The Four Color Map Theorem - Numberphile

The Four Color Map Theorem (or colour!?) was a long-standing problem until it was cracked in 1976 using a "new" method... computers!

We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.

NUMBERPHILE

Videos by Brady Haran

[jimthompson]

It's not just useful for drawing maps, either: the same principle allows cell towers from interfering with each other, by using four sets of frequencies. Using four sets of frequencies, no adjacent cells have to use the same frequencies.

[proxy]

I love how all this started with some guy filling out a map with colors and noticing that he only needed 4

[timsullivan]

I never thought I would say this of a mathematician, but I don't believe I could ever tire of listening to James Grime. I actually find myself smiling far more often than was likely ever the case back in my school days. Thank you Dr Grime.

[a]

omg watched this mindlessly 3 years ago when i was in high school then here i am studying graph theory in college coming back to see how it actually works like an hour before midterm

[davidyoung]

I recall an issue of Scientific American back in about 1974 (more or less) that had an article that purported to show 7 amazing recent discoveries. One was that the best first move in chess was shown to be h4, another was a logical way to disprove special relativity, another was that Di Vinci invented the toilet and another was that someone came up with a map that required five colors. I can't recall the year of the publication, but I can recall the month. The magazine came out on April

[iancopple]

11:06 I'm currently studying Actuarial Science at the University of Illinois (same awesome school as Appel and Haken). You wouldn't think the Four Color Map Theorem would show up in an insurance internship, but I showed this theorem to a few of my coworkers and they made a colorblind-friendly map of the U.S. for me to use in a project. Thank you Numberphile!

[thepip]

What if it was in 3d? like, with colouring 3d spaces instead of 2d shapes. Maybe filling hollow glass chambers with coloured liquid. How many colours would that take? Would there be a limit?

[ontario]

6:03 careful dude you're gonna summon the devil

[abidhossain]

0:04
"It's easy to state"


I see what you did there..XD

[bentleystorlie]

I learned this in a book called "betcha can't" (which actually has a lot of the problems I've seen on Numberphile). But the story was that the father died and his five sons inherit his land. In the will it says they can divide it up however they want, but each plot needs to be all one piece and must share a border with all four other sons' plots.

[mazingzongdingdong]

everytime i take a test i imagine that he's looking over me and kinda guiding my way to success lol

[TheOfficialCzex]

Enclaves and exclaves can not be considered as the theorem requires *contiguous* regions. The term "map" in the theorem refers to a physical map as opposed to a political map. This could be confusing to grasp after watching this video as they refer to real-world examples as well as abstractions.

[aurelia]

"Let's try making a map that requires five colors"
*second map drawn only has four sections*

[XiaoyongWu]

While watching this, I thought at 3:23, you could leave the last quarter circle border unclosed and make a bigger circle around everything. With the existing coloring, it looks like that would need a fifth color. But, after more thinking, it's doable by some shifting on the colors used earlier

[patrickhodson]

0:16 Dang foreigners colored Michigan two different colors lolol

[p.mil.]

14:14 look above the o there are 2 yellows

[richarddeese]

Really interesting video; great job! I myself spent a lot of 'doodling' time back in the 80s trying to find a counter example. I also don't like the computer proof for the same reason you stated: it doesn't teach us anything but that some result is true. We don't know WHY it's true. But to me it boils down to a topology problem, not a color problem. I state it thus: The greatest number of closed figures which can be drawn on any 2D surface such as a map or globe in such a way that every figure touches every other figure along a side, is four. You'd literally have to put another figure into the third dimension, making it go above or below the 'plane' to connect it to other figures, thus forcing a 5th color. You simply can't do it any other way. That is what makes the 4 color conjecture true... but, of course, that is not a proof in itself. But I can tell you that I'm done doodling with it. I'm satisfied that eventually, someone will prove it with geometry or more likely topology. Rikki Tikki

[Robertlavigne]

Thanks for the nerd snipe numberphile. Every time I see this theorem stated I always end up taking a stab at finding a weird case to disprove it. Today I was so close to calling a math friend to show him my counter example, before realizing I had a colour wrong.

[sophieeula]

i don’t know why i’m watching these math videos at 3 am bc i truly don’t understand them but everyone in the vids seem to so i keep on comin back

[zombiedude]

Back in the windows xp days, I'd make images in paint by making one arbitrary continuous path both ends on an edge of the image. The curve would intersect itself at many points, but never intersect itself multiple times at the same point. I found that you could always cover the "map" created using these restrictions with exactly 2 colors.