Welcome to another Mathologer video. Today we'll take a well-earned break
from all the heavy-duty algebra of the last couple of videos. Today it's all
going to be super visual and super accessible, promise. Okay, to start with
let me first get you hooked. Three colors: red, yellow, blue. Put down a row of ten
hexagons and color them randomly. There that's one possible coloring. Draw a row
of nine hexagons underneath. The color of one of these new hexagons depends on the
colors of the two hexagons above it. See whether you can guess what the rule is.
Yellow and red add to blue. Red and yellow add to ... also blue. Yellow and blue
add to red. Easy, right? I'm sure you've guessed
already that whenever we are adding two different colors the result is the third
color, and so on. Blue plus yellow is red. Yellow plus red is blue. Okay, what if we
have to add a color to itself? Well, red plus red is red. That's the most natural
rule. Agreed? Agreed! Draw a row of eight hexagons underneath
what we've got so far and, using the same rules, just keep
filling in colors and drawing rows until it all ends, like this. So you end up with
an equilateral triangle composed of hexagons. Important observation:
once the colors of the top row have been chosen, all the other colors are pinned
down. Here a couple of examples of triangles resulting from different
choices of colors in the first row. There and that one and that one and that one
and that one. Very pretty. Before we take a closer look, let's play
a speed game. In a moment I'll randomly color this
first row. Then I'll count down from 5 and ask you to decide what the resulting
color of the bottom hexagons is. Get it right and you win, hmm, a lifetime
subscription of Mathologer videos. Yes, okay, it's free anyway but you win
bragging rights. What are you going to do? Of course, you can just guess for a one
and three chance. On the other hand, since the first row will determine all the
colors of the hexagons below, you can just calculate the color of the bottom
hexagon. IF you're really really quick (or if you pause the video :) Anyway, time to go.
Ready? Well, ready or not here we go! Okay, make your guess. What's the color of
the bottom hexagon. Five, four, three, two one. And the answer is ... yellow. Did you get
it without pausing the video or just guessing? Sounds impossible? Well it turns
out that there is a super surprising shortcut for this calculation. It turns
out that the color of the bottom corner of our triangle is simply the sum of the
two colors in the top corners. There, yellow plus yellow at the top is yellow at the
bottom. Yellow + blue at the top is equal to red at the bottom. Blue plus red is
yellow. Blue plus blue gives blue. It really works. Pretty amazing, isn't it?
Choosing different colors in the top row gives very different colors of our
triangle and yet the colors of the top two corners alone determine the color of
the bottom corner. Weird, hmm? Well, it gets weirder. What if instead of ten hexagons
at the top we started with nine hexagons Well, then the short cut doesn't work
anymore. For example, stripping a slanted column
off of the width 10 triangle over there gives us a width 9 triangle and here the
shortcut clearly doesn't work: blue plus blue should be blue not yellow, stripping
off another row gives a width 8 triangle. Again the shortcut doesn't work. Keep on
going. Hmmmm, In this case, blue plus blue is blue, as
expected. However, if instead we had stripped
things like this we get a width seven triangle that
doesn't work. Width six doesn't work, and neither
does width five. Okay so why is 10 special? And are there any other special numbers.
To hunt for clues let's have a closer look at this larger triangle. Notice all
the smaller solid color triangles here and there? It's raining little triangles.
Yeah raining raining raining. Maybe just this reminds you of something.
Hmm, can you think of a famous mathematical supermodel dressed in
little triangles? :) No? Okay have a look at this special super symmetric example. In
this case the first row is entirely yellow except for a single red hexagon
in the middle. I first encountered the strange shortcut phenomenon in an
article by mathematicians Erhard Behrends and Steve Humble in the Mathematical
Intelligencer. It reminded me of the famous Sierpinski triangle fractal in
which it is also raining little triangles. Beautiful pattern isn't it. And
I was also reminded of the pattern on a giant snail shell that I bought a couple
of years ago in a butcher shop here in Melbourne. Butcher shop? Yep you can find
the strangest things in Australian butcher shops. And, of course, the whole
summing two above gives the one below business should be very very familiar.
Yep our good old friend Pascal's triangle is based on the same sort of
growth principle. Steve Humble one of the authors of the article I
mentioned earlier created the three color game as a mathematical outreach
activity in 2002 and discovered the shortcut while demonstrating it to kids.
