It’s the holiday season, a time of year to bring people together and to do something a little bit different so Mathologer, here I’m Matt Parker from stand up maths. Hey, this Is sam from Wendover Productions and half is interesting hi everyone this James Grime from the singingbanana channel Which Brady reporting for service from numberphile objectivity and various other channels. Hey everyone, my name is Steven Walsh My channel is Welch labs. I’m from the channel Looking Glass Universe. Grant told me he, was sending, me a Puzzle and a mug. Hey Grant, I am here I’ve got a mug, and some paper and some markers and I’m ready to do your puzzle I really should know how to solve this mug, because i’m the guy that makes and sells them with Matt Parker so I’ve been instructed not to read the directions before Starting I’ve been. Hey Grant so a friend just gave me this mug you are gonna be challenged And I’m just gonna kind of make you do this on camera to embarrass you We’ve got three different houses here three different cottages and then three different utilities the gas the power and the water draw A line from each of the three utilities to each of the three houses so nine lines in total okay without letting Any two cross, no two lines crops, is right here if you, wanted to just go straight from power to the house right Okay, interesting that is quite a challenge so nine lines that don’t cross that doesn’t even sound possible. I’ve got, my mug I’ve got my utilities mug here I’ve even got real coffee in the mug i mean that look at that that’s attention To, detail i’m willing to give this a, go i’m just Worried I’m gonna muck it up I tend to make bit of a poker square of these things when I when I truck Say, well let’s just fill in as many as i can and see what happens i’m sure this will end terribly. So there’s one? There’s the other There we go. Gas line it’s, gonna be easy we’re gonna go like this, wow sound effects are crucial, I’m not gonna go around the green one don’t want to fall for that I can do another one and now up to five four, go I’m looking at my display over here I should have put it over there but, oh well. Oh that’s good of it That’s your go to the second is okay? There’s no ibly this is easy enough And so we just need to get from here to there. I have one two three four five six seven lines two to go. So I have that one connected to that one I Mean that one connected to that one. Oh now, we get into trouble, okay, now I start to see the problem. And there I have made my fatal error in not paying attention I have boxed in this house right here as you can see there’s no way to get to it. Gas needs to get to number 1 and 2. And that’s the problem because we’re cut Off i kind of want to try it on paper, okay it’s getting really Awkward to draw on a mug i think what i’m gonna, do is i’m gonna go to a Piece of paper this this kind of property that you can, make lines go from here to here and also all the way around Makes it seem like i should, be drawing a spear Something like that Okay, let me i need, bigger, lines bigger bigger space But now i’ve just blocked off how is this possible this isn’t getting anywhere let’s try, again Water i need, to the first and second What i really messed it up okay, to make that at least look, easier i’m gonna go around here Around around around around around to to go around the mug with, the gas here, so i’m just gonna go all the way Around i’m gonna go around Let’s go underneath the handle here So now it’s closed We just need to figure out how, to get that red in there house number three is all done and good look at that house number three good to go so this house has all three and That house has all three but this one in the middle doesn’t have gas Alright let, me try something, new Let me just try an experiment here let’s let’s. Be let’s be empirical What’s really nice about the mug Is that it’s shiny so if you use a dry erase marker you can undo your mistakes you rub it off Posit, okay, so there’s some very pleasing math within, this puzzle for you, and me to dive into but first let Me just say a really big thanks to everyone here, who, was willing to be my, guinea pigs in this experiment Each of the runs a channel that i respect A lot and many of them have been incredibly kind and helpful to this channel So if there’s any there that you’re unfamiliar with or that you haven’t been keeping Track with, they’re all listed in the description so most certainly check them out, we’ll get back to all of them in just a minute Here’s the thing, about the puzzle if you try it on a piece of paper you’re gonna have a, bad time But if you’re a mathematician at heart when a puzzle seems hard. You don’t just throw. Up your hands and walk, away Instead you try to solve a meta puzzle of sorts see if you can, prove that the task in front of you is impossible In this case how on earth do you, do that how, do you prove something is impossible For background anytime that you have Some objects with a notion of connection between those objects it’s called a graph often represented abstractly with dots for your objects Which i’ll call vertices and lines for your connections, which i’ll call edges Now in most applications the way you draw A graph, doesn’t matter what matters is the connections but in some peculiar cases Like this one the thing that we care about is how it’s drawn and if you can draw a graph in the plane without crossing Its edges it’s called a planar graph So the question before us is whether or not our utilities puzzle graph Which in the lingo is fancifully called a complete bipartite graph k33 is planar or not And at this point there are two kinds of viewers those of you who know About euler’s formula and those, who don’t those, who? Do might see where this is going but rather than pulling out a formula from thin air and using it to solve the meta puzzle i Want to flip things around here and show. How Reasoning through, this conundrum step, by step can lead you to rediscovering