072 = The Perfect Paint Set and Rubik's Reset

Welcome to a problem squared.

This is the problem solving podcast where we solve your problems.

And this podcast is a bit like the Earth's orbit.

Yeah.

In that it goes on and on.

It just goes on and on and on and on.

And everybody is along for the ride.

So your co-hosts are Beck Hill, a comedian, author, TV presenter, and a bit like the Earth's orbit in that she's just the right amount of eccentric.

Ah, thank you.

You're welcome, I think.

And that's what gives us the seasons, plus the tilt.

You're not tilted.

That's okay.

And I'm Matt Parker, mathematician, author, YouTuber, and I'm a bit like the Earth's orbit in that at my heart, it's all just angles and careful calculations.

Yeah.

Yeah.

Yep.

Good intro.

Thank you.

I like it.

I was trying to think of other.

I originally thought about doing because it got real wintry real quick in the UK.

Yeah.

And I was going to do a like a seasonal intro and I got distracted by the Earth's tilt.

And then I was like, well, why don't we just do the orbit?

Yeah, why not?

Yeah.

It's almost circular orbit, the Earth's orbit, but just a little off, a little eccentric.

And that's what makes it.

Yeah.

Yeah.

And

there's a leap podcast.

We do an occasional leap podcast.

You're right.

Because we do two, one every two weeks.

Which means it's not always.

Not quite two a month.

Yeah.

Every now and then we have to do a leap third one in a month.

Yeah.

On this episode, I'm going to try and create a hypothetical set of paints.

I show that the Rubik's Cube is more repetitive than you expected.

And there's going to be some anti-orbital business.

Business.

Nicely done.

Bec.

How are we doing?

I'm great.

We're in the same room.

We are in the same room.

It's always fun to have the same room recording.

That's so nice.

I'm so the cupboard man.

Just when you think you're out.

Yeah, when you're in Australia.

I'm really looking forward to that.

Oh, yeah.

I'm off to Australia soon.

So, yeah, I'll be phoning in from summer.

Yeah.

Good to have fun.

You tell how excited I am.

Go sun yourself.

Go sun yourself.

No, I'm good.

I...

Now, two episodes ago, I think it was, I talked about painting my bookcase.

Actually, it may have been in the I'm a a Wizard.

That's the quality of content that we put in the Patreon.

That's right.

It's literally about paint drying.

That's right.

Yeah.

And I didn't stop at the bookcase.

Oh, after I did that, I was like, I'll do the rest of the wardrobe.

And so I started filming a time-lapse of it.

And

I asked the landlord if they minded me painting it.

And now, in fairness, they did come back and say, as long as you paint it a reasonable color.

It's a building wardrobe.

It is made of the cheapest MDF and it has black mold constantly coming out of it.

Yeah.

It's like it is, we've been in this flat.

It's terrible.

We've been in the flat for almost a decade.

But like we have a good relationship with our landlord and we like our flat and you know they're good to us.

We're good to them.

We don't complain too much.

They don't ask too much of us.

And normally if something is broken then they replace it for us as a landlord should.

Now we could complain about the wardrobes, be like, you need to replace them.

Yeah.

Man, they're getting the job done and we clean out the mold.

or somewhere yeah keep it

so when they said as long as it's a reasonable colour i figure

that as long as i return it to a reasonable color oh you took it to be ultimately well i mean as long as it's a reasonable color one day i figure that even if they look at it and go that's not what we agreed to we'd be like yeah hey one person's reasonable but while we're here you're not the one looking at it yep and also genuinely i think by the time we move out they'll they'll gut the whole place anyway because it does because it previously mentioned

now.

How many colors did you end up using?

So I used the same color palette that I used for the bookcase.

Yep.

So there's five colors.

Nice.

Yep.

I kept to the same palette.

Oh, good.

But I just got like black outlines.

So it has a bit of a Mondrian feel.

Yeah.

You know, the outer side.

Geometric squares, lines.

And I wanted to do a similar kind of graphic cartoon type vibe.

So.

Can I just clarify?

Yes.

Are you painting just the outside of the wardrobe or the interior as well?

Just the outside.

Okay.

And I sanded and sanded and cleaned it because it hadn't actually been done since we'd moved in.

And so it was a pretty gross condition.

Lots of marks and scuffs and stuff.

Sanded that down.

I gave it a nice lick of white neutral paint.

And then I've painted the drawers in the block colours.

And then I've given the whole wardrobe like a black outline.

So it looks almost drawn.

Like a cartoon wardrobe.

yeah wow because as as we've discussed i'm gonna start putting out more youtube videos next week oh yeah yeah yeah it's the year of youtube and so i wanted to have an interesting background ah the wardrobes going on so i thought i'll do the wardrobes and then you can get a little bit of color and stuff in the background i put some funky patterns on some of the side as well it's very me it's very me yeah and so i filmed it and then i put the video up on my instagram as a reel and then i put up like the little thing from the landlord saying as long as it's a reasonable color and then it

I knew what I was doing.

Yep.

And then obviously at the end of it, it shows the transformation between how they look before and how they look after, which is amazing.

I've just

transformed.

Yeah.

And then I sort of said, hopefully, their version of reasonable

is the same as mine.

One person's reasonable.

Yeah.

Yeah, exactly.

Now, let me tell you, I've made a lot of videos in my time.

Yep.

I've put in a lot of effort.

Yeah.

There's a lot of talent, a lot of skill.

A lot of interaction with people online.

Arguably stuff that is much more worth your time to watch.

Yeah, yeah.

Than a time of me painting it's had 1.5 million views there you go and it's got like 40 000 comments at the moment right yep people are yeah very heated oh i can imagine lots of people calling for my eviction now oh wow yeah yeah now i would i find that very funny considering it would be a heck of a lot cheaper for the landlord to turn around and say I didn't agree to this.

You dingus.

You need to paint it back.

Fix it.

Now, they could say paint it back now.

I don't think they would.

