Find the total number of ways of coloring the eight regions

In summary, the discussion is about solving a problem related to graph theory using the deletion-contraction theorem or elementary counting. The participants also discuss the possibility that the OP may have already studied graph theory and the purpose of the question. The final solution involves the product of eight numbers, six of which are 2, and mirrors the deletion-contraction approach in reverse order.
  • #1
feesicksman
2
0
Homework Statement
We have a figure which has been divided into 8 regions (A to H) as shown below. Each region must be colored in exactly one color. There are four colors to choose from: yellow, red, green and blue. There is only one other constraint: neighboring regions must not be of the same color.
Find the total number of ways of coloring the regions A to H.
Relevant Equations
I think this problem can be viewed as a vertex coloring procedure (with at most 4 colors) in graph theory.
image.jpg

The eight regions
 
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  • #2
Please show your efforts to work on the problem. Thank you.
 
  • #3
It may help to first work through a smaller, simpler problem.
 
  • #4
If you know some graph theory, have you come across chromatic polynomials?
 
  • #5
OP has not returned. The solution can be found by elementary counting, I don't think that categorising the problem as graph theory helps.
 
  • #6
pbuk said:
The solution can be found by elementary counting,
Really? That looks a bit tough to me. Using the deletion-contraction theorem I got 768 quite quickly.
Besides, it would seem the OP has done some graph theory, so likely this question is intended as an exercise in using the theorem.
 
  • #7
haruspex said:
Really? That looks a bit tough to me. Using the deletion-contraction theorem I got 768 quite quickly.
I got the same answer accumulating the product of 8 numbers in my head, 6 of which are 2:
How many ways can you colour D? (d)
Given a colouring for D, how many ways can you colour E? (e, so what is de?)
Given a colouring for D and E, how many ways can you colour B? (b, so what is deb?)
...

haruspex said:
Besides, it would seem the OP has done some graph theory, so likely this question is intended as an exercise in using the theorem.
I'm not so sure - it could just be that they searched the interweb and found "vertex coloring procedure".

Anyway they don't look like they are interested in this any more.
 
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  • #8
pbuk said:
I got the same answer accumulating the product of 8 numbers in my head, 6 of which are 2:
How many ways can you colour D? (d)
Given a colouring for D, how many ways can you colour E? (e, so what is de?)
Given a colouring for D and E, how many ways can you colour B? (b, so what is deb?)
...
I'm not so sure - it could just be that they searched the interweb and found "vertex coloring procedure".

Anyway they don't look like they are interested in this any more.
Very neat. Interestingly, it mirrors the deletion-contraction approach, in reverse order.
To be looking up vertex colouring, the OP first had to recognise the duality between vertices and regions.
 

FAQ: Find the total number of ways of coloring the eight regions

What is the problem statement for "Find the total number of ways of coloring the eight regions"?

The problem typically involves determining the total number of distinct ways to color eight regions, which could be part of a geometric figure like a circle divided into sectors or any other configuration, using a given set of colors under certain constraints, such as no two adjacent regions having the same color.

What methods can be used to solve this coloring problem?

Common methods to solve such coloring problems include combinatorial techniques, graph theory (such as using chromatic polynomials), and sometimes group theory, particularly when dealing with symmetries and rotations. The exact method depends on the specific constraints and the arrangement of the regions.

What are the typical constraints in a region coloring problem?

Typical constraints include no two adjacent regions being the same color, a limited palette of colors, and sometimes additional conditions like symmetry or specific patterns that must be followed. These constraints greatly influence the number of valid coloring ways.

How does symmetry affect the total number of ways to color the regions?

Symmetry can significantly reduce the total number of distinct colorings. For example, if a figure can be rotated or reflected in such a way that it looks the same, these symmetrical configurations should be counted only once. Group theory and Burnside's Lemma are often used to account for these symmetrical arrangements.

Can you provide an example problem and its solution?

Sure! Suppose you have a circle divided into 8 equal regions and you want to color each region using 3 different colors such that no two adjacent regions have the same color. Using combinatorial methods or graph theory, you can determine the total number of valid colorings. For instance, if we use the chromatic polynomial approach for a cycle graph with 8 vertices (representing the regions), we find that the number of valid colorings is given by the chromatic polynomial evaluated at 3 colors. The calculation involves more detailed steps, but the final result gives you the total number of ways to color the regions under the given constraints.

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