And after that, he drew a bath.

## An Argyle Agglomeration

From Andréa.

And I swear I remember when their State Tourism Office had this as their motto! (Probably around the time the University of Illinois used “Four colors suffice” on their postmarks.)

Okay, I get the Missouri joke; but still wonder why it’s California and Nevada in particular who are having this conversation.

@Danny Boy probably the artist found them easiest to draw recognizably.

Re: the dentist. The patient is a zombie. Zombies eat brains.

How a whole brain gets stuck between someone’s teeth is questionable but probably also down to “how do I convey that brain matter is on the floss?”

Also dentists/hygienists don’t use dental floss.

Why 1962? I’m guessing it’s arbitrary. I might have picked 1968, when “Night of the Living Dead” was released.

https://en.wikipedia.org/wiki/List_of_zombie_films

Dental hygienists absolutely use floss. Or they should.

Thanks. I could see it was a zombie but couldn’t tell what the dentist/hygienist was doing. Now that I know, it’s an Ewww for me.

That said, was it submitted as an Oy? Or just a bonus maybe-CIDU? Because I’m not seeing an Oy.

“Also dentists/hygienists don’t use dental floss.”

This may be true now; but it wasn’t true 15 months ago.

At my last cleaning, my hygienist let me know they were no longer flossing after a cleaning.

(At the same cleaning, I was surprised to find that my relatively young hygienist (45?) of the last 15 years had chosen to retire from the practice due to the pandemic.)

Does it need to be states? I’m thrown off because Missouri asking California and Nevada to visit and St. Louis and Branson would be like me asking people to visit my tonsil or lower intestine.

“Four colors suffice”?????

“This may be true now; but it wasn’t true 15 months ago.

At my last cleaning, my hygienist let me know they were no longer flossing after a cleaning.”

That’s

yourdentist office. Not all dentists in general.The Missouri/misery pun is as ancient as they come: https://www.youtube.com/watch?v=NztfOSyCCFM

Like a few of you, I was perplexed by the “dentists/hygienists don’t use dental floss” comment. I have not missed a six month checkup in over 30 years, and every single time the hygienist has used dental floss, including the two times I have been during the pandemic.

`me asking people to visit my tonsil or lower intestine.`

Do those invitations get accepted much?

woozy: “Four colors suffice”?????

For a long time it was conjectured that you needed at most four colors to color a map (assuming that all the countries are contiguous, on a single surface, and that you have a requirement that no countries which share a border can be the same color; and probably some other conditions that I’m forgetting). It was finally proven [*] by mathematicians at the University of Illinois in 1982.

I understand the reference to the Four Color theorem, but what is the connection between that and the Missouri state motto?

`I understand the reference to the Four Color theorem, but what is the connection between that and the Missouri state motto?`

These were two different local claims to fame that got promoted on some official postage meter postmark imprints. (Which I happened to see at a job where we processed a lot of mail from official or educational sources.) Sorry I left the linkage unspoken.

My hygenists have always ended the cleaning with a flossing. This cleans some more of the cleaning gunk out, and let’s them check for bleeding when flossing.

Coming soon: A Missouri Loves Company craft beer, a cooperative effort by several craft brewers in Missouri. https://drink314.com/2021/04/28-missouri-breweries-team-up-for-new-missouri-loves-company/

Did the dentist have to use mental floss to get the brain out?

The worm one makes me think of chicken fingers.

Winter Wallaby: Contiguity and single surface are not requirements; if there are noncontiguous groups of countries you simply apply the theorem to each. But the maps have to be on a plane or sphere; tori and Moebius strips need more than four.

The Crankshaft example may work in print, but not in sound. Is there any English dialect in which bored and board are not homophones? How would the other two detect the malapropism?

Why does Nevada have arms and hands but California doesn’t? Do we really want to know how California is holding up the paper?

Treesong: is there a simple counterexample for a torus? Not seeing it. If you retract the torus to a sheet that wraps at both the sides and top/bottom, it seems like if instead you paste arbitrarily many sheets that wrap only at the side (which are spheres) vertically any counterexample for the torus should provide a counterexample for the sphere. (If you just paste one, the new top and bottom don’t have to align, but since there are only finitely many different ways to color the boundary it seems like if you paste enough copies you must eventually reach a matching boundary and then you’ve colored the torus.)

… but I’m hardly an expert on this stuff

Dave in Boston: A torus requires seven colors. http://faculty.smcm.edu/sgoldstine/torus7.html

Oddly, planar coloring (or equivalently, sphere coloring) is one of the hardest surfaces. IIRC, I was able to read the proof for the torus in college in a short amount of time. Plane/sphere was actually the last single surface to be solved. I believe having two spheres, with each country having components on each sphere, is still unsolved.

Treesong: It seems odd to object that “continguity and single surface” are not requirements because you can just decide to color different parts of the same country differently. That’s just making contiguity and single surface requirements by re-defining country to be the continguous, single-surface components.

Early this morning I unfortunately experienced a browser crash and lost an almost-completed comment on the four-color thread. Mostly pointing out that — although the most intuitive way of presenting the ideas, and indeed historically the origins of thinking about it, were in terms of regions on a geographic map — in modern mathematical treatment you would more generally expect to see it treated within graph theory. There you are coloring the vertices of an undirected planar graph and requiring that no edge links two vertices of the same color. Then the niceties about how much of the restrictions need to be explicit and how much can be trusted to default understandings seem to evaporate, as the defaults of graph definitions match up better to what we need.