Today I'd like you to imagine that you are a mathematician who, just like Steve, has
stumbled across this mathematical gem. so As a true mathematician you are now
cursed to not be able to sleep until you've come up with an explanation for the
shortcut and how it relates to the snail pattern and to Pascal's triangle and
Sierpinski's triangle. Well, let's find out together, shall we? In
five easy chapters. Okay time to investigate. Why does the
shortcut work for width ten triangles. And, are there any other special numbers? Well
we already saw that nine, eight, seven, six and five are not special. And that's
where we stopped. Hmm, I wonder why? :) Let's have a look at four: blue + yellow is red.
Probably also just a fluke. Right? Wrong! If you keep experimenting with widths four
triangles you'll find that the shortcut always works. Four is also special. Okay,
let's say it's been a long night and we're all pretty brain-deaded. In this
state can we still show that 4 is special? Yes, no problem. There are four
hexagons in the first row and three possible colors. This translates into
three to the power of four that's 81 different ways to color the first row.
this means that they are exactly 81 with four triangles and even if I'm half dead
I can quickly throw together a computer program that will in an instant list all
these triangles and check that our shortcut works for all of them. Actually
if you're not quite so brain-deaded then we can use symmetry and permuting
colors to reduce the number of triangles we need to check. Ao a little challenge
for you: How many essentially different widths four triangles are there? Let us
know in the comments. Of course, checking all those triangles doesn't tell us why four
is special, we still haven't proved that ten is special and we haven't checked many
other widths. So it's time to go deeper. With my computer program I can also
quickly make up larger widths triangles and check that among all the numbers
from 1 to 100 the only numbers that appear to be special are 2, 4, 10, 28
and 82 - hmm - for 2, 4, 10, 28, 82... tricky! Nope, not tricky at all. Did you
spot the pattern? These numbers are all one up from a power of 3. 4 that's 3 plus
1. 10 3 squared plus 1,28 3 cubed plus 1, 82 3 to the power 4 plus 1. And let's not
forget 2 at the beginning. 2 is 3 to the power of 0 plus 1. The plot thickens and
based of what we've got so far we conjecture that exactly the numbers one
up from powers of three are special. Now let me show you something absolutely
beautiful a proof that all these numbers are indeed special. this one is really
good, promise. Let's begin where we began by showing that width 10 is special. So let's
start with any old width 10 triangle. Now focus on these hexagons here. Then these
three hexagons are the corners of this widths 4 triangle and since width 4
triangles are special, the top two corners must add to the corner at the
bottom. The same is true here and here and here and here and here. And now, well
I probably don't even have to say it, right? Can you see what's happening. we
just showed that over there any two adjacent colors add to the color below
and what this means is that these highlighted heagons combine into a
widths four triangle. And, therefore, because four is special the top two
corners add to the bottom corner. But since the top corners of the width four
triangle are also the top corners of the original widths ten triangle, it
follows that the top two corners of a width 10 triangle add to the bottom
corner. Always! In other words, 10 is special.
Ta da. An argument like this makes my day. So on a scale from one to ten how
beautiful an argument is this? Yep we are doing polling now :) Let me know in the comments
what you think. To prove that twenty eight and indeed all the other powers of
three plus one are special, we just repeat this argument over and over. Here
JUST the quick animated first iteration of the argument that shows that twenty
eight is special, using the fact that four and ten are special. So nice. Qnd a little challenge for you:
Can you think of a second way to argue? Hint: switch the roles played by four and
ten. Anyway, I suspect that at this point most people
would declare the mystery solved. TIme for a cat video? Well, that's not
what we do here on Mathologer, right? Is it really just a big coincidence that
widths four is special or is there a deeper reason? Qlso we know that two four ten
etc special and we suspect there are no others, but how can we be certain? And
what's up with all these similar phenomena? Sirpinski and snail shells and whatnot?
