A Problem Squared - 072 = The Perfect Paint Set and Rubik's Reset

Episode Date: November 6, 2023

In this episode...  🎨 What's the best hypothetical paint set?  🧊 Can a single turning pattern return a Rubik's cube to where it started?  🛰 And some Any Orbital Business!  If you want to weigh in on the Reasonable / Unreasonable debate, and watch Bec in her element painting a wardrobe, follow this link: https://www.instagram.com/reel/CyJwjphMesl/?utm_source=ig_web_copy_link&igshid=MzRlODBiNWFlZA==  And if you want to see more from the colour theorist, Peter Donahue, who consulted on Bec's problem, find him on instagram: https://www.instagram.com/art.pete.repeat/?hl=en  For Martin Bellander's incredible colour graph, follow this link: https://blog.martinbellander.com/post/115411125748/the-colors-of-paintings-blue-is-the-new-orange  Please do send us your problems and solutions to the website: aproblemsquared.com.  If you want more from A Problem Squared, find us on Twitter,  Instagram. and of course Patreon.

Welcome to our Problem Square. 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, just 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. Oh, 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 And I was going to do a Seasonal intro And I got distracted by the Earth's tilt

And then I was like Why not just do the orbit Yeah, why not It's almost circular orbit, the Earth's orbit But just a little off, a little eccentric And that's what makes it And there's a Leap podcast We do an occasional Leap podcast You 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 any orbital 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. Oh, it's so nice. I'm so, the cupboard man. Oh boy. 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 i'm i'm good i now two episodes ago i think it was i talked about painting my bookcase actually i may have been in the i'm a wizard and that's the quality of content that we put in the patreon yeah it's not only 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 and i yeah i asked the landlord if they minded me painting it and now in fairness they did come back and say as long as you painted a reasonable color it's a building wardrobe it is made of the cheapest mdf and it has black mold constantly that's not

good coming out of it yeah it's like it is've been in the, 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. Yeah. We don't complain too much. They don't ask too much of us. Yeah.

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 in the wardrobe. So when they said, as long as it's a reasonable color, 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 gut the whole place anyway. Because it does need. Because it's a desperate need of refurb. 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 outline so it has a bit of a mondrian yeah you know the 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, gave it a nice lick of white neutral paint. And then I've painted the drawers in the block colors 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 we've discussed, I'm going to start putting out more YouTube videos next year. Oh, yeah, yeah, yeah. It's the year year of youtube and so i wanted to have an interesting background oh the wardrobes 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 for the landlord saying, as long as it's a reasonable color. And then I knew what I was doing. 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. 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. 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 lapse of me painting. It's had 1.5 million views. There you go.

And it's got like 40,000 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 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 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've paid on time, looked after the place, not complained too much. The wider context is important. reasonable tenants for the last decade who've 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 just destroying the house. Yeah, they're not reading things. They're not reading on the video. Literally says that to start with. There you go. Been here nearly a decade. So I can see 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 really i would kick you out if you're my turn you're the reason that we don't trust tenants 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 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 actually in fairness, 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. This is the thing. 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 very funny yep i thought like i knew what i was doing with that sense you weren't but people were questioning the notion of land ownership lots of people telling me why didn't i just buy a house simple answer 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 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 no thanks i've late i've just been writing my book. That's my answer every time. Getting real close. Although when this 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 additions, but it's a whole thing. So really, once the book is done, it's done. 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 Timon 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, Let me get the exact wording.

Mathematically inclined, they said, could you cover the cake in hexagons? Which is mathematically very straightforward. Yeah, because you've got essentially two surfaces for each cake, sorry. You've got the bit around the side for the cylindrical bit and then the top. And then you just do the same for the other layers. Yeah. 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 made octagons. And so would you like to see the hexagonal covered cake where instead of hexagons, it's octagons? Oh, no. They've just given 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 bits of confetti on it or something. But you can just imagine the cake decorator's frustration when 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. Yeah, just stack them, hoping for the best. And I've got a collection of 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 stamp mesh. Like the mesh built into it.