As the wikipedia article on the four-color theorem (or “conjecture” as they used to feel obliged to call it!) illustrates by first showing regions in a geographic map but almost immediately going into graph-theoretical development.

`"In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short: Every planar graph is four-colorable."`

Incidentally, I was happy to see in this article that Winter’s date of 1982 for the proof is right for the revised and

correctedpublication of the proof, but the announcement by Appel and Haken at Illinois was originally in 1976 — which is a better fit for when I was working that job where I would see a lot of academic postmarks!—

That was the gist of this morning’s lost comment. To note some of the subsequent dialogue, I would note for Dave in Boston and Treesong that the delightful physical models, actually colored in, seen in WW’s link, are not really needed if you wanted to try sketching a torus or mobius-strip example. The site does mention flattening some of the softer actual 3D models, and Dave says something about pasting a lot of sheets together — but that is overkill.

For that sort of sketching and illustration, you can accurately model a torus, or a little differently a mobius strip, with a single rectangular piece of paper. Ususally this is done with the whole sheet, but you could also just draw a smaller rectangle on the page – useful if you want to put in connecting lines, or jot notes. But the main thing is to IDENTIFY the top and bottom edges, and also IDENTIFY the left and right vertical edges. That is, you can draw a position or line, or picture an ant crawling on the surface, and if it goes off the right-side vertical edge it immediately reappears at the same height on the left-side vertical edge.

That construction, with the top and bottom edges identified and separately the left and right edges identified, is a model for a torus surface. And if you connect the left and right only, and reverse the height when you crawl off one side and reappear on the other, you have the model for a mobius strip.

… And then, on the torus model, and using vertex-and-edge graphs rather than filled-in regions on a map, you can make a minimal 7-colors-required graph pretty easily. (This would be a “counterexample” if we had a “six-color conjecture” for toroidal surfaces saying that six colors always suffice. ) Just make 7 big dots then start drawing lines to connect each pair. As usual lines may not cross, and you can’t freely or loosely go up in the air or show overpasses (as on circuit diagrams), but you

cango off one side and come back in the other, as long as you respect the ordering of the loose ends.Okay, just one more! From Ww’s linked “Seven-Color Tori” page, one of the marginal additional links is https://en.wikipedia.org/wiki/Heawood_conjecture which deals with the generalization to different surfaces. That is about a lower bound, for higher-order genus. “On the sphere the lower bound is easy, whereas for higher genera the upper bound is easy and was proved in Heawood’s original short paper (1890) that contained the conjecture.” (The four-color theorem for the sphere or plane is an upper-bound question.)

Related (somewhat) to the four-color theorem: Write the letters A, B and C in a line, spaced apart. Somewhere below them, write the numbers 1, 2 and 3 also in a line, spaced apart. Now draw nine lines (curves actually; they don’t have to be straight and can go the long way around) connecting each digit to each letter: A to 1, A to 2, A to 3, B to 1, etc.

You must do this without crossing any lines.

It’s impossible to do this on a plane or a sphere without crossing any lines. But you can do it on a torus.

MiB , I recall that puzzle cast as three homes in a new block, and needing connections to three utility substations. There is for some reason a restriction that the service lines cannot cross, despite that not being the case in real world construction. Your electric conduit can of course go under or above the gas or water pipes, for instance. But that is the premise of the puzzle.

Solving it n a torus is a good way to try out the technique Mitch describes for identifying opposite sides. Simpler than the complete graph on 7 points!

WW: thanks. My proposition (posted with inadequate coffee) is rubbish – it shows that given a torus you can construct a different torus that’s 4-colorable, but that’s not the least bit interesting even if it’s correct.

mitch4: I was going to post about it really being graphs yesterday and then I didn’t for some reason and the conversation moved on. thanks for taking up the slack 🙂

MiB: When I was a kid, I had a puzzle book with that problem. The solution was to run one of the connections underneath one of the other homes. I felt cheated.

Taking up Dana K’s suggestion of drawing a solution to MiB’s utilities puzzle on a torus, here is my hand sketch of that, with the torus represented in a plane rectangle as explained previously.

The connecting lines are labelled with the letter and number of the vertices they join. In retrospect, I could have done a color coding or something like that to make it clearer.

A1, B1, and C1 are simple direct connections, as are C1, C2, and C3. [That only names 5 distinct lines, I’m aware.] A2 and B3 (which I apparently forgot to label) stay in our focus area on the near side of the donut surface, but do a bit of curving around obstacles. Connection B2 goes up and down, and is a vertical belt around the donut’s thinner dimension, going in back of it. And the 9th and last one, A3 (labelled 3A by mistake at one of its ends) , goes around the the inner-tube’s floating diameter.

That description already assumes you see how the rectangle gets wrapped up to be a torus. If it helps, there are a few more remarks at the post where the scan image appears, https://wpdemos.blog/2021/05/09/utilities-puzzle/

And here in Nancy, the top three panels show how a flat rectangle can be wrapped as a torus by identifying top and bottom edges, and left and right edges.

(After that, it tries to impose that topology on her reality.)

Wow, that’s a very sophisticated Nancy comic! Take

that, Mad Magazine!