As I said, we've been there for nearly 10 years.

we've got a good relationship with them but if they were like look we're not okay with this please paint it back now yep fine yeah that would be a heck of a lot cheaper than it would to kick out people who have been reasonable tenants for the last decade who have paid on time looked after place not complained too much the wider context is important people are imagining you're a new tenant and you're you're just destroying the house yeah they're not reading things they're not reading

on the video no literally says that to start with there you go been here nearly a decade so i can see why why there are multiple interpretations.

Everyone's arguing their interpretation in the comments.

I think it's a bit much to call for someone to lose their home.

You have no idea how many people have been like, I would kick you out if you're my tenant.

You're the reason that we don't trust tenants.

It's like, really?

This?

A painted wardrobe?

At first, I was like, I was responding to people.

And then after a while, I was like, oh, no.

And then you get the people who, there's a lot of lovely people saying the same thing I'm saying.

And just being like, hey, she's been there for nearly 10 years.

And And actually, Infan has a lot of people saying, I'd be fine if my tenant did this.

I think the takeaway is if you want organic online interaction, just stoke some kind of class warfare between the landowning

gentry.

I had, I did not, I thought maybe people would go, oh, I don't like it.

It's gross.

Or whatever.

Fine.

Maybe some people would go, I don't think that's what your landlord meant.

I thought it would stop at that.

That would be funny.

Yep.

I thought it was like, I knew what I was doing with that sense.

But people were pushing the notion of land ownership.

Lots of people telling me, why don't I just buy a house?

It's like a tree fancy,

yeah, yeah.

Don't do that to someone else's property.

I'm sorry, they're not paying to live in it right now.

I am.

And then there was someone who was like, Don't blame society for your mismanagement of money.

You're failing to pull yourself up by your reasonably colored bootstraps.

Yeah, that's right.

So that's what I've been up to is dealing with

a heck of a lot of inciting the

downfall of capitalism.

Yeah, honestly.

Nah, I mean, I would take things that happen online as a indicator or metric, more of a temperature of the society we live in.

I wouldn't take any of it literally.

That's a good way of putting it.

Yeah.

Yeah.

I think the details are irrelevant, but the sentiment is probably realistic.

Yeah, I mean, that doesn't make me feel any better.

Wow.

Have you been?

Good, great.

Haven't stoked any massively controversial online things of late.

I've just been writing my book.

That's my answer every time.

Yeah.

Getting real close.

Although when this podcast comes out, I will have had to have handed in the second draft.

I will have been out of time.

So the second draft, as people are listening to this, is definitely in.

Amazing.

I'm sure future Matt will stick to that deadline.

But one of the advantages of a book running over so late is that as new things come up, I can put them in the book.

Books are very final, and you can do other editions, but it's a whole thing.

So, really, once the book is done, it's done.

Yeah.

And I got an email from someone who said that they'd seen my ongoing campaign about street signs in the UK that inaccurately show a football being made all out of hexagons.

And that's very funny because you can't make a football all out of hexagons.

You cannot arrange hexagons to cover a ball.

And a lot of people send me things like people trying to make football cakes

and just messing it up.

Totally messing it up.

That said, you can use hexagons to cover a surface.

So if you were just trying to cover the side of a cake in hexagons,

it would be fine.

Yeah.

No worries.

Classic tiling.

So Timmin and their partner getting married.

They went to the cake decorator and said, Can you make us a cake?

And because we're both, let me get their exact wording, mathematically inclined, they said,

Could you cover the cake in hexagons?

Which is mathematically very straightforward?

And yeah, because you've got essentially two surfaces for each cake.

Sorry.

You've got

the bit around the side,

and then the top.

And then you just do the same for the

other layers.

Yep.

So they're like, look, put some hexagons around the side.

Can't go wrong.

Easy to put together.

The baker had some problems.

And they sent them a photo of the cake in progress.

And it turns out when they heard hexagons, they thought octagons.

They They made octagons.

And so, would you like to see a hexagonal covered cake where instead of hexagons, it's octagons.

Oh, no.

And they're just giving it a good go.

They've tried to artistic their way out of it.

The thing is, if you saw that, just in general, you'd be like, oh, yeah, that's like it looks like it's got a bits of confetti on it or something.

But you can just imagine the cake decorators' frustration where they're told just hexagons and then they all fit together, but somehow they've made octagons and they've just kind of put them all near each other.

We will share this.

Yeah, like if you're putting some balls, like yeah, yeah, that's just stacked them, hoping for the best.

And I've got a collection of like shapes that are mislabeled.

So I've got hexagon biscuits, but they're actually octagons.

Like this is a surprisingly common mistake.

Do you know what?

There needs to be a rolling pin with hexagon like the mesh built into it, yeah.

Mesh.

So you can just roll it across your

hexagons.

There you go.

And then turn the whole sheet of fondant into label it

done there's something from masgear.com oh my goodness and uh so at the moment at the moment it's not in the book but i've got another oh and how long until this episode comes out a couple weeks to finish the draft and see if i can get this cake into the book we will see it's currently not in there i've got a bit of a gap in the introduction at the moment i might try and sneak it in there If you want to find out if it is in the book, if, ooh, I don't know if I will be allowed to have pre-sales yet when this episode comes out.

So, one of two things will happen: we will either link to pre-orders for my book or there'll be a cheeky link.

So, hypothetically, I'd be the kind of person to leave a link to a website where you could sign up to hear about pre-orders before anyone else.

And that might be in the show notes instead.

Hypothetically, not technically pre-orders, but if you want to pre-pre-order, there will be a link available so you can be first in the queue.

Our first problem was sent in on the problem posing page, which anyone, literally anyone with internet access, can go to at a problemsquared.com.

And someone named Nettle, N-E-T-A-L,

wrote in to say they have an art supplier's problem, which is 100% a Beck problem.

And they have a very friendly opening.

They say hi with three eyes.

Hi.

Hi.

Which cancels down to negative I for people playing along at home.

And Nettle is buying sets of colored pens, pencils, paints, et cetera.

And they've noticed it's much cheaper per color than buying them individually.

Oh, so if they buy a set of them, And then divide it between the colors,

it's cheaper per color to buy a bunch at once than to buy individual paintings or paints or whatever.

That makes sense.

They've discovered buying in bulk.

