Problems like four-colour theorem

  • Thread starter jobsism
  • Start date
  • Tags
    Theorem
In summary: 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.In summary, Ben found a counterexample to the four-color theorem and is now trying to find more problems like these. Some other nice problems are in http://www.math.hmc.edu/cgi-bin/funfacts/main.cgi?Subject=00&Level=0&Keyword= where the solutions can be extremely difficult.
  • #1
jobsism
117
0
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??
 
Physics news on Phys.org
  • #2
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...
 
  • #3
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?
 
  • #4
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
 
  • #5
jobsism said:
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...
 
  • #6
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...
 
  • #7
jobsism said:
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?
 
  • #8
Ben Niehoff said:
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
 
  • #9
jobsism said:
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.
 
Last edited:
  • #10
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.
 
  • #11
Ben Niehoff said:
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.
 
  • #12
jobsism would enjoy the game of "Y" or mudcrack Y and also poly-Y.
 

1. What is the four-colour theorem?

The four-colour theorem, also known as the four-colour map theorem, states that any map can be coloured using only four colours in such a way that no two adjacent regions have the same colour.

2. Who first proposed the four-colour theorem?

The four-colour theorem was first proposed by Francis Guthrie in 1852. However, it was not until over a century later, in 1976, that the first valid proof was published by Kenneth Appel and Wolfgang Haken.

3. Why is the four-colour theorem significant?

The four-colour theorem is significant because it is a fundamental problem in mathematics and has sparked much debate and interest among mathematicians. It also has applications in fields such as computer science and cartography.

4. Has the four-colour theorem been proven?

Yes, the four-colour theorem has been proven to be true. However, the first proof, published by Appel and Haken in 1976, was controversial as it used a computer to check thousands of cases. A simplified proof was published in 2005 by Robertson, Sanders, Seymour, and Thomas, which did not rely on a computer.

5. Are there any similar problems to the four-colour theorem?

Yes, there are several similar problems to the four-colour theorem, such as the five-colour theorem, which states that any map can be coloured using only five colours. Other examples include the Heawood conjecture and the Hadwiger-Nelson problem, which also involve colouring maps or graphs with a limited number of colours.

Similar threads

  • General Math
Replies
8
Views
984
  • Art, Music, History, and Linguistics
Replies
21
Views
1K
Replies
10
Views
1K
Replies
16
Views
2K
  • Introductory Physics Homework Help
Replies
25
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
646
  • Programming and Computer Science
2
Replies
37
Views
3K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Replies
1
Views
827
Back
Top