WEBVTT
Kind: captions
Language: en

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- This week's guest video
was pitched to me as a puzzle

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and I couldn't solve it.

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So please welcome
physicist Jade Tan-Holmes

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who's going to show you the same puzzle

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and then solve it with maths.

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Jade, take it away.

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- Here's a puzzle for you.

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You have a painting, a piece of string,
and two pins stuck in a wall.

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Can you hang the painting in such a way

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so that if you remove
either one of the pins

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the painting falls?

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The thing to focus on here is
how to tie the string around the pins,

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not any of the physics of the objects.

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So if you tie the string in the normal way
one would hang a painting,

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this obviously won't work.

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It's pretty easy to think of ways

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we can tie the string so it falls
when one of this pins is removed.

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But the challenge is to make it fall
if either one is removed.

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Pause the video now if you'd
like to try it for yourself.

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So what we're working towards is

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a way to tie the string around both pins
as a whole but neither one individually.

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So there are a few different
ways to solve this puzzle.

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But we're going to do the one
which takes us through knot theory.

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Knot theory is the mathematical
study of tangled systems.

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As this is a problem about
tangling things together,

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chances are knot theory
will come in handy.

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With puzzles like these
it's sometimes hard

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to know where to even begin.

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A common technique mathematicians use

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is to strip away everything but
the most important features.

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So, let's think about
what we can get rid of.

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What parts of the problem
aren't providing us

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with any useful information?

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Let's look at the painting.

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It's cool, but is it
really telling us anything?

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It's just joining the
two ends of the string.

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If we get rid of it and
tie the string together

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we can reframe the question as

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can we hang the string
so that if either one

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of the pins is removed,
the string will fall?

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It's just a simpler version
of the same question.

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Sorry, fashionable chicken.

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[chicken squawks]

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What else is unnecessary info?

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Let's take a look at these pins.

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Basically nothing matters about them

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except how they interact with the string

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and the fact that they don't
interact with each other.

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We don't care about their
positions, size, shape or colour.

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We don't even care that they're pins.

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Mathematically, it makes
sense to want to deal

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with just one type of
object instead of two.

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Imagine reading a story that's written in

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both English and Chinese.

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Sure, both languages
make up the whole story

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but it'd be much easier to work with

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if it was all the same language.

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This is exactly what
we're going to do now.

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Translate our problem all
into the same language,

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the language of knot theory,

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by converting our pins into strings.

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All the properties we care
about are still there.

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If we tie them shut they
can't interact with each other

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and they can trap our original string.

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In knot theory a closed string
is called a knot

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and a group of entangled
knots is called a link.

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Now we can reframe the question again as

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can you tie the three knots so that
when together they're interlocked,

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but if you remove one,
the others fall apart?

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So far we've gotten rid
of a bunch of information,

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but we haven't really
isolated what's important.

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What should be focus on if
we want to solve the puzzle?

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Well, there's really only one way a string
can interact with another.

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It can cross over them
or it can cross under.

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The crossings are the only
things we have control over

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and are the key to unlocking the puzzle.

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The crossings basically
determine one knot from another.

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But the thing is,
the same crossing can look very different.

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This makes knots hard to reason about
in an organised way.

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It's like if you had all
the components of a story

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but had no idea what order
they were meant to be in.

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But if we organise the crossings

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we can reason much more
clearly about the knot.

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Let's take this random link of two knots.

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If I start on a random string,
at a random point

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and start following the string,
drawing on my paper as I go,

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we can keep track of all the crossings
in an organised way.

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We keep going until
we come back to where we started.

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We've just created something
called a braid diagram.

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Knot theorists use these
to model all kinds of links.

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The point of braid diagrams is
to keep track of the crossings

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while getting rid of all the
unnecessary information distracting us.

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In a braid diagram,

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the lines representing our
strings are called strands.

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If we label the strand
positions one, two, and three,

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when the strand in position one
crosses over the strand in position two

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we can label this crossing X,

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and when the strand is position one
crosses under the strand in position two,

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we'll call this inverse X,

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as it basically undoes what X just did.