Yeah, if you buy like a set of 12 paints.

That's cheaper than buying 12 separate colors.

That makes sense because you could pay a premium for a single one.

Yeah.

And there's some economy of scale to buy all 12 at once.

Great.

I don't see how this is a problem so far.

However, oh, there's more.

Some colors will inevitably get used up faster than others.

And at some point, a whole new set will be needed to replace worn-out colors.

So the least used colors will build up duplicates.

This thing, if they use up all of a popular color and they just go and buy a whole new set,

they've now got too much of the unpopular colour.

Correct.

They need to paint a wardrobe.

Yep.

So they want to know what is a good ratio of colors for a hypothetical set such that all the colors would run out simultaneously.

And I love

because I've been buying my paints and sets and then using up the main ones and then I go to replace them.

I'm like, it's still cheaper just to buy a whole set.

Now,

the problem with this problem is that, so for instance, it really depends on like, it's going to change depending on what type of artist you are and what you like.

I asked the question on Twitter.

I just said, artists, what colors do you tend to run out of the fastest?

Yep.

Of pens, paints, you know, pencils, whatever.

And a lot of artists said, when it comes to pencils, they ran out of blue first.

Blue.

And that's because you do your initial sketch in blue and then go over it either in ink or

colours or.

So you're not just using blue as a color.

You're using the blue pencil like as infrastructure.

Yes.

Yeah.

Apparently that's because it's a much easier color to remove, like in Photoshop or something if you're scanning it.

And it was far less likely to turn out in photocopying, if you're scanning or photocopying things.

So it was easier to not see

when you were reproducing it so i i will say this as someone who doesn't do that i know a lot of artists i know a lot of artists use blue to do the initial sketchings in photoshop like you know when i've seen like web comic artists put up work in progress it is often blue outline even though it's all digital i mean doing it in a different color makes sense because then when you're going over it you know what lines you've already done and what you know you can see what you're going over but you don't do this i don't do it because i don't well as you famously say sketching is cheating

that's what i say yeah you do you're like do it in pen final one first that's that's your motto said it once and i'll say it again my motto is measure never cut forever there you go

um no to be fair when i'm making flip charts Again, you listeners, if you're not sure what that means, just Google Beck Hill Wardrobe.

Look on flip charts.

You'll see.

When I'm sketching those out, I tend to do them in pencil first and then I erase the pencil afterwards.

So, yeah, it depends on what your medium is.

It depends on what sort of paint.

Certain paint allows itself more to layering than other paint.

And then what you are painting.

Like if you draw or paint landscapes, you're going to be using certain colors more than you would if you're not.

The greens, the blues.

Yeah.

Yeah.

So it's a difficult one to provide.

an overall answer for, but I thought I'd give it a crack.

That's the motto of this whole podcast.

So the first thing I did is I reached out to Peter Donahue,

who is art.pete.repeat

on Instagram.

He's very popular and he has a bunch of reels on Instagram because he is a color theory educator.

Oh.

So he looks into color theory.

He'll show the way that color is perceived in light with crayons in pencils and his reels are so fascinating.

So I posted this problem to him and just said, look, I understand it's going to be difficult.

And he also agreed.

But he pointed out something I hadn't even considered, which he said, I think the key thing to understand would be tinting strength.

Tinting strength.

Yep.

So when it comes to paint, every pigment has a different tinting power that certainly impacts how long it takes to use up.

It would take longer to use up a tube of thallow blue than cobalt blue.

Classic cobalt.

There's other variables to consider as well.

What hues are the most used on average in art?

What lightness levels?

And so

tinting levels.

tinting is just that's how opaque the paint is or how much you have to use to build up the same level of colour yeah so if you've got particular pigment in there let's say if i was going to mix a bit of let's say one type of blue yep with a type of red yep

ie it to get like a mauve colour yep i a certain types of blue if i had a tiny bit might change the red a lot yep There's other types of blue where if the pigment's different, I would need a lot more strong to get the same.

So, you're going to use up different amounts of different colors

based on their impact.

Yeah.

So, as we know from printing,

that works with CMYK.

Oh, yeah.

Four colors.

Yeah, which is, which is cyan, yellow, K is black,

and magenta and magenta, yeah.

So, why don't they just sell paint sets with only those four colors?

You could.

You can make the rest yourself.

There you go.

Buy a barrel of each of them.

You've got to remember that CMYK, when it comes to printing, for white, it just uses less.

So you don't have a white paint, a white ink.

White paper.

It just uses less of the ink there to create a lighter color so that the paper comes through.

Whereas if you were wanting to paint something and you're using CMYK and you wanted to...

Okay.

Yeah.

Then it comes down to the medium because you can't use a thinner level of paint if it's like a thick acrylic.

Worked it out.

Yeah.

Five colors.

Correct it.

But then I was like, hang on.

You still got the same problem, though, because with CMYK, you're going to be using certain of those colors way more than others.

Right.

Yep.

So I reached out to Richard Moros, who's the CEO of Moo.com.

They

print business cards and greeting cards.

And I said, what do you use the most of?

K.

K is the most common.

They run out of that.

Yeah.

First thing.

Black.

So you could be like, oh, well, well, then I guess we need CMK, K, K, K, K, and white.

And white.

So I was like, hmm.

All right.

Let's tackle this from a different angle.

Why don't we look at the most popular colors used?

Right.

Yeah.

So I found a blog

by someone called Martin Bellender.

And the article was The Colors of Paintings, Blue is the New Orange.

And they created, you're going to love this, Matt.

They created a code to scrape.

4,434 pages in the BBC search browser of classic art images.

That's amazing.

I was just thinking, how would I scrape images from Google Images or something to analyze?

But they found a website that's got paintings on it.

A lot.

A lot of paintings.

From over the centuries.

They downloaded all the functions.

They had a lot of data information.

Yeah.

Good work.

So it was all

130,000 images, totaling 2.5 gigabytes.

130,000 images.

Yes.

Good work.

They tried to format the years as well.

so that they could get an idea of what year

the paintings were done.

