# How Knot To Hang A Painting

– This week’s guest video

was pitched to me as a puzzle and I couldn’t solve it. So please welcome

physicist Jade Tan-Holmes who’s going to show you the same puzzle and then solve it with maths. Jade, take it away. – Here’s a puzzle for you. You have a painting, a piece of string,

and two pins stuck in a wall. Can you hang the painting in such a way so that if you remove

either one of the pins the painting falls? The thing to focus on here is

how to tie the string around the pins, not any of the physics of the objects. So if you tie the string in the normal way

one would hang a painting, this obviously won’t work. It’s pretty easy to think of ways we can tie the string so it falls

when one of this pins is removed. But the challenge is to make it fall

if either one is removed. Pause the video now if you’d

like to try it for yourself. So what we’re working towards is a way to tie the string around both pins

as a whole but neither one individually. So there are a few different

ways to solve this puzzle. But we’re going to do the one

which takes us through knot theory. Knot theory is the mathematical

study of tangled systems. As this is a problem about

tangling things together, chances are knot theory

will come in handy. With puzzles like these

it’s sometimes hard to know where to even begin. A common technique mathematicians use is to strip away everything but

the most important features. So, let’s think about

what we can get rid of. What parts of the problem

aren’t providing us with any useful information? Let’s look at the painting. It’s cool, but is it

really telling us anything? It’s just joining the

two ends of the string. If we get rid of it and

tie the string together we can reframe the question as can we hang the string

so that if either one of the pins is removed,

the string will fall? It’s just a simpler version

of the same question. Sorry, fashionable chicken. [chicken squawks] What else is unnecessary info? Let’s take a look at these pins. Basically nothing matters about them except how they interact with the string and the fact that they don’t

interact with each other. We don’t care about their

positions, size, shape or colour. We don’t even care that they’re pins. Mathematically, it makes

sense to want to deal with just one type of

object instead of two. Imagine reading a story that’s written in both English and Chinese. Sure, both languages

make up the whole story but it’d be much easier to work with if it was all the same language. This is exactly what

we’re going to do now. Translate our problem all

into the same language, the language of knot theory, by converting our pins into strings. All the properties we care

about are still there. If we tie them shut they

can’t interact with each other and they can trap our original string. In knot theory a closed string

is called a knot and a group of entangled

knots is called a link. Now we can reframe the question again as can you tie the three knots so that

when together they’re interlocked, but if you remove one,

the others fall apart? So far we’ve gotten rid

of a bunch of information, but we haven’t really

isolated what’s important. What should be focus on if

we want to solve the puzzle? Well, there’s really only one way a string

can interact with another. It can cross over them

or it can cross under. The crossings are the only

things we have control over and are the key to unlocking the puzzle. The crossings basically

determine one knot from another. But the thing is,

the same crossing can look very different. This makes knots hard to reason about

in an organised way. It’s like if you had all

the components of a story but had no idea what order

they were meant to be in. But if we organise the crossings we can reason much more

clearly about the knot. Let’s take this random link of two knots. If I start on a random string,

at a random point and start following the string,

drawing on my paper as I go, we can keep track of all the crossings

in an organised way. We keep going until

we come back to where we started. We’ve just created something

called a braid diagram. Knot theorists use these

to model all kinds of links. The point of braid diagrams is

to keep track of the crossings while getting rid of all the

unnecessary information distracting us. In a braid diagram, the lines representing our

strings are called strands. If we label the strand

positions one, two, and three, when the strand in position one

crosses over the strand in position two we can label this crossing X, and when the strand is position one

crosses under the strand in position two, we’ll call this inverse X, as it basically undoes what X just did. When the strand in position two

crosses over the strand in position three we’ll call this Y. And when the strand in position two

crosses under the strand in position three, we’ll call this inverse Y. This notation is called a braid group. And actually everything we need to know