Are there any beautiful connections? Ready to go deeper? The basic rule of two adjacent colors
in one riw summing to give the color immediately underneath just cries out
for us to have a look at the tip of Pascal's triangle. Right? Any two numbers
in this famous number triangle add to the number right below. Here four plus
six that's ten, five plus one that's six. Okay, this addition process is very
similar to our coloring scheme but there are also obvious differences: in Pascal's
triangle there are no hexagons, we're adding numbers instead of colors and
everything starts from the tip rather than from a row of numbers. Hmm, okay, you
want hexagons? Well here are a couple. What else? You want the whole thing to
start from a row and not from the tip? Not a problem at all. So we can think of
Pascal's triangle growing from an infinite row of zeros with a single one thrown in somewhere in
the middle. And that should also remind you of the Sierpinskish color triangle we
saw earlier which starts from a row of yellows with one red exception in the
middle. Interesting, huh? Anyway to keep things uncluttered and to escape from
this frame here, I'll hide all the zeroes anyway and we return to the familiar
picture of Pascal's triangle. Okay, just remember that the zeroes are
still there but hidden. Now what about colors? Well, the first idea that
comes to mind is to color the hexagons according to one of the natural ways to
split the integers into a finite number of classes. For example, coloring all the
odd numbers dark and the even numbers white, we get this. That looks promising
and it's even more promising when we zoom out to reveal a larger slice of our
triangle. Whoa pretty impressive, huh? In fact, if you
keep zooming out, our ever more detailed even odd triangles will converge to the
famous Sierpinski triangle fractal. Anyway, it's most definitely raining
little triangles in there, just like in our three color
triangles. Another way to interpret our odd-even coloring is to say that we
color according to what remainder a number has on division by two. After
division by two the possible remainders are zero and one. For odd numbers the
remainder is one and we color dark. For even numbers the remainder is zero and
we color in white. What we then get is Pascal's triangle using what is called
base two modular arithmetic. Fancy words but it's just even and odd. So wherever
you see one and zero next to each other there will be a 1 underneath, 0 plus 1
equals 1. That translates to even plus odd equals
odd. Similarly 0 plus 0 equals 0 and then there's the slightly weird 1 plus 1
equals 0 which amounts to odd plus odd equals
even. Okay we found a natural numberish way of
coloring with two different colors. What happens now if we play our game with the
even-odd rule. Are width ten triangles still special? Hmm, here a few random width ten
triangles you get this way. There's one, there's another one, there's another one,
there's another one, there's another one and ... Well, is 10 still special? No,
definitely not. If it were, black and white on top should
give black at the bottom. Right? 1 plus 0 equals 1 not the 0 we got there. But don't
despair, there are also special widths for our even/odd coloring. However
instead of 1 plus powers of 3, this time it's 1 plus can you guess it ... powers of 2? It is easy to prove that all these
numbers are special using the same collapsing argument as for our red
yellow, blue triangles. This lovely and simple construction also suggests a very
natural model that can be used to at least partly explain the formation of
snail shell patterns. A snail shell grows in thin layers that are added onto its
lip. In our mathematical model these layers are the horizontal rows of
hexagons. Then the color of the hexagons and each new layer is determined by the
simple rules for adding odd and even numbers. This is all very neat and is
based upon a biological mechanism that can be observed where the
characteristics of existing cells determine the characteristics of newly
formed adjacent cells. So Nature has already found a lovely
application of the games we are playing here. Do any of you apply similar this
plus that mathematics in anything you do in your working life? Let us know in the
comments. Okay, so this is what Pascal's triangle
looks like when you color it according to the remainder after division by two.
For our original game we use three colors. So let's see what happens when we
use remainder on division by three. Of course, now the possible remainders are 0
1 and 2 and let's respectively assign them the colors yellow red and blue. And this
is what you get. There, very pretty again. Now compare
this triangle to the nicely symmetric triangle that resulted from our original
three color game. Yep so close :) Not quite the same but
we're definitely onto something. Okay let's have a close look at the growth
rules for our remainder three Pascal triangle. So with Pascal to add a and b we simply
go a + b mod 3. Now in terms of colors this doesn't quite correspond to
what happens in our original color game. For example, in our original color game
red plus red equals red, but Pascal gives the answer as blue. However, there's a
super simple algebraic adjustment to get perfect agreement. Can you see what we
need to do? Well, first notice that at least the middle
rules always work. But to make things work completely
these 2s need to be replaced by 1s and these 1s need to be replaced by
2s and there's a really simple tweak to do that. Just go - (a+b) here.
Let's check that this tweak really has the desired effect. Okay - 0
well that's 0. Nothing changes here which is great. What about the top rules? Well
now we have minus - 2. But in mod 3 arithmetic - 2 is the same as -2 + 3 which equals 1. Got it? And on the bottom row we now get - 1
which is equal to - 1 + 3 = 2. This really works!