And then they plotted a graph, basically, based based on I feel like there were some steps in between there's some steps in between there are I'm gonna post that because this is very much your tab in terms of describing what I'm sure they analyze because you can analyze each one automatically to pull out the color of every pixel exactly which would give you pretty good information about what it's

similar to how you worked out how much ink would be used in a book in your book in my book yeah so for previous for new listeners when one of Beck's books came out I took the PDF files and then just analyzed it to work out how much was blank page and how much was ink in the letters or diagrams.

Yes.

And so they've kind of

the same thing, but they would do it.

I just did it for ink or no ink because it was printed in Grayscale.

They will have done it like just by splitting out the colors or something clever.

That's nice.

Yeah.

And so they've come up with this incredible looking graph, which we'll post on socials.

And it goes back from 1250 to 2010.

That's good stuff.

And it shows that generally over the years, most of the time, it's sort of an orange color.

There's a dip around the 1500s where there's a bit more blue.

But then as we get closer to 2010, you get a lot more blues and greens being used.

Is that just paint technology?

Ah.

Well, they said there's quite a few things to take into account.

One is that

Blue may have become more affordable in recent years.

It's quite a hard color.

There was actually a whole thing about blue M ⁇ Ms because finding the fake blue colour was kind of a new deal.

So blue's become much more easy to get hold of and affordable because it's easy to get hold of.

They also pointed out that a lot of these images are photos of the paintings.

They're not the actual paintings.

No.

So it could also be the lighting of the painting.

And maybe in some cases, black could be showing as blue

or misred as that.

So we could be getting mixed ideas from that.

That's true.

They also said that obviously different types of paint fade in time and become more yellow.

If you think about a lot of older pictures, a lot of them are more yellow than they would have looked originally when they were painted.

The varnish will have yellowed.

That could also be having a huge effect on the results.

So at first I thought, oh, I could look at these results.

I could look at what more recently, what paint is being used in certain art and sort of maybe create a palette based on what seems to be the most used colors versus the least.

But then, because of that, I was like, there's still a lot of variables.

So,

I tried one more thing.

I reached out to Sea White Brighton, which is an art supplies company based in Britain.

They're actually where I get my flip charts from.

Oh, they're your flip chart suppliers.

Yeah, yeah, because they also make paint.

And so, I reached out to them and said, I don't suppose you could let me know just

which paints sell the most

versus that.

And I thought, yeah, I'll wang it out there.

See about half an hour later, I got an email back with an Excel spreadsheet.

And this is where I'm going to need your help, Matt.

It's because you're one of their most valued customers.

So they sent you an Excel spreadsheet.

Yes, they did.

It's got 50 rows.

We're down in row 50.

And each row is a volume and color of paint.

So row 34 is a 500 ml burnt sienna container of acrylic paint.

So this is not trade sensitive data.

They've sold 409 of those.

Okay.

You would need to pull out the colors from the text for each product.

You'd need to pull out the volume and then you'd need to group and weight the sales figures by the volume of paint and then sort by color.

Yeah.

Yeah.

Should we have a little break?

Yeah, yeah, yeah.

Let's have a quick break.

And I reckon I can I can do something something really, really quickly.

Okay, I've got something.

So I pulled out all the volumes from the descriptions of each product.

Yeah.

So it meant that I could multiply the volume of each product by the number of units sold to get the volume of paint sold.

Okay, yep.

Which I've converted into liters.

So everything is now liters of that colour.

Oh, handy.

Pulled out the colors and I've combined any that have exactly the same name, but I've left things like primary yellow is separate to yellow,

separate to lemon yellow.

They're all different yellows.

And then I've I've worked it out as a percentage of total sales by volume.

So the most common color sold by volume, coming in at 29.4%

of all volume of paint sold by this shop in whatever the time period may be, is white.

Yep, that figures.

Like priming canvases.

Priming, but also.

Most people painting wardrobes.

But also to lighten the colors because

again, like printing, it's not like you do a thinner layer.

You get it for free.

If you're printing a printer, there's a painting you got to mix in the water.

You got to mix it in, yeah.

Followed by, not closely, 12.1% of all paints sold by volume was black.

And again, similar thing.

Do you want to have a guess what the most popular color by volume that's not white or black?

Ooh, okay.

I'm not going to go based on exact name.

No.

You can

pick a category.

But I'm going to say red.

Clothes.

Cadmium red is number four on the list.

Ooh.

At 6.8% of sales.

Wow.

It is beaten.

Primary yellow.

Figures.

Primary yellow was.

I was between red and yellow.

1,537 liters of primary yellow.

1,344 liters of cadmium red.

So the ranking goes white, then black, then primary yellow, cadmium red, ultramarine.

Oh.

Third most popular colour.

Followed by cyan magenta.

So these are the classic building blocks.

Burnt umber.

Brown.

Phallocyine.

Don't know what colour that is.

Phallocanine.

That could be it.

Thanine.

Something like that.

That one's the blue.

And also, given this is one shop, if there's one local artist who has a signature color and bulk orders their paint through their shop, that would swing the results.

True.

Yeah.

So this is a point in time.

I don't know if colors that are used are seasonal, but this is a specific time interval for a specific shop and location and everything.

So,

not totally generalizable.

I mean, see white of Brighton, they're just outside of Brighton.

So, they're either going to sell a lot of blue because they're near the sea.

Anyone who's doing landscapes, and I imagine during autumn, you're going to get a lot more yellows and reds.

I will say, though, after you're out of the top 10, and top 10 includes black and white, all the paints are pretty similar.

They're all 2.5%, 2.5%, 2.5%, 2.

Like they're selling the same amount of all of these.

Even once you get down to the more obscure ones they're 1.9 percent 1.6 percent there's not a lot of variation but you could use this breakdown if you wanted to put together a paint set you could generate an image with circles of each of the colors size to match the frequency of sales it would be a picture of the set oh so you're saying you might have a 500 mil tube of white yeah but 200 mils of the other ones now generally in a set they're all the tubes are the same size That's where we're going wrong.

We're breaking ground here.

Different size tubes.

Different size.

Yeah.

I mean, it doesn't package nicely.

Hey, hey,

stacking is a whole separate issue.