about our problem is encoded in these letters. The way these letters work

is that the X and inverse X cancel and the Y and inverse Y cancel. But they only cancel out if

they are next to each other. So coming back to our problem we want the knots to be

interlocked as a whole but none of them locked

individually to another. So if we think about this

in terms of our braid group, removing a knot is like

removing all crossings involving that knot. If we remove the green string

all the Xs will be removed. And if we remove the pink string all the Ys will be removed. What we want is a configuration

where as a whole nothing cancels. But if we remove either term,

everything cancels. Let’s see what we can do. We’ve got three strings. Our system is symmetric,

so what we do first really doesn’t matter. Let’s just start with an X. Remember the pins can’t

interact with each other, so we have no choice but to

do another X or inverse X. We don’t want to have an X

and an inverse X next to each other because we don’t want

anything to cancel yet. So we’ll do another X crossing. This is just like wrapping

one string around the other. We could keep going like this with the Xs

but we need to include the other string, so let’s do some Ys. Now because we want our terms

to cancel out in the end, that means every X needs an inverse X and every Y needs an inverse Y. If we do two inverse Y crossings now they’ll immediately cancel

out the Ys we just did, which isn’t what we want. So let’s try two inverse Xs. Now we can do two inverse Ys. So let’s take a look at where we’re at. If you remove all the Y terms, the remaining X terms

cancel each other out. And if you remove all the X terms, the remaining Y terms

cancel each other out. But when everything is together,

nothing cancels. That’s exactly what we want. Now let’s work backwards

to the original puzzle, chicken and all. We can use the braid

diagram to model our link. Let’s see if it has the property we want. That when all three of them

are together they’re locked, but if you remove one

the others fall apart. Yup! Now let’s bring back the

chicken and the pins. Okay guys, this is it. So now we’re going to map

our braid group notation onto our system of pins. So, remember from the braid group diagram that basically a double X corresponded to

wrapping one string around the other. And a double inverse X corresponded to

undoing that move. So here we’re going to say that a double X corresponds to

wrapping around a pin clockwise and a double inverse X

corresponds to undoing that move which would just be wrapping

around the pin counterclockwise. So, double X. Double inverse X. Double Y. Double inverse Y. So let’s do it. Double X Double Y. Double inverse X, wrapping

around counterclockwise. Double inverse Y. So now the moment of truth. [drumroll] [thudding] BOTH: Mathematics! So this definitely wasn’t the easiest or most straightforward

way to solve this puzzle. So, why did we do it this way? Well, now we have this whole framework for solving this problem for

any number of pins easily. If we want to add

another pin to the puzzle this would be a nightmare to brute force. But we can just work backwards

from our braid group, and it’s simple. Today we used knot theory

to solve a fun puzzle, but it actually leads to some

very fundamental mathematics. We were searching for a

way to wrap the string around both pins as a whole,

but neither one individually. But more broadly we had to keep track

of the relationships between smaller parts of a bigger system. This kind of connection has led to

deep insights in many areas of science. Just recently, people have been exploring

the design of quantum computers using particles that behave just like

the braids that we saw in this video. So if you want to win

a Nobel prize someday better start tying some knots! – Thank you Jade!

Go subscribe to Up and Atom. I would recommend starting with her video

on quantum tunnelling. And next week, a video that

may leave you breathless.

Thanks Tom. It was awesome to be a part of this đ

Was gonna come up with a pun but I can see there's already a bundle.

Imagine using knot theory to solve string theory, that will help us understand the spacetime fabric! đ©

Just tie a loop hanging from either pin and put the painting though both

How to explain a simple concept in the most convoluted way possibleUnnecessary dramatisation of pointles thing. Damn people have too much time to waste.

The part around 6:15 with x – 1 over 2 and xÂŻÂč – 1 under 2:

As I understand it, 1 isn't the green string, 2 isn't the purple string and 3 isn't the pink string.

1 is the left part of the paper, 2 the middle part and 3 the right part.

