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• So some months back Grant Sanderson of the channel 3blue1brown invited me to take part

• in a collaboration where various science and math youtubers attempt a twist on a classic

• puzzle where the original goes like this: There's three houses and three utility stations:

• gas, water and power.

• Connect every house to every utility, simple enough, except oh no if you cross two utilities

• they explode so do it with no crossings.

• People usually manage pretty well up to line 7 or 8 but number 9 seems impossible.

• The timing didn't work out for me to take part in the original collab, where the same

• puzzle is printed on a mug, but I finally saw the video, and I was surprised to find

• that my answer was different from anyone else's.

• You see, I wanted to leave room for future infrastructure, so, sure we need to untangle

• everyone's gas water and power but we also want internet that connect every house to

• every other house, and a separate secure network that connects every utility station to every

• other utility station.

• So if you don't know the original puzzle try that first, and if you do know it then

• you might want to check out the mug variation first, and if you've already figured that

• one out then you might want to pause and see if you can figure out the Vi Hart Infrastructure

• Special and make everything connect to everything, that's 15 lines, three each for gas, water,

• power, social network, and secure network, without any of them crossing.

• But on a mug.

• And one more thing before we start.

• The official version of this mug is sold on mathsgear and comes with a dry erase marker

• so you can make mistakes, but I'm a sharpie person and a lover of beautiful solutions

• so we need something that makes enough sense that I can draw in all 15 lines using permanent

• marker without worrying about making a mistake.

• Ok so here's the thought process.

• First I look at the puzzle and see what associations come up.

• If you're a science or math person you might think, oh, this has to do with graph theory,

• there's lines connecting vertices and that's like the definition of a graph.

• For the mug variation, you might think, oh, maybe it has to do with topology, because

• there's that joke about how a topologist thinks a mug is the same as a donut because

• topologically you can transform one into the other, as seen in this piece by Henry Segerman

• and Keenan Crane.

• So the main associations come up, and depending on how much you know about them these secondary

• associations might come up, or maybe you just start trying things out, or maybe your biggest

• association ishey, that's the classic impossible utilities puzzle,” which might

• stop you from looking further than that.

• The classic puzzle is supposed to frustrate you with your inability to get that last line

• in there until you either mathematically prove it's impossible or come up with some other

• creative answer like that the last line goes through a house, or over a bridge, or through

• a hole in the notebook paper and around the back, which is kind of what the torus will

• allow us to do.

• I mean, mug handle.

• So I know I can solve the original three utilities problem if I can solve the four utilities

• problem.

• And I'd know it's possible to connect all six into what we call a “complete

• graph if it's possible to connect seven into a complete graph on the torus.

• And maybe this bridge of reasoning can land somewhere near the four-color map theorem

• which is connected to the seven color map theorem for toruses,

• And I just happen to have a seven color torus right here to demonstrate.

• Every color section of this torus touches all six other colors, this is by Susan Goldstine

• and I'll link a tutorial in case you're into mathematical beadwork.

• Ok so now we just need to connect the bridge.

• Each color area here touches every other area, see, it's likewell we'd better bagel

• it,

• So take the orange area, it touches yellow on one side, red on the other, then next to

• yellow is green which touches the orange part over here, and then purple, well let's do

• pink, yeah so all the colors have two neighbors on their sides and they touch two other colors

• on one end and two other others on the other end so they all touch all six other colors.

• Maybe next time we'll use colored frosting and do a bundt cake.

• Anyway then I just take the dual, which means I make every area a vertex and then I can

• connect them all to each other, so it's really handy that I know a lot about dual

• graphs.

• And there we go, I can connect 7 dots to each other without crossing, which means I could

• ignore one vertex to get 6, and then delete the extra lines until we've solved the original

• mug puzzle.

• So this went through my head within seconds of seeing the mug because all this math stuff

• is close to the surface of my brain, I have experience using these facts and making them

• accessible, but it's not yet a solution, it's a sketch of a proof and a method of

• finding a solution, but we don't want just any solution but a pretty one that's simple

• to draw.

• And that means I want to take advantage of symmetry.

• So when I wanna put six points with symmetry on a torus I've got some options, we want

• something reminiscent of the original puzzle even though on the mug they're all squashed

• together.

• But I'll keep the idea of having three on top and three on bottom, and with something

• like this I can see the symmetry and triangulation, we connect these to each other and then we

• connect the top ones to the bottom, so that'd be like everything is squares if we unwrapped

• it, and then just do triangles to it, all symmetric like so it's easy to visualize,

• every square is the same, I mean it gets squished when you do topology to it but I can visualize

• it as a pattern that's all the same, and now we have more than enough lines to connect

• the utilities to all the houses.

• If the mug had the little icons printed around the handle maybe it'd be more obvious how

• to start mapping it back on, or maybe if the icons were spaced around the mug and if the

• inside of the mug actually had a hole in the bottom so you could loop through that way,

• and then we wouldn't even need the handle.

• But from here we can scoot over to the handle and see just one more interesting thing that

• the more common solutions don't take advantage of and that we'll need more colors to make

• clear, so let's go ahead and do this.

• Ok, we'll start with internet because that's important and it's just two straight lines

• that circle around the mug.

• Then we'll triangle up our other utilities between them, and now it's just three lines

• left and we'll just pretend they're looping through the cup so they're nice and organized,

• and then scoot them over to the handle, And now here's the funnest part.

• If just one line is going over the handle it might as well be a flat bridge.

• But we've got three lines, and notice that they don't match up.

• On a flat bridge you'd have to worry they'd block each other from connecting, But our

• solution uses all of the 3d roundiness of the handle by rotating around it so the lines

• come out apparently in a different order on the other side.

• So there's my answer to the three utilities mug puzzle, and also the four utilities puzzle,

• see if you can rediscover it yourself without looking, and if you need another mug challenge

• try adding a fourth house.

So some months back Grant Sanderson of the channel 3blue1brown invited me to take part

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Four Utilities Puzzle (and how to ruin a bagel)

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林宜悉 posted on 2020/03/30
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