But that's an interesting take, Matt.

And I like it.

Can we tumble-pack them into a box?

So rather than going, well, you've got 12.

See, because at the moment.

What are you thinking?

You're getting like 12 whites.

Yeah, that's

what I was thinking.

Because at the moment, Sea White of Brighton, they do sell paint sets.

And what was very interesting is that in their 15-pack of paints, they have three tubes of white and two tubes of black.

Ah, so they are kind of doing this.

Yeah.

So normally in a pack, you get 12 or 24.

Yep.

They've gone, okay, instead of 12 of each color, let's chuck in some extra white and some black, take it up to 15.

Oh, okay.

Yep.

I mean, using the spreadsheet, you could give me a number of tubes and it could generate the color breakdown of those.

Oh,

this is not perfect, but it'll do for a first pass.

So, how many tubes would you like in your?

I guess that's good for a standard 12.

Okay, your standard 12, you'd have four whites, so you'd have four whites, one black, and all the rest of ones: one yellow, one cadmium red, one ultramarine.

And you work way down the list until you've done another eight.

So, what they've done, they've put in cadmium green and lemon, and they haven't put in primary cyan and red magenta.

Other than that, they've gone with the what's the most commonly ordered.

Interesting.

But I liked your suggestion of what you could do is just

have

different ratio of amounts of color.

So if you did a 24-pack.

I can give you my ultimate 24-pack right now.

Give us your ultimate 24-pack.

Here's what you get.

You get seven whites, three blacks, two primary yellows, two cadmium reds, and then you get one of each of the following: ultramarine, primary, cyan, red magenta, burnt umber, that weird blue we keep talking about, that's dance PTH, the yellow orca, naph carmine, cadmium yellow, cobalt blue, and cadmium green.

Done.

There's your set.

Amazing.

See, now I understand that this is a paint set based on what everyone is buying right now.

Yes.

And so therefore...

If you're a real average painter.

It won't apply to you specifically.

And unfortunately, what that means is the best you can do is just keep an eye on what you buy most of.

It probably won't apply to anyone specifically.

No.

But on average, it'll be the least worst.

Yeah.

Which it sounds like they're kind of close to doing that anyway.

Yeah, yeah, yeah.

So we're back where we started.

They probably started with this and then just tweaked it based on what people actually want.

Yeah.

There's method to the madness.

But I'm not going to start listening to what people actually want at this point in my career.

I would say this is the breakdown.

I want to do one final thing to work out, does this fit what I do?

Yep.

So I recently started painting, actual like on canvas and stuff.

And I was like, well, let's do a painting and we'll see what, what I use up most of.

Yeah.

So I thought great experiment.

So thank you because I knew you were going to do extra work on this.

Uh-oh.

Oh, you brought a painting.

How'd you smuggle a painting in?

Is this for me?

Yeah, this is for you.

Oh, my goodness, Beck.

I've done a painting for you.

You've done a painting?

Okay, so for the listeners, it's wrapped up in brown paper.

I don't want to damage it in the unwrapping process.

I'm in.

Oh, my goodness.

You've painted the photo you took of me at the end of a long, hard day in

our accommodation in New York.

I think you've really captured the emotion.

Yeah.

That's really good.

That's good.

I didn't realize as I slumped into a chair in

New York Brooklyn apartment without enough ventilation.

And

I was sitting for a portrait of this quality.

Thank you, Beck.

To be fair, I didn't know either until this problem came in.

So, yeah.

That's hilarious.

I was wearing that t-shirt yesterday.

We were off by one for me wearing the t-shirt when you gave me the painting, which would have been pretty special.

You have once again included without permission my bare feet.

No, I made sure I didn't get there's one foot and it's the top of the foot.

Oh, okay.

That's all right.

I made sure I couldn't, you couldn't see all the feet.

I wasn't going to give those for free.

Beck had a bad habit of putting

everyone's feet on the internet for free.

You'd accidentally do it once.

That's great.

Ah, and you've signed both paintings in the painting.

Whistler's Matthew 2023 by Beck Hill.

We will put a photo of me holding this up on the socials.

And what I will say, I have got a question.

I used a ton of white.

What was the break?

Thanks.

So much white.

And that was just my head.

Yeah, you're super pale.

So, yeah, ton of white.

Got through my yellow and my red pretty quick.

Bit of blue and my black I got through as well.

But yeah, but that's, that was most of the colours.

Your dog has started eating the brown paper.

Skylab is now eating the brown paper.

It's a present for everyone.

Well, Bec, I mean, obviously.

We need to wait until we get the official thing from Nettle in on if they're happy with this.

But I'm prepared to give this an official painting.

Thank you.

I appreciate that very much.

Our next problem comes from Loritz Bolterer.

Apologies, I've definitely mispronounced that.

Lorit says, hello, Becca Matt.

I have a problem that has been bothering me for quite some years now.

I noticed that when I repeat a turning pattern on the Rubik's cube, that after some time the cube returns to the original shape.

I tried it out with a few simple patterns and it always worked.

So they give some examples of the patterns that they do.

Then they put them in Rubik's Cube notation, but we'll worry about those if we get to them.

Okay, all right.

I always wondered if that works for every turning pattern and couldn't find anything on that.

Could you look into it?

I'd be very appreciative.

They had a further question.

He's sneaking in an extra question.

Always an extra question.

Just one more thing.

Is there a correlation between number of repetitions and style of the moves?

Love listening to your podcast.

I'm currently at episode 057 and sick with a lot of time to catch up.

Oh, no.

Well, obviously maybe they're home ill listening to the podcast playing with the Rubik's Cube.

Yeah.

We'll see if we can chew them up with a solution to their problem.

Yeah.

So we'll have a think about it.

I'm holding a Rubik's Cube.

Everyone imagine that.

Yep.

Unlike a collection of paints, there's the same amount of every color.

It looks a bit like my bookcase.

It looks a bit like your bookcase.

I'll give you that.

