Octahedron Problem: How Many Color Patterns?

  • Thread starter reese92tsi
  • Start date
In summary: The symmetry is the catch here...Welcome to PF!In summary, if each side of a tetrahedron is an equilateral triangle painted white or black, five distinct color patterns are possible: all sides white, all black, just one side white and the rest black, just one side black and the rest white, and two sides white while the other two are black.
  • #1
reese92tsi
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0
If each side of a tetrahedron is an equilateral triangle painted white or black, five distinct color patterns are possible: all sides white, all black, just one side white and the rest black, just one side black and the rest white, and two sides white while the other two are black. If each side of an octahedron is an equilateral triangle painted white or black, how many distinct patterns are possible?

I was able to come up with a good answer but I'm not sure if its right or not.
 
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  • #2
3!.3!.28/26 I think...

The symmetry is the catch here...
 
  • #3
Welcome to PF!

Hi reese92tsi! Welcome to PF! :smile:
reese92tsi said:
I was able to come up with a good answer but I'm not sure if its right or not.

Show us (with your reasoning, of course)! :wink:
 
  • #4


tiny-tim said:
Hi reese92tsi! Welcome to PF! :smile:


Show us (with your reasoning, of course)! :wink:
Thanks for the warm welcome

all sides black: 1 pattern
all white: 1

7 white 1 black: 1
7 black 1 white: 1

6 white 2 black: 3
6 black 2 white: 3
-2 white faces adjacent at an edge
-2 white faces adjacent at a vertex
-2 white faces neither adjacent at an edge or vertex

5 white 3 black: 3
5 black 3 white: 3
-3 white faces adjacent at edges
-3 white faces adjacent at only vertices
-2 white faces adjacent at an edge and 1 white face
adjacent at either vertex of the set of the 2 edge adjacent white faces

4 black 4 white: 7
-4 white faces sharing a common vertex
-3 white faces sharing a common vertex and 1 white face
adjacent at only the vertex of the rightmost and leftmost
white face of the 3
-3 white faces sharing a common vertex and 1 white face
adjacent to only the edge of the right most face of the 3
-The mirror to the pattern above
-3 white faces sharing a common vertex and 1 white face adjacent to
only the edge of the center white face of the 3
-4 white faces adjacent only at vertices
-2 sets of 2 white faces which are adjacent at an edge
touching each other at only 2 vertices

So I've got 23 patterns
 
  • #5
reese92tsi said:
5 white 3 black: 3
5 black 3 white: 3
-3 white faces adjacent at edges
-3 white faces adjacent at only vertices
-2 white faces adjacent at an edge and 1 white face
adjacent at either vertex of the set of the 2 edge adjacent white faces

Hi reese92tsi! :smile:

0 1 2 and 4 white look good, but I'm not convinced about 3 …

what does 3 white faces adjacent at edges mean?

and have you checked that there are no mirror images left out?
 
  • #6
I'm sorry about the phrasing. "3 white faces sharing a common vertex" is probably a better way of phrasing it.

Going over the patterns in my head it looks like I've found them all (still not positive).

chaoseverlasting said:
3!.3!.28/26 I think...

The symmetry is the catch here...

Can you explain this formula in further detail, I'm not really a math buff (would love to become one). I think puzzles/logic problems are a lot of fun.
 

Related to Octahedron Problem: How Many Color Patterns?

1. What is the Octahedron Problem?

The Octahedron Problem is a mathematical problem that involves determining the number of distinct color patterns that can be formed on the faces of an octahedron using a specific number of colors.

2. How many colors are typically used in the Octahedron Problem?

The Octahedron Problem is usually solved using 6 or 8 colors, as these are the most common numbers of colors used in coloring problems involving polyhedra.

3. What makes the Octahedron Problem difficult to solve?

The Octahedron Problem is difficult to solve because of the large number of possible color patterns that can be formed on the faces of an octahedron. As the number of colors used increases, the number of possible patterns also increases exponentially.

4. How is the Octahedron Problem related to other mathematical problems?

The Octahedron Problem is related to other mathematical problems such as the Four Color Theorem and the Seven Color Theorem, which involve determining the minimum number of colors needed to color a map or graph without any adjacent regions or vertices sharing the same color.

5. What is the significance of the Octahedron Problem?

The Octahedron Problem has been studied by mathematicians for decades and has led to the development of new mathematical concepts and techniques. It also has real-world applications in fields such as graph theory and computer science.

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