Is that correct?

E.g. the second knot is X because the string coming from the left side (1) is over the string coming from the right (2) side. 1 over 2. Right?

Because if 1 would be always the green string and 2 the purple string, then the second knot would be XÂŻÂč (green string (1) under purple string(2)), but the second knot is X.

Is that correct? Or did I completely misunderstanding this?

I could listen to her all day.

This makes my brain rage

double X double Y double Ds

Just hang it.

How could you knot like this video?

Live and learn

Never thought maths could describe knots!

Noone:

Nobody:

Literally not a soul:

Tom Scott: Here's how to hang a picture so it falls

thanks, really cool

I've seen this years ago.. on different channel

Wait is this what they mean by string theory? đ

Works the same as the old scam game "fast and loose"

Haha

knotBrilliant

Got it

Knot what I expected

I can add this to my ancient temple traps!

I figured it out with the pins and the painting but then you changed the whole thing. Why?? Made me feel smart for coming up with a solution then changed it all so it didn't work anymore. (My method relied on the weight of the painting)

And then I couldn't come up with another solution bc holy hell the rest of the video didn't make sense to my dyscalculia brain….

I havent seen a single comment mentioning her tatas

So knot theory is not a theory

This blew my mind more than I thought it would

Halfway through am like SKIP to the end

Well, I just put the image on top of both pins but the solution is fine too đ

Math. Not "Maths."

Timmy: I wish we were string so we could study knot theory!

Cosmo and Wanda:

Poof2:28

What about the chicken?

Put both the pins next to each other and tie a know it the string

I just thought about putting the pins over the painting and string so if you remove own, it disengages said part of the string and it falls.

So this is string theory

My solution is to put the painting on top of the pins and not tie the string at all.

why not just do XYX*Y*?

oh wow that was way more complicated than I ever thought it would be

4:43 wouldn't it be better to alternate the breaks for better visualization?

head explodesđDamn it.. i thought this was a video about decorating your house or something smh, i'll still watch it though

Actually, I think I'm gonna use a nail and a hammer, thank you very much.

Wot?

This is awesome!

Gotta bust a knot

Wee wah woo wah

Get a frame

So you answered a question that has nothing to do with the original problem, yup that's mathematics.

'Can you hang the painting such that if either pin is removed the painting falls?'

Yes, place both pins at the bottom of the painting and let it rest on them. Problem solved.

The X shape string support is the best for

framed pictures and mirrors.Because the movable cross point in the crossed X string allows adjustment.

Someone needs to tell me what the name of that painting in the thumbnail is called.

Paused the video. Here's my guess.

1) Tie the strings in a single loop.

2) Thread the loop through both holes so that you have two ends of the loop going out through the back.

3) Hang both ends on the pins.

When you take out a pin, the corresponding end of the loop will go loose, and the painting will slide off that end.

Now to check the video to see if this was the solution.

"People have been exploring the design of quantum computers." (8:38)

I think that's an appropriate reference, coz those are, you know, based on STRING theory? :>

You Rock Jane….. Insane… !!!

String theory, nice.

But…why

did my brain really just explode because of a string and two pins?

I feel like this explanation over-complicates things. If you simply think of the problem in a true bare bones fashion and eliminate all knot theory it can be broken down into the question "In what formation are both pins on the "outside" of the string (from the perspective of the other pin) but it remains in place?". You can do this because in every situation where all pins are completely outside of the string the painting is insecure. This can easily be solved by wrapping the whole string around one and tying a loop to the other as you can see at 7:36 the green pin is NOT inside the string, then you backtrack along the string secured to the red pin so that what is attached to the red pin ends up as a loop and now puts the red pin "outside" the string. Oh god this took me 20 seconds to come up with but 20 minutes to explain concisely. As they say you need to truly understand something to teach it and i do not. I am an engineer its my job to be able to make things work and understand how i did it, not how the solution was originally or should be derived.