Yeah. So you can just roll it across your fondant. Roll out some hexagons. There you go. And then turn the whole sheet of fondant into hexagons. And label it octagon roller. Done. There's something for massgear.com.

Oh, my goodness. And so at the moment, it's not in the book, but I've got another, oh, how long until this episode comes out? A couple of 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 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 a 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 supplies problem, which is 100% a Beck problem. And they have a very friendly opening. They say hi with three I's.

Hi. Hi. Which cancels down to negative I, people playing along at home. And Nettle is buying sets of colored pens pencils paints etc and they've noticed it's much cheaper per color than buying them individually oh so if they buy a set of them yeah and then divide it between the colors it's cheaper per color to buy a bunch at once than to buy individual pencil 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 the popular color and they just go and buy a whole new set. Yeah. They've now got too much of the unpopular color. 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 this 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 colours 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 other colours or paint. So you're not just using blue as a colour, 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 wow so it's easier to not see yep when you're reproducing it so i will say this as someone who doesn't do that i know a lot that. I know a lot of artists do. I know a lot of artists use blue to do the initial sketchings in Photoshop. You know, when I've seen web comic artists put up work in progress, it is often blue outline. I mean, that'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 you can see what 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 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 new listeners if you're not sure what that means just google back hill wardrobe look up 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 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 colours more than you would if you were doing. The greens, the blues, gone. 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 Donoghue, 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 and 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 fallow 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. Tinting is just that's how opaque the paint is or how much you have to use to build up the same level of color. 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 with a type of red.

To get. Like a mauve color. Yep. 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. To get the same. Right.

So you're going to use up different amounts of different colors. Yeah. Based on their impact. Yeah. So as we know from printing. Yep. That works with CMYK. Oh, yeah.

Four colors. Yeah, which is cyan, yellow, K is black. Yeah. 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 gotta remember that cnyk when it comes to printing for white it just uses less so you don't have a white paint no a white ink you white paper it just uses less of the ink there to create a lighter color yeah 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. I've worked it out. Yeah. Five colors. Cracked it. But then I was like, hang on, you've still got the same problem though

because with CMYK you're going to be using certain of those colours way more than others. So I reached out to Richard Moross, 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, then I guess we need CMKKKK 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.

Right. 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. A lot of paintings. From over the centuries. They downloaded all the thumbnails. Possibly with date information.

Yeah. Good work. So it was almost 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 on i feel like there

were some steps in between there's some steps in between there are i'm going to post the out because this is very much your town 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 colors in the painting. 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. and so they've kind of done 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 but there's a bit more blue but then as we get close to the 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 yeah in recent years it's quite a hard color there's actually a whole thing about blue m&ms because finding the fake blue color was kind of a new deal. Yeah. So blue's become much more easy to get hold of and affordable because it's easier 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 misread 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 has been used in 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 was like, there's still a lot of variables. So. So. I tried one more thing. I reached out to C White of 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. 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 hang it out there see about half an hour later got an email back with an excel spreadsheet and this is where i'm going to need your help, Matt. That's because you're one of the most valued customers. So they sent you an Excel spreadsheet.

Yes, they did. It's got 50 rows. We're down on 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. Yeah. Should we have a little break? Yeah, let's have a quick break. And I reckon I can do something really quickly.

Okay, I've got something. So I pulled out all the volumes from the descriptions of each product. Yeah. 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 color. Oh, handy. sold to get the volume of paint sold okay yep which i've converted into liters so everything

is now liters of that color 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 is separate to lemon yellow they're all different yellows and then 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. Wow. 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, if you're painting, you've got to mix in the white.

You've got to mix it in, yeah. Followed by, not closely, 12.1% of all paints sold by volume was black. Again, figures. 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 of color. But I'm going to say red. Close. Cadmium red is number four on the list. Ooh. At 6.8% of sales. Wow.