And so there are, you know, you've got the six different colors and you've got to get all nine stickers of the same color on the same face for each of the colors to solve it yeah and you can do that by twisting different bits of it around yeah and so the question you can just take the stickers off but you can't just

down upon it is it is frowned upon in the biz so the question is

if you keep doing the same pattern over and over will you get back to where you started and actually if you just imagine turning one face if you turn that four times it goes back to where it started yep they're asking something more complicated.

So I've just done four twists in a row and it's mixed it up a little bit.

I've taken the one on the right and turned it anti-clockwise.

I've turned the one underneath anti-clockwise, the bottom one.

And then the one on the right clockwise and the one underneath clockwise.

And it's still mixed up.

So I'm going to do that a bunch of times.

And after a little while, it'll come back to where we started.

There we go.

So I didn't do it and reverse it.

I just kept doing the same thing over and over and over again.

Yep.

So the question is, if you pick any combination of moves, any any twists, whatever you want to do, or even rotating the cube, if you did the same thing systematically over and over and over again, will you definitely come back to where you started?

And so the Rubik's cube only has a certain number of arrangements it can be in.

There's not infinitely many.

No, there's a lot, though.

There's a lot.

Yeah.

43 quintillion.

Yeah.

When you say it only has a certain number, that when you say only.

Only, only.

Compared to infinitely many, it's tiny.

Yeah, sure.

So, you know, there's a big difference between a finite number and, you know, infinitely many positions.

And it's 43 quintillion.

I've written it down here: 252 quadrillion, 3 trillion, 274 billion, 489 million, 856,000 flat.

And you can actually work that out by looking at each of the pieces.

So you mentioned stickers a second ago.

Yeah.

You're saying you peel the stickers off.

What you'd actually do practically is like each of these little sections of the cube is like a mini cube.

That's That's a solid bit of plastic.

So if you want to, you'd actually just pull like this one.

I've actually got a second rubik cube.

I can just put cube off.

Yeah.

I can pull the cube out.

Here we go.

Pull it apart, rearrange them all, put them back in again.

And that is.

That's cheating.

That is cheating.

But that's how you think about the Rubik's Cube.

It's not actually stickers.

It's not nine stickers of each color.

What you're actually doing is you're arranging the eight corners to be in the right spots.

with the correct rotation and the 12 edges to be in the right spots with the correct rotation.

And so what you can actually do is if you just multiply together the number of ways you can arrange the corners, the number of orientations they can have by the number of ways you can arrange the edges, and they can have two orientations each, then you have to factor in there's actually not every arrangement's possible.

No, because you can't have a piece that has green and blue because they're on opposite sides of the cube.

Correct.

But even more than that, even these pieces, you would never end up with like one corner the wrong way around.

So I've just rotated a single corner physically.

I like I pulled it out and turned it and put it back in again.

This is now an unsolvable cube.

This cannot be solved.

So if you want to upset someone,

take out a corner of the cube of their Rubik's Cube, turn it slightly,

so it looks like there's one color that's out and they will never be able to solve it.

You will never see a Rubik's cube with a single piece.

By piece, I mean a corner or an edge the wrong way around.

Because you can't change one by itself.

Something else always changes with it.

All of that factored in.

So, if you multiply together the number of ways you can arrange all the pieces, you then got to divide it a bit for the, there are some impossible ones.

So, you can't actually have every possible combination of the orientations of the corners, for example.

Yeah, so you wouldn't be able to work out how many different ways the cube could be just by taking the six different colors

and saying this is nine of each, nine times six of each, you know, colour.

No, and then being like,

what are all the different patterns they could be in if you would line them up?

Because a lot of those wouldn't be possible.

But all that aside, there's only a finite, there's only a set number of places.

And you could give them all a name.

You could name this arrangement.

I mean, 43 quintillion is a lot of names.

You can give them a number or something, but you could do that.

So now, if you think about it, any combination of moves you do is going from one of these possible states to another one.

And the same starting place and the same moves will always take you to the same final arrangement.

Because it's not like something's different is going to happen.

You do the same twists, the same Rubik's Cube, always get the same result.

And you keep doing that.

And what's going to happen?

If you keep doing it, you can't always get a new arrangement because sooner or later, you're going to run out of arrangements because there's only finitely many of them.

So eventually, you're going to hit an arrangement that you've seen before.

And then you're going to be stuck in a loop.

So if you start off with it solved, you'll eventually get back to it being solved again.

Well, so far, you'll eventually get back to one arrangement that you went through in the chain at some point.

If you're doing the same move each time.

Same move over and over and over again.

But you're absolutely right.

It turns out you do get back to the starting one because you can think about the same thing in reverse.

There's only one starting one and a certain move that will get you to the next one.

So actually you're guaranteed to come back to where you started.

Yeah, you'll loop around.

You always, always, always, always loop around.

So the answer is yes.

Because there's only a finite number of states.

And each time you do do an operation, you always get the same output.

And so I'm now saying it a bit more generally because there's a bunch of fun bits and maths where

you can deduce that you will eventually come back to where you started, or you can come up with other conclusions or deductions because there's only a finite number of things that can happen.

And if you're always going from one to the other, eventually you're going to run out of places to go and you're going to have to loop back on yourself.

So if it's unsolved,

but you get eventually get back to back to the unsolved state where you started.

Yes.

Yes.

So you can use that to your advantage if you want to pretend to solve the Rubik's Cube.

Because, you know, if you start with a solid.

That's how I play with Rubik's Cube.

I just keep moving it until my hands get bored.

For me, it's a fidget toy.

That's true.

Yeah.

Which it is.

I mean, for everyone, it's a fidget toy.

Yeah.

It's just different

results at the end of the fidgeting.

That's true.

And my first ever viral video on my channel, my stand-up mass channel, was me talking about the finite number of states of a Rubik's cube and I talked about and so I said if you do a move where you take the the from the point of view of you holding the cube the left hand side and move it towards you and the right hand side to move it away from you and then you turn the whole cube around once if you keep doing that eventually the cube looks quite mixed up yeah it looks like a mixed up Rubik's cube now you keep going halfway around there's a cool pattern yeah i think i've done this one a few times just playing with them And then you keep going.

And then, so I, my advice was to get there.

It takes 12 of the moves to get back to where you started.