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When the strand in position two
crosses over the strand in position three

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we'll call this Y.

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And when the strand in position two
crosses under the strand in position three,

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we'll call this inverse Y.

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This notation is called a braid group.

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And actually everything we need to know
about our problem is encoded in these letters.

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The way these letters work
is that the X and inverse X cancel

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and the Y and inverse Y cancel.

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But they only cancel out if
they are next to each other.

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So coming back to our problem

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we want the knots to be
interlocked as a whole

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but none of them locked
individually to another.

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So if we think about this
in terms of our braid group,

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removing a knot is like
removing all crossings

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involving that knot.

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If we remove the green string
all the Xs will be removed.

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And if we remove the pink string

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all the Ys will be removed.

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What we want is a configuration
where as a whole nothing cancels.

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But if we remove either term,
everything cancels.

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Let's see what we can do.

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We've got three strings.

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Our system is symmetric,
so what we do first really doesn't matter.

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Let's just start with an X.

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Remember the pins can't
interact with each other,

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so we have no choice but to
do another X or inverse X.

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We don't want to have an X
and an inverse X next to each other

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because we don't want
anything to cancel yet.

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So we'll do another X crossing.

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This is just like wrapping
one string around the other.

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We could keep going like this with the Xs
but we need to include the other string,

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so let's do some Ys.

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Now because we want our terms
to cancel out in the end,

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that means every X needs an inverse X

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and every Y needs an inverse Y.

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If we do two inverse Y crossings now

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they'll immediately cancel
out the Ys we just did,

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which isn't what we want.

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So let's try two inverse Xs.

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Now we can do two inverse Ys.

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So let's take a look at where we're at.

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If you remove all the Y terms,

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the remaining X terms
cancel each other out.

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And if you remove all the X terms,

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the remaining Y terms
cancel each other out.

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But when everything is together,
nothing cancels.

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That's exactly what we want.

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Now let's work backwards
to the original puzzle,

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chicken and all.

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We can use the braid
diagram to model our link.

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Let's see if it has the property we want.

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That when all three of them
are together they're locked,

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but if you remove one
the others fall apart.

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Yup!

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Now let's bring back the
chicken and the pins.

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Okay guys, this is it.

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So now we're going to map
our braid group notation

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onto our system of pins.

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So, remember from the braid group diagram

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that basically a double X corresponded to
wrapping one string around the other.

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And a double inverse X corresponded to
undoing that move.

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So here we're going to say that

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a double X corresponds to
wrapping around a pin clockwise

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and a double inverse X
corresponds to undoing that move

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which would just be wrapping
around the pin counterclockwise.

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So, double X.

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Double inverse X.

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Double Y.

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Double inverse Y.

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So let's do it.

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Double X

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Double inverse X, wrapping
around counterclockwise.

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Double inverse Y.

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So now the moment of truth.

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[drumroll]

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[thudding]

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BOTH: Mathematics!

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So this definitely wasn't the easiest

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or most straightforward
way to solve this puzzle.

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So, why did we do it this way?

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Well, now we have this whole framework

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for solving this problem for
any number of pins easily.

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If we want to add
another pin to the puzzle

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this would be a nightmare to brute force.

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But we can just work backwards
from our braid group,

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and it's simple.

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Today we used knot theory
to solve a fun puzzle,

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but it actually leads to some
very fundamental mathematics.

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We were searching for a
way to wrap the string

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around both pins as a whole,
but neither one individually.

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But more broadly we had to keep track
of the relationships between smaller parts

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of a bigger system.

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This kind of connection has led to
deep insights in many areas of science.

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Just recently, people have been exploring
the design of quantum computers

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using particles that behave just like
the braids that we saw in this video.

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So if you want to win
a Nobel prize someday

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better start tying some knots!

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- Thank you Jade! 
Go subscribe to Up and Atom.

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I would recommend starting with her video
on quantum tunnelling.

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And next week, a video that
may leave you breathless.