It is beaten. Primary yellow. Figures. Primary yellow. I was between red and yellow. 1,537 litres of primary yellow, 1,344 litres of cadmium red. So the ranking goes white then black, then primary yellow, cadmium red, ultramarine. Oh.

Third most popular colour, followed by by cyan magenta so these are the classic building blocks burnt umber brown the low c i don't know what color that is that could be it 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, Sea White of Brightonizable 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 two and a half percent two and a half percent two and a half percent two and and a half percent, two and a half percent, two and a half. They're selling the same amount of all of these. Even once you get down to the more obscure ones, they're 1.9%, 1.6%. 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 of 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 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. We tumble pack them into a box. So rather than going, well, you've got 12. Because at the moment. What are you thinking? Are you getting like 12 whites? Yeah. That's what I was thinking. Because at the moment, CY to 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. Oh, 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. So we'll do a 15 pack. Yep. I mean, using the spreadsheet, you could give me a number of tubes and it could generate the color breakdown of those. Ooh. This is not perfect, but it'll do for a first pass. So how many tubes would you like in your kit?

I guess let's go 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 are ones. One yellow, one cadmium red, one ultramarine. And you work really 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 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 starts PTH, the yellow orca, naphth, 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.

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 when i so i've 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're going to do extra work on this. Uh-oh. Oh, you brought a painting.

How did you smuggle a painting in? Is this for me? Yeah, this is for you. Oh, my goodness, Bec. I've done a painting for you. I'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 a New York Brooklyn apartment without enough ventilation. And I was sitting for a portrait of this quality.

Thank you, Bec. To be fair, I didn't know either until this problem came in. 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've once again included, without permission, my bare feet. No, I may try to get, there's one foot and it's the top of the foot. Oh, okay. That's all right then. I made sure 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. And you 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, that was most of the colors.

Your dog has started eating the brown paper. The dog is now eating the brown. Skylab is now eating the brown paper. It's a present for everyone. Well, Beck, 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. Loritz 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 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. Always 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 cheer them up with a solution to their problem. Yeah. So we'll have a think about it. I'm holding a Rubik's Cube. Yeah. We'll see if we can cheer them up with a solution to their problem. Yeah. So we'll have to 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 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.

And you can do that by twisting different bits of it around. Yeah. You can just take the stickers off, but let's look down upon. 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 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 just 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 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. Yeah. So the question is, if you pick any combination of moves, any twist sort of you want to do, or even rotating the cube, if you did the same thing systematically over and over and over again, well, 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 that only has a certain number, 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 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 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's Cube. I can pull the cube out. Here we go. Pull it apart. Rearrange them all. Put them back in again. And that is 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, they can have two orientations each. Then you have to factor in there's actually not every arrangement

that'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 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 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 colours and saying this is nine times six of each colour. Yeah, no. And then being like, what are all the different patterns

they could be in if you were to 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 different is going to happen.

You do the same twist, 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 finally 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 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 in 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 yep yep but nope but you get eventually get back to back to the unsolved 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, if you start with a solid one. 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 Sound 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, from the point of view of you holding the cube, the left-hand side and move it towards you, and the right-hand side and 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 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 solved 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. Yeah. 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 used the Rubik's Cube notation R U. And so you've got the six faces. You've got right and left.

You've got front and back, up and down. And so if you just do the letter, it means clockwise. And the letter with a little apostrophe, a little dash prime that means counterclockwise okay and so they're saying you turn the right one clockwise just 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. Right side away from me. One rotation and then the top one rotation.

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? And if people want to try it themselves afterwards 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, 10, 11, 12, 13, 14, 15. Now the middle edge bits are all wrong.

Yeah. The corners are fine. Yes. So 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 occasionally occasionally it'll look almost of uh 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 as 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 7 and 15 you're like yeah it's 7 times 15 yep is it you know it's going to be 105 times 105 repetitions because that's 7 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 there might be smaller ones. So something like six. So six and nine, six times nine is 54. 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 smallest set of prime factors that includes the prime factors from both numbers. Got it.