And so I used to do nine of them because now it looks really mixed up.

And now you can be like, oh, Rubik's Cube.

And you're like, let me have a little look at that.

I reckon it's probably, there we go.

And solved.

So if you want to look like you can solve the Rubik's Cube, you just need to start with a sort of Rubik's Cube.

Yeah.

Start doing the same thing over and over.

And it'll look like you're mixing it up and then solving it.

Yep.

But maybe try and disguise the fact you're just doing the same thing.

The video I filmed over 10 years ago was me in my bedroom of me showing that you can get back to where you started with the Rubik's Cube.

So when this question came in from the person whose name you said before,

it was very nostalgic.

So it was an absolute pleasure to answer it.

I mean, they didn't bother Googling it or looking it up because I did a video over 10 years ago.

Over 10 years ago, that may as well not exist on the internet.

And I can't answer their follow-on question because they said, is there a correlation between the moves you're doing and how long it takes to get back?

And one of their examples they gave, they used the YouTube, they use the Rubik's Cube notation R U.

And so you've got the six faces, you've got right and left, you got front and back, up and down.

And so if you just do the letter, it means clockwise and the letter with little apostrophe, a little dash, little prime, that means counterclockwise.

Okay.

And so they're saying you turn the right one clockwise, which is that, and then you turn the top one clockwise.

Yeah.

Yeah.

For their style, they were saying like right, right, and then up.

Right up and then right counterclockwise.

Yes, they were doing right up.

So you're sort of moving your right hand from turning from the right bit

one rotation and then the top

closer to me on the right.

Yeah.

Yeah.

If I do this 105 times, which I will do, I'll eventually get back to where I started.

And the question is: why is it 105 times if I don't mess up?

But if people want to try it themselves afterwards, you'll notice after a while that the edges are taking 15 moves to cycle through.

And can you see these edge pieces, the middle ones, these are all back where they started.

So every seven times I do the move, they scramble and fix themselves.

Yes.

But if I go on to 15 now, so that was seven, eight, nine, ten, eleven twelve thirteen fourteen fifteen now

the middle edge bits are all wrong but the corners are fine yes so every time every multiple of seven these edge middle bits will be fixed and every multiple of 15 the corner bits will be fixed and the corners are taking seven moves to cycle through.

And so you've got to keep going until both of them have finished a cycle simultaneously.

And the lowest common multiple of 7 and 15 is 105.

So that's the reason why this takes 105 moves.

It's because it's cycling the edges and the corners on periods of 15 and 7.

And so you can actually work out how long any combination of twists will move by looking at how long it takes the different bits to cycle around.

And if I've messed up, we'll never get to the final one.

Oh, oh, look at that.

It's practically...

no, we're off again.

So, occasionally, occasionally, look almost of the top, the edges are lined up then for a brief second.

105.

Yay!

So, some of them take a very long time to get back to where you started.

Some of them are much, much quicker, but it can be done.

Yeah, but you need to look at how things are changing and compare the cycle length of the corners compared to the edges.

You can get into the group theory of this is the area of mathematics that will tell you more about how long the different cycles take, but for the most part, just do it and then find out.

But it's not always going to be a case of just timesing those two numbers together, is it?

Not necessarily, no.

Because, like, seven and 15, you're like, yeah, it's seven times 15.

Yep.

Is it?

You know, it's going to be 105 times.

Yeah, 105, yeah, yeah.

105 repetitions because that's seven times 15.

Yep.

But are there cases where it wouldn't be just those two numbers?

Yeah, because what you're looking for is what's called the lowest common multiple.

Right, yeah.

Which you can get, you can get a common multiple by multiplying them together.

Yes.

But it might be smaller ones.

So something like six.

So six and nine.

Six times nine is 54.

That's that's a common multiple of six and nine.

It's not the lowest.

The lowest is 18, because that's three sixes or two nines.

Yeah.

And students who are in high school may have to do this already.

And what you need to do is you need to look at the prime factors of each number and then work out the smaller set of prime factors.

that includes the prime factors from both numbers.

Got it.

So if you had corners for, you know, every six times you turn, you get the same corners.

Yep.

And every

nine times you turn, you get the same edges.

Then, and by edges, we're saying the bit that isn't a corner or the very center bit, then you would only need to repeat it 18 times.

You're back where you started, yep.

Huh.

Right.

Okay.

Yeah.

So what you can do is start do something over and over and count how long it takes the corners to get back where they started, how long it takes the edges to get back, and then find the lowest common multiple, and that will give you the total time.

Well, Matt, I think I understood that.

Okay,

I think I explained it.

Yeah, I don't ever want to think about it again.

Excellent.

But I think Lawrence will be very happy with that.

I feel like they'd be very happy with that explanation.

So there you are.

That's

how you can use the finite nature of states to answer a question, and you can pretend you can solve the Ruby Scube.

now we're up to acrylic oil what

it's but the and a medium that starts with b

I was giving different types of paints I know you've taken over with your color chat I'm sorry this is an orbit themed so other business from previous podcasts to case people up on there's not not a lot

a bunch of people had comments about when we solved the problem of if you're riding a bicycle, how many cars would you need to stop at a crosswalk to get your bike across the road to cancel out the carbon benefits of cycling?

I thought we did a pretty good job.

We looked at how much carbon you produce breathing.

We look at how much you produce in the car and how many cars are idling, all that jazz.

Now, a lot of people, upset's not the word, but as a theme of people.

sending in comments for us.

They wanted to point out that cars are terrible and produce a lot of carbon dioxide.

Yes, and it's not the cyclists' fault that they're holding up the traffic.

Yeah.

And so I want to say, obviously, I'm very non-car-centric infrastructure.

Yeah.

And I'm a big fan of cycling.

Yeah, you are.

And I've got opinions about climate change.

But, but I thought we answered.

I think we just took a neutral approach to answer the question as asked.

Yeah, I mean, normally I can get pretty funny if I feel like it's not being ethically correct.

Yes.

Yeah.

And I felt that I still thought it came, we still came away saying cycling's better.

I think we just did it on a neutral to mildly cycle positive answer.

Yeah.

