Problems like four-colour theorem

[jobsism]

I recently came across the four-colour theorem, and found it so interesting that I spent hours trying to find a counterexample(even though I knew there couldn't be one!)...now I'm so obsessed with trying out problems like these...can anyone suggest me problems like these in topology, which are simple to understand and yet so enticing??

 

[micromass]

Get a book in graph theory, you should find many problems like this in such a books...

Some other nice problems:
- The isoperimetric problem: determine a plane figure of the largest possible area whose boundary has a specific length.

- Kakeya needle problem: find the smallest convex set in the plane in which you can reverse a needle (see: http://www.math.hmc.edu/funfacts/ffiles/20005.2-3.shtml )

Other nice problems are in http://www.math.hmc.edu/cgi-bin/funfacts/main.cgi?Subject=00&Level=0&Keyword= These problems are all very easy to understand, but the solution can be extremely difficult...

 

[Ben Niehoff]

On the surface of a torus, you have a seven-color theorem. There's one you can play with.

And speaking of the four-color theorem, most world maps actually use five colors. Aside from just being easier (you don't have to put as much thought as to what color goes where), why would they need to do this?

 

[jobsism]

Thank you guys for the suggestions! I'm going to try out these problems...as for the isoperimetric problem, isn't circle the answer? I read somewhere that for a specific perimeter, the circle possesses the largest area...

And Ben, I think more than 4 colours are used in maps because 4 aren't enough for specific details...like maybe in colour-coding regions prone to earthquakes, I think upto 7 colours are used for specifying which regions are more prone than others by a range of values...for each range, a colour is used...i may be wrong too...;D

 

[micromass]

Thank you guys for the suggestions! I'm going to try out these problems...as for the isoperimetric problem, isn't circle the answer? I read somewhere that for a specific perimeter, the circle possesses the largest area...

Ah, yes, but how to prove such a thing. The proof is not evident...

 

[jobsism]

Oh..I see...perhaps it goes the same way as the 4 colour problem...taking all possible sets of curves and eliminating everything (forgive me if I'm wrong..I dint actually view the proof), leaving the circle alone...

 

[Ben Niehoff]

And Ben, I think more than 4 colours are used in maps because 4 aren't enough for specific details...like maybe in colour-coding regions prone to earthquakes, I think upto 7 colours are used for specifying which regions are more prone than others by a range of values...for each range, a colour is used...i may be wrong too...;D

No, I don't mean fancy features like color-coded earthquake regions. I mean just the colors that are used to fill in each country. Why would mapmakers need more than 4?

 

[jobsism]

No, I don't mean fancy features like color-coded earthquake regions. I mean just the colors that are used to fill in each country. Why would mapmakers need more than 4?

In that case, I've no idea...maybe its to increase the beauty or sth...I mean 10 colors would be more colorful than 4, ryt?;D

 

[Unrest]

In that case, I've no idea...maybe its to increase the beauty or sth...I mean 10 colors would be more colorful than 4, ryt?;D

When I was a kid all the commonwealth countries were pink. So that restricts your choices for the other countries who's colors don't usually matter but can't be pink. Probably also helps to clarify that two nearby countries with small ones sandwiched between them aren't actually the same thing.

 

[Ben Niehoff]

No, neither of those reasons.

The four-color theorem assumes that each region to be colored is connected (i.e., that each region is all in one piece). However, on a real map of the world, some regions are NOT connected. That is, some countries actually have two separate pieces with other countries in-between (usually the result of wars). Those two pieces need to be colored the same color, which is sometimes impossible with only four colors.

 

[brocks]

No, neither of those reasons.

The four-color theorem assumes that each region to be colored is connected (i.e., that each region is all in one piece). However, on a real map of the world, some regions are NOT connected. That is, some countries actually have two separate pieces with other countries in-between (usually the result of wars). Those two pieces need to be colored the same color, which is sometimes impossible with only four colors.

Never mind countries; what about water? You need to reserve a color (I recommend blue) for water, so you can easily distinguish large lakes from countries. If you do that, you still need four more colors unless it's a very simple map.

 

[Helios]

jobsism would enjoy the game of "Y" or mudcrack Y and also poly-Y.