So if you had corners for every six times you turn, you get the same corners. Yep. And every nine 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. Yep. 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 doing 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 larts will be very happy i think they'd be very happy that explanation so there you are that's that's uh how you can use the finite nature of states to answer a question and you can pretend you can sell the ruby scoop Now we're up to acrylic oil. What?

It's. And a medium that starts with B. I was giving different types of paints. I know you've taken over with your colour chat. I'm sorry. This is an orbit themed. So other business from previous podcasts to catch people up on. There's not a lot a bunch of people had comments about when we solve 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 look at how much carbon you produce breathing we look 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 cyclist's fault that they're holding up the traffic. Yeah, true. 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 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 yeah and i i felt that i still thought it came we still can't wait saying cycling's better i think we just did it on a neutral to mildly cycle positive answer yeah and people wanted a very cycle 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 more carbon dioxide yeah because you're slowing down everyone else and so actually it's worse than just your carbon dioxide yeah

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. And 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 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 a lot of 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 they won't rent one back to you. I just didn't learn to drive, guys. I just didn 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 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 are asking for people to send us their wizarding problems as voice notes which makes sense if you listen to the podcast. It turns out a lot of our Patreon supporters are wizards. Yeah. And they offer us gifts that come with non-trivial drawbacks. By gifts you mean like spells? 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. Please don't. Don't give us whiz favors. Sounds illegal, doesn't it? It does. Buddy, have you to go whiz favors. Give us some whiz favors. Please don't. Don't give us whiz favors. Sounds illegal, doesn't it? It does. Buddy, have you got any whiz favors?

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 whatsits. Not. Completely at random. Yes. Not crypto thingy Whatsits. Not cryptographically secure. The pseudo random number generator baked into Excel.

Lazy man. And this episode we'd like to thank John B. Vann or Don Bevan, potentially. Most likely. Yummers Makey. Yarmus Mackey. 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, wow. Okay. Next one is Ellie. Elle Lye.

I'm just going to split it between the L's. Elle-lie. 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, yeah. And name. And name. That's also handy. We write the name on there. Yeah.

But we do half the names each. So lucky dip, who writes your name? Oh, but we both sign them. We both sign them. Yeah. Yeah, but 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, but we both signed them. We both signed them. Yeah. Yeah, but 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. You can't do it. You can't have it. You know the 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. Oh, that's very true. Really appreciate it, guys. Not going to get my deposit back. You've been listening to A Problem Squared with Bec Hill and myself, Matt Parker. And of course, we need to remember

Kepler's law of planetary motion. That is Lauren Armstrong Carter, who keeps everything moving just the way it should. Oh, 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've 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? 12.

Oh, when I said higher. Definitely off. Definitely higher. Definitely off. Okay, right. New guess. Do you want me to give it a shake with that help? No. Why would you do that?

There you go. Made it so much worse. There's a finite number of dice in there. Oh, you're weighing it. It's really heavy. Yeah. Oh, I should just weigh it and then. That's ultra light glass. I in there. Oh, you're weighing it. It's really heavy. Yeah. 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. I'm going to take it off. You've actually got an iron bar down your arm to support the weight. Yeah. All right. So it's roughly.

Oh, Beck is really investigating this. Yeah. It's roughly seven, no, six die high. So for the listeners, Beck is doing 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 dice. To work out the surface of a circle.

Nah, it's guesswork. Yeah, you're not going to let me do that, though, are you? That's why it's called the circumf guess. Oh, sorry. I don't want to have to look it up. I know. I'm literally going to guess. If you, though. That's why it's called the circumference guess. Also, I don't want to have to look it up. I know. I'm literally going to guess. If you can have that back.

I'm taking the dice back. And 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.

The rest is trivial. Max threshold.