And people wanted a very psycho positive answer.

Yeah.

And no, actually, the one, there was a few people pointed out that it's not just a case of if you drive instead, you produce the carbon dioxide that your car produces.

Every additional car causes more congestion, which causes more and more carbon dioxide because you're slowing down everyone else.

And so actually, it's worse than just your carbon dioxide

because you're making everyone else a little bit slower and more carbon dioxide from that.

And so yes, there are a lot of complications out there in

cycling is wonderful and we should probably produce less carbon dioxide.

Yeah.

In an ideal world, the default is cycling

and someone choosing to drive a car is someone saying, I'm going to put this much CO2 in the air.

Yeah.

Rather than

not producing as much CO2 as you would normally.

And, you know, and even if we had carbon neutral car transport, which is well within our technological reach, there would still be arguments for walkable cities and the like.

So

I think there's, there's a lot of, you know, important arguments for how we structure our cities and living.

And I'm very aware there's a lot of very important use cases for cars.

I own a car.

I drive around in the car.

There are many journeys and chores.

that I wish to undertake that the car is the best solution to those problems.

But I think

riding a bike is excellent.

I don't have one because someone bought all the cars and now I can't drive.

And they won't rent a hunt back to you.

Just didn't learn to drive, guys.

Just didn't learn to drive.

But arguably, that makes me better than everyone.

That's true.

And that's why you bring it up all the time.

All the time.

When I'm like, can I have a lift, please?

Yeah, actually.

Yeah, bro.

Can you drive me home from Edinburgh?

Oh, no.

Actual conversation.

But hey, we came up with this podcast.

That's very true.

Very, very true.

And we have a request for our Patreon supporters.

We do a bonus podcast called I'm a Wizard for anyone who supports us on any particular level of Patreon.

But we're asking for people to send us their wizarding problems as voice notes.

Which makes sense if you listen to the podcast.

Turns out a lot of our Patreon supporters are wizards.

Yeah.

And they offer us gifts that come with non-trivial problems.

You mean like...

like

wizard gifts.

Classic wizard gifts, you know.

Yeah, but whiz favors just sounds weird.

No, I'm going to go whiz favors.

Give us some whiz favors.

Don't give us whiz favors.

Sounds illegal, doesn't it?

It does.

Buddy, if you've got any whiz favours.

It at least sounds frowned upon.

All the details are on patreon.com slash a problem square.

Speaking of Patreon, we

like to take a moment to thank three of our supporters at random.

Picked completely at random.

Yes.

Not crypto thingy, Watsits, but.

Not cryptographically secure.

No.

The pseudo-random number generator baked into Excel.

Yeah, I know.

Lazy Man.

And this episode, we'd like to thank John B.

Van.

Or John Bevan, potentially.

Most likely.

Yamaz Makey.

Wow.

James Mackey.

Wow.

I don't know who exactly will be offended by your pronunciation, but someone out there will be.

Probably James Mackey.

Oh, I bet.

Wow.

Okay.

Next one's Ellie.

Ellie.

I'm just going to split it between the L's.

Ellie.

The well-known bandit.

Yeah.

And a reminder that if you would like one of our Patreon Christmas cards, either an e-card or one of the hard copy ones our special wizard-level Patreon supporters receive, then you will need to sign up before the end of November.

because we're going to be sending them out and we'll need addresses and stuff.

And details and details.

In name.

In name, that's also handy.

We write the name on there.

Yeah.

So we do half the names each.

So, lucky dip, who writes your name?

Oh, but we both signed them.

We both signed them.

Yeah.

Yeah, we're not taking requests for who writes the names.

We could do every second letter.

I don't think you should even mention that.

Oh, too late.

You know that thing you hadn't considered?

Yeah.

Can't do it.

You can't have it.

You know, that thing you didn't know you wanted?

Sucks to be you.

So, yeah, do that.

I mean, you are welcome to sign up to Patreon just for that.

And then quit.

And quit.

That's fine.

We'd rather you didn't, but if that's what you've got to do.

I'd rather you do that for one month and quit than don't do it at all.

Yeah, you're more likely to forget.

No, we're not banking on that.

I am.

The business model of a gym.

I think you'll find that I don't have enough money to buy a flat.

So

really appreciate it, guys.

Not going to get my deposit back.

You've been listening to A Problem Squared.

with Beck Hill and myself, Matt Parker.

And of course, we need to remember Kepler's law of planetary motion.

There is Lauren Armstrong Carter, who keeps everything moving just the way it should.

Ah, you're like the sun.

Yeah.

So bright and shiny.

You trace out an equal amount of area per unit of time.

Couldn't put it better myself.

Okay, back.

Oh, we got the jar.

We've got it.

I know.

We're in the same room as the jar for once.

And I see it, and it's much more full than I did from memory.

What did you guess last time?

13.

Oh,

higher.

Definitely higher.

Okay, right.

New guess.

Do you want me to give it a shake?

Will that help?

No.

Why would you do that?

There you go.

Made up so much.

There's a finite number of dice in there.

Oh, you're weighing it.

It's really heavy.

Yeah, I'm watching it.

Oh, I should just weigh it and then...

That's ultra-light glass.

I'll do it like they do at the fairgrounds where I'm holding it like, yes, but I'm actually sitting on a scale.

Yeah, yeah, yeah.

Take it off.

You've actually got an iron bar down your arm to support the weight.

Yeah.

All right.

So it's roughly.

Well, Becca's really investigating this.

Yeah.

It's roughly seven, no, six die high.

So for the moment, a lot of them are an actual.

Becoming some working out.

We did share a photo of this, so you can look that up on our socials.

I know there's maths.

It looks like a jar full of circles.

To work out the surface of a circle.

Nah, that's guesswork.

Yeah, you're not going to let me do that, though.

That's why it's called the circumfere guess.

Also, I don't want to have to look it up.

I know.

I'm literally going to guess.

You can have that back.

I'm taking the dice back.

And I'm going to.

Let me give you a dice roll.

And your guess is

600.

Lower.

Cool.

Now I've got an upper and lower limit.

There you go.

Right.

The rest is trivial.

Max threshold.