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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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四大功用之謎(以及如何毀掉一個百吉餅)。 (Four Utilities Puzzle (and how to ruin a bagel))

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    林宜悉 發佈於 2021 年 01 月 14 日
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