a very charming and very general piece of math To start as you’re drawing Lines here between homes and utilities one really important thing to keep note of is whenever you enclose a new region that is some area that the paint bucket tool, would fill in Because you see once you’ve enclosed a region, like that, no new, line that you draw Will be able to enter or exit it so you have to be careful with these In the last video remember how. I mentioned that a useful problem-solving tactic is to shift Your focus onto, any new constructs that you introduce trying to reframe your problem around them Well in this case, what can, we say about these regions right now i have up on the screen and in complete puzzle Where the water is not yet connected to the first house and it has four separate regions But can, you say anything about how. Many regions A hypothetically complete puzzle would have what about the number of edges that each region touches, what can you say there There’s lots of questions you might, ask And lots of things you might notice and if you’re lucky here’s one thing that might pop out for a new, line that you draw to create a region it has to hit a vertex that already has an edge coming out of it Here think of it like this start by imagining one of your nodes as lit up, while the other five are dim and then every time you draw an edge from a lit up vertex to a dim vertex light up the, new, one So at first each new, edge lights up one more vertex But if you connect to an already lit up vertex notice how This closes off a new region and this gives us a super useful fact, each new, edge either increases the number of lit up nodes by one or it increases the number of enclosed regions, by one This fact, is something that, we can, use to figure out the number of regions that a? Hypothetical solution to this would cut, the plane into can, you see how When you start off there’s one node lit up and one beaten all of duty’ space By the end we’re going to need, to draw. Nine lines since each of the three utilities gets connected to each of the three houses Five of those lines are going to light up the initially dim vertices So the other four lines, each must introduce a new region So a hypothetical solution would cut. The plane into, five separate regions and you might say, okay, that’s a Cute fact but, why should that make things impossible what’s wrong with having five regions Well again take a look at this partially complete graph notice that each region, is bounded by four edges And in fact for this graph you could never have a cycle with, fewer than four edges Say you start at a house then the next line has to be to some utility and then a line out of that is going to go to another house and You, can’t cycle back to where you started immediately because you have to go to another utility before you can Get back to that first house So all cycles have at least four edges and this right here gives us enough to prove the impossibility of our puzzle Having, five regions, each with a boundary of at least four edges would require more edges than, we have available Here let me draw. A planar graph that’s completely different from our utilities puzzle but useful for illustrating what, five regions with Four edges each, would imply if you went through each of these regions, and add up the number of edges that it has Well you end up with five times four or twenty and of course this Way over counts the total number of edges in the graph since each edge is touching multiple regions But in fact each edge is touching exactly two regions so this number twenty is precisely double counting the edges So, any graph that cuts, the plane into, five regions, where each region is touching four edges would have to have ten total edges But our utilities puzzle has only nine edges available So even though, we concluded that it would have to cut, the plane into, five regions it would be impossible for her to do that So there you go bada-boom bada-bing it is impossible to solve this puzzle on a piece of paper without intersecting lines tell me that’s not a slick proof, and Before getting back to our friends and the mug it’s worth taking a moment to pull out A general truth sitting inside of this think back to the key rule, where each, new Edge was introducing either a new vertex by being drawn to an untouched spot or it introduced a new enclosed region That same logic applies to any planar graph, not just our specific utilities puzzle situation In other words the number of vertices minus the number of edges plus the number of regions remains unchanged No, matter what graph you draw, namely it started at two so it always stays at 2 in this relation True for any planar graph is called euler’s characteristic formula Historically, by the way the formula came up in the context of convex polyhedra, like a cube for example Where the number of vertices minus the number of edges plus the number of faces always equals two So when you see it written down. You often see it with an f for faces instead of talking about regions Now before you go thinking of me as some kind of grinch that sends friends an impossible puzzle and then makes them film themselves trying to, solve it keep in mind i didn’t, give, this puzzle to people on a piece of paper And i’m betting the handle has something to do with this. Ok, otherwise, why, would you have brought a, bug over here This is a valid observation Maybe use the mug handle, oh? Yeah, i think i see okay i feel like it has to do something with the handle And that’s our ability to hop one line over the other i’m gonna start by i think Taking advantage of the handle because i think that that is the key to this you know what i think actually a sphere is the wrong thing to be thinking about i Mean like famously a mug is topologically the same as a Doughnut so to solve this thing you’re Gonna have to use the “torus-ness” of the mug you can have to use the handle somehow That’s the thing that makes this a torus mm-hmm