the way they present it in schools
is really not helpful because they just say that the area of a triangle is 1/2
base times height, and that misses out the important thing. "It's not the fact
that triangles take up half their box that matters. What matters is the
beautiful idea of chopping it with the line and how that might inspire other
beautiful ideas and lead to creative breakthroughs in other problems -
something a mere statement of fact can never give you." So I want to make the
point here that when you're learning new things in mathematics you should always
be trying to work out how people came up with that idea, like try to see the
derivation and understand it because that's the important part. The formulas
and stuff and conclusions are really nice to know, like they can be
useful in many ways, but often what's more important is how that formula was
derived, how people came to that conclusion. So if you want to be really
good at mathematical problem-solving, try to work out those things. All right I
hope this is helpful to you guys. If you enjoyed this video then give it a like
and consider subscribing. Now I'm not sure what Olympiad question I'll be
doing for the next video in this series. If you've got a particular one in mind
that you'd like to see me go through then let me know in the
comments.