S.W.M.B.O., who isn't a math freak, knows I'm a math freak, and for gift-giving occasions uses her intuition and reads reviews on amazon.com to buy me cool math books. For Christmukkah she scored a real winner, a new book about a Math Thing (a technical term) I've been fascinated with since I was a kid.
Above is a map of the continent of Euforia, on planet Yobbo (where the pierced + tatted teenage dropout punks Lenny + Spike were last known to be loitering).
The Vleeptron Geographic Society has lots of different colors of ink, so they don't have any trouble making a map like this so that countries which share a common border are never colored the same color.
On Planet Earth the shapes and borders of sovereign nations are so weird and loony; often if you know a lot of history, every screwy squiggle is the result of some ghastly war.
In the USA, every schoolkid learns the campaign war cry "Fifty-Four Forty or Fight!" Well, we didn't fight, and our northern border is way south of Fifty-Four Forty.
But these strange shapes are all about old fights over land. (Quite often they fight because God Himself has clearly stated that a certain piece of land belongs to a specific group of people. Sometimes the Same God clearly stated that the Same Piece of Land belongs to two different groups of people.)
If you look at a map and see a really insane-looking curly-cue country or state or county, you can pretty much assume that once upon a time, a whole bunch of luckless young guys in uniforms had to shoot at each other and step on land mines and get maimed and die, and the crazy shape is the compromise the Leaders (not in uniform) grudgingly settled on when both sides finally got exhausted and were running out of live soldiers.
Much too weird and screwy, actually, so we have to impose some Rules on these crazy maps.
Rule 1. All borders must have non-zero Length.
No Point Borders like Four Corners in the USA (see Fig. 2). I have been to Four Corners; there's a little park there with a couple of picnic tables, and I can verify that it is possible -- easy, even -- to urinate on four states with just a single whiz, and I'm sure lots of other guys and the occasional ambitious and uninhibited girl before and after my brief visit have done the same.
Rule 2. All entities (nations, counties, states) must be a single contiguous blotch of land. No separated exclaves (I just learned that word today, I think it's neat).
Here are two examples of real, true, authentic Exclaves. First, there's Russia (see Figure 3).
Forget offshore islands, we're not talking about them. Offshore islands never present border-coloring problems.
If you're in Russia, it's impossible to walk to every other part of Russia. You have to go through another sovereign nation or take an airplane or a boat to get to Kaliningrad, which belongs to Russia every bit as much as St. Petersburg belongs to Russia. Only Kaliningrad is separated from the rest of Russia by about 400 miles of land you could walk through, but it's land that belongs to another nation. That's an Exclave.
Figure 4 shows the USA state of Michigan. The part shaped like the palm of the human right hand is called the Lower Peninsula, and the other entirely separated splotch of land is called the Upper Peninsula.
So our Rule 2 forbids Exclaves. Vleeptron, Yobbo, Hoon and Mollyringwald are sensible planets and have no Point Borders, and no Exclaves.
Now we can get back to my favorite problem and my new book.
Whether it's the nations of Eurasia, or the counties of England, or the states of the United States of America, or the provinces of Canada, or the cantons of Confederatio Helvetica -- no matter what crazy geopolitical map you can possibly imagine -- what's the fewest possible ink colors you need to color the map so that no two countries with a common border are colored the same?
Vleeptron Geographic used 5 colors for Euforia. Could they get away with just 4? Could they get away with just 3?
Exists there a map that demands no fewer than 5 ink colors? Exists there a map with entities and borders so screwy and convoluted that it absolutely requires 6 ink colors?
Whatever you think is the Answer, Prove It. Don't just bullshit your way through it or try to impress us with Long Words, like Synchropolychromatic. Prove It.
Oh, also, suppose you live on a planet which isn't a sphere, but is shaped like a Doughnut. How many ink colors do you absolutely need for any map on the surface of your Donut Planet?