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 is “hey, 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 like… well 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.