TL;DR:

Make it so that if a pin is removed all that is left is a loop around the other by backtracking.

This seems important for many things…

This was soooo amazing. Not because of the result but because of the great way you showed how to work on difficult tasks systematically. Great job

brain

hurt

I remember solving this puzzle a long time ago but forgetting the solution because its …

Knot… very intuitive.At 8:10 it shows that mess of letters, but removing the x and x^-1 gives y^-1 zz y z^-1z^-1, which doesn't cancel out everything. I'm curious as to why the extra y and y inverse are there, as they don't really need to be. Drawing the braid diagram without the extra y's, however, does not appear to work. Why does this happen?

Some nice spirit of the law music

It's 'not'. You idiot.

I heard mathematics and I dip.

7:51

me: falls off chair

7:15

Sheâs got nice jugs

Or just tie it exactly how it is in thumbnail, except upside down

(With the ends of the string on the pins)

Wow, this was a really fascinating video! It is very interesting to see the puzzle laid out mathematically. The mathematical approach to it helped in understanding how the solution worked and why it worked that way. It is always great to see problems from a mathematical perspective!

Why?

Guessing they were originally gonna call this string theory but that was already taken

Use a string that can support 0.999 weight of the picture. Removing either pin ensures that it will break and the image falls!

I noticed that the meaning of X-1 is actually

tiedto the previous X. What I mean by that is you can represent String1 going under String2 by X, if next time the same string goes over again, then that becomes X-1; and if it goes under, then X-1.So a string going over or under another string can still represent X depending on what previous sequence was. (See @5:20, string goes over, then under, which is physically different, but has same notation ie X)

4:15 '

underthe strand in position 2'draws string 1 passingoverstring 2…and again at 4:29 with 2 and 3

EDIT: 6:06 I HATE THIS GAH

What a great life hack!

Thanks for nothing Tom Scott

That was f*cking brilliant!

Why do math just look for 2 mins and them do it. Your brain itâs your calculator. Run brain simulations.

Up and a Tom?

The secret is to set fire to the pushpins and inhale the smoke.

Mathematics!

Did I just watch a 9 minute video about knots

a slipknot

In the douchiest way possible…who cares?

Me: There's an app for that.

Does Jade have a remnant South African accent?

at 7:47 it looks like the string on the right falls off the pin, rather than unravelling from it. The string is still a closed loop, so shouldn't it catch on the pin and still be hanging?

It's nice to see how mathematics is totally subservient to reality and that is only a speck of dust in the palace of human existence.

does pot = part?

"Pause the video now if you'd like to try it for yourself" … OK

20 seconds later … Solved.

Difficult?

Knot.

/shrug

Whilst video is paused…whole string around left pin, on right pin go on left hand side, have the tip of the looped string come around and be held in place by the string going back to the left pin….like winding a guitar string on a guitar. Removal of Right pins leads to picture falling, removal of left pin would release tension on the string….let's see how wrong i am…

the title gave me a very different idea đ

put two very thick knots into to the string. then use those big thick knots kinda like hooks. place only the knots, not the actual string, over a pin. the tension keeps both knots pulled against the sides of the pins and it stays up. but pull out one pin and the angular momentum of the falling painting and the lack of tension pushes it off the other.

This went from a fun puzzle to a math problem where the solution is just arranging it into the borromean rings. How.

Well i could put the painting on the wall in a way that it would fall if one of the pins was removed. I just wouldnt use the string. Just put the pins on the wall in a horizontal "line" just a bit shorter than the painting and put the painting on the pins. Problem solved.

This is fantastic, i think my love for mathematics was just reignited

The craziest title ever

So… Could knot theory be applied to knitting/crochet?

How long did it take me to solve this one? Well, how long is a piece of…

Oh, never mind.

Certainly demonstrates the Advantages afforded by the Use of Explosives.

I've never been great at theory, but this is knot theory so it's ok