So we've done it. This algebraic rule has captured a mathematical soul of our
original three color game. As a first application of this clever insight, let me
show you how you can see pretty much at a glance why four was special in the
original game. Remember, originally we showed this by listing all possible 81
widths four triangles and checking that the shortcut works for all of them. We
can do much better than that using just a little algebra. We start with four
random colors a, b, c, d and then we can simply calculate the remaining colors
like this... But, of course, in mod 3 arithmetic three
times anything is just the same as zero and so .... Fantastic, b and c just cancel out
and the number at the bottom only depends upon the numbers a and d in the
top corners, in the right way. So our simple calculation confirms, once again,
that 4 is special for our original setup. But actually, this calculation tells us
something about Pascal's triangle as well. Instead of our original three color
game, take a look at the Pascal 3 color game. Then the algebra is exactly the
same, except all the minuses here turn into pluses. And again the three times
somethings in the bottom hexagon disappear. So for this Pascal 3 color
game as well the number 4 and therefore all those other numbers 10, 28, etc. are
special. What about the odd-even, black-white mod 2 Pascal game. In the case of this
game, we only need to worry about this smaller part of the triangle. Why? Because
2 times anything is even and so it reduces to 0 mod 2 and so the 2b term
in pink zaps to 0. This proves with 3 is special
for the mod 2 game and then by our collapsing argument we can see that all powers of 2
plus 1 are special. Of course, we can play the same games
using any number of colors. With m colors we can then use the sum modulo m or its
negative as our generation rule and then the same algebra and our contraction
argument can pin down the corresponding special numbers. For this we need to
identify those entries here for which all the middle coefficients are
multiples of m. As we've already seen for m is equal to 2 the first time this
happens is here and as we've also already seen for m is equal to 3 the
first time this happens is here. And now a challenge for you, tiny little
challenge. Figure out the smallest non-trivial special numbers for m is
equal to 4, 5, 6, etc. until you get sick of it. Ok, a hint: just focusing on the
blue entries can you see another Pascal triangle? If
you're keen to learn absolutely everything else there is to know about
these games check out the Intelligencer article by Erhard Behrends and Steve
Humble that I mentioned earlier. In particular, in this article you can find a
proof that the special numbers we've spotted are really the only special
numbers. This proof is based on the 1909 article by the Indian mathematician Balak
Ran in the Journal of the Indian Mathematical Club. How on earth did they
find that one? So, using the special numbers that result
from our additional rules, we can fashion ourselves some nifty keys with which to
explain those striking self-similar patterns. Let's have a closer look at the
triangle over there. A reminder, we got this triangle using our original three
color addition rules starting from a row of yellows with a single red hexagon
right in the middle. Okay so the whole thing starts with this row of hexagons
at the top. Since yellow plus yellow equals yellow
it's clear that we'll get those two huge yellow oceans on the left and right
right. Now what about the mystery white region in the middle. Let's figure out
what yellows we get there. In order to do that I'll make myself one template each
for the special numbers 4, 10, 28 and so on. I now place the template so that the
two upper corners sit on yellows in the outer regions. Whenever this is done, we
know that the bottom corner of the template must also be yellow. Let me
show you. All make sense, right? But just to pinpoint
exactly where the self-similarity of the pattern comes from, think about the
yellows being built in this order: When we place one of the templates so that
it's top two corners lie in the left and right oceans of the yellow, the template
generates one of the centered yellow triangles, like this. This takes care of
all these centered triangles, there, all all of those. But now focus on this off-center
region here. What's happening here? Well we already have large lakes of
yellow on the left and right plus a starting row all yellows with one
hexagon of a different color in the middle. So within this frame we can now
generate centered triangles exactly as before, there, and so on. This really gives
a very good intuitive feel for where this self-similar pattern comes from,
doesn't it? And it's not that hard to turn this intuition into rigorous proofs
for the fractal nature of these patterns. There's one very surprising feature of
the original three-color game that I did not mention yet. When you rotate one of
the triangles colored like this you get another triangle of the same type. So
what this means is that this new triangle also grows from its first row.
This feature is shared by all the other negative mod m colorings, but not by the
straight Pascal mod m colorings, except mod 2. Can you think of the simple
explanation for this phenomenon? Anyway, to finish off, let me just show you an
animated introduction to the most natural three-dimensional counterpart of
Steve's three color game that also shares the 3d counterpart of the
rotation feature. And
that's all for today.