let’s take the green and go Over the handle here okay? And then the red can kind of come under nice My approach is to get as far as you can with As far as you can as if you are on a plane and then See, where you get stuck so look i’m gonna draw this too, here like that and Now i’ve come across a problem because electricity Can’t be joined to this house this is where you have to use the handle so whatever you Did do it again but go around the handle, so i’m gonna go down here I’m gonna loop Underneath come back around, and back to where i started And now i’m free to get my electricity messy there you, go and then i’m gonna go on the inside of the handle go all the way around the inside of the handle and finally connect To, the gas company to solve this puzzle just drawing the m. And there’s three more connections to go so let’s just make them one Two and i will have to connect those, two guys right just watch it In through the front door out. Through the back, door done No, intersections Maybe you think that it’s cheating, well sort of topological puzzles so it means the relative positions of things, don’t matter what that Means is we can, take this handle and move it here Creating another connection, oh? Oh, my, god am i done is this over i think i might’ve gotten 24 minutes granny says to take 15 minutes There you go i think i’ve solved it you haven’t success but but, not impossible hard but not impossible this Isn’t it maybe perhaps not the most elegant solution to this problem and if i drew this line here you’ll think, oh? No, he’s blocked that house there’s No, way to get the gas in but this is why it’s not a mug right because if you take The, gas line all the way up here to the top. You then take it over and into the mug if you draw The line under the coffee it wets the pen so when the line comes back out, again, the pens not working anymore you can Go, straight across there in and join it up and because it wasn’t drawing you haven’t. Had across the lines Baby, by the way funny story so i was originally given, this mug as a gift and i didn’t really know Where it came from and it was only after i had invited people to be a part of this that i realized the origin of? The mug maths kheer is a website run By, three of the youtubers i had just invited matt james and steve small world given just how. Helpful these Three guys, were and the logistics of a lot of this really the least i could, do to thank them, is give a Small plug for how, gift cards from matt’s gear could, make a pretty good last-minute christmas present Back to the puzzle though this is one of those things where once you see it it kind of feels obvious the handle of the Mug can, basically be used as a bridge to prevent two lines from crossing, but this raises a really interesting mathematical question We just proved that this task is impossible for graphs on a plane so where exactly does that proof break down on the surface of a mug and I’m actually not going to tell you the answer here i want you to think about this on your own and i don’t just mean saying Oh it’s because euler’s formula is different on surfaces with the whole really think about this Where specifically does the line of reasoning that i laid out break down When you’re working on a mug i promise you thinking this through will give you a deeper understanding of math Like, anyone tackling a tricky problem you will likely run into walls and moments of frustration But the smartest people i know actively seek out new, challenges even if they’re just toy puzzles They, ask, new questions they aren’t afraid to start over many times and they embrace every moment of failure So, give this and other puzzles and earnest try and never stop, asking questions But grant i hear you complaining how, am i supposed to practice my problem-solving if i don’t have Someone shipping me puzzles on topologically interesting shapes, well let’s close things off by, going, through a, couple puzzles created By, this week’s mathematically oriented sponsor brilliant dork So here i’m in there intro to problem solving course and going Through, a particular sequence called flipping pairs and the rules here seem to be that we can, flip, adjacent Pairs of coins, but, we can’t flip, them one at A time, and we are asked is it possible to get it so that all three coins are gold side up Well clearly i just did it so yes And the next question, we start with different configuration, have the same rules and rask the same question can we get it so that all three of the coins are gold side up and You know there’s not really that many degrees of freedom, we have here just two different spots to click so you Might quickly come to the conclusion that no you can’t even if you, don’t necessarily know the theoretical reason Yet that’s totally fine so, no and we kind of move along? So next it’s kind of showing us every possible starting configuration that there is and asking for how Many of them can, we get it to a point, where all three gold coins are up Obviously i’m kind of giving Away the answer it’s sitting here four on the right because i’ve gone through this before but if you Want to go through it yourself this particular quiz has a really nice resolution and a lot of others in this course do build up Genuinely good problem-solving instincts so you can, go to brilliant org/3b1b to them know that you came From here or even slash 3 b 1 b flipping to jump straight into this quiz and you can Make an account for free a lot of what they offer is free but They, also have a annual subscription service if you want to get the full suite of Experiences that they offer and i just think they’re really good i know a couple of the people there and they’re Incredibly thoughtful, about how. They put together math explanations water goes to one and then wraps around to the other and Naively at this point, oh, wait i’ve already messed up Then from there water can, make its way to cut it number three. Ah i’m trapped i’ve done this wrong, again