# Thread: How Many Combinations are Possible for Octahedron?

1. ## How Many Combinations are Possible for Octahedron?

Tough question. I think the question wants you to consider different orientations of the same pattern as one pattern.

Question: If each side of a tetrahedron is an equilateral triangle painted white or black, five distinct patterns are possible: all sides white, all black, just one side white, just one black, and two sides white and two black. If each side of an octahedron is a white or black equilateral triangle, how many distinct patterns are possible?

2. Yes, it appears to want you to consider all types of symmetry.

All White - Only one way to do that.
All Black - Only one way to do that.
One White (same thing as 7 black) - You can paint any one of the 8 surfaces, but various symmetries show they are all the same. - Just one distinct way.
One Black (7 white) - Again, just one.
Two White (6 black) -- What say you? More than one distinct way?

3. Originally Posted by tkhunny
Yes, it appears to want you to consider all types of symmetry.

All White - Only one way to do that.
All Black - Only one way to do that.
One White (same thing as 7 black) - You can paint any one of the 8 surfaces, but various symmetries show they are all the same. - Just one distinct way.
One Black (7 white) - Again, just one.
Two White (6 black) -- What say you? More than one distinct way?
There seem to be four total patterns, if you allow for reorientations, by coloring 8 triangles one color (8 black, 8 white) or just one triangle one color (1 white, 1 black), as you say. All black, all white, any one white triangle, and one black triangle.

For two triangles, I think three patterns are possible for each color. So, we would have three for white and three for black. Considering the four we counted before, that would bring our total to 10 (4+3+3), unless I've counted incorrectly. Does that look right?

4. Originally Posted by bennyJ
For two triangles, I think three patterns are possible for each color. So, we would have three for white and three for black.
I have to agree with this. We have:

With reference to the provided drawing:

Well, that covers a lot of he problem. Why quit now?

8W0B - 1
7W1B - 1
6W2B - 3
5W3B - ?? - Still some thinking to do.
4W4B - ?? - Still some thinking to do.
3W5B - ?? (by symmetry)
2W6B - 3 (by symmetry)
1W7B - 1 (by symmetry)
0W8B - 1 (by symmetry)

What say you of 5W3B?

5. Originally Posted by tkhunny
I have to agree with this. We have:

With reference to the provided drawing:

Well, that covers a lot of he problem. Why quit now?

8W0B - 1
7W1B - 1
6W2B - 3
5W3B - ?? - Still some thinking to do.
4W4B - ?? - Still some thinking to do.
3W5B - ?? (by symmetry)
2W6B - 3 (by symmetry)
1W7B - 1 (by symmetry)
0W8B - 1 (by symmetry)

What say you of 5W3B?
I don't plan on stopping. It will start to get challenging soon, though. Do you think there are two or three distinct patterns for each color of the 3/5 pattern?

To my count, at least two distinct patterns, considering symmetry, seem to be three in a row and a kind of checkered pattern in which only two of the three similarly colored triangles share one vertice.

6. Originally Posted by bennyJ
I don't plan on stopping. It will start to get challenging soon, though. Do you think there are two or three distinct patterns for each color of the 3/5 pattern?

To my count, at least two distinct patterns, considering symmetry, seem to be three in a row and a kind of checkered pattern in which only two of the three similarly colored triangles share one vertice.
Does anyone have any guesses for how many total patterns there might be?

8. Originally Posted by tkhunny
I have to agree with this. We have:

With reference to the provided drawing:

Well, that covers a lot of he problem. Why quit now?

8W0B - 1
7W1B - 1
6W2B - 3
5W3B - ?? - Still some thinking to do.
4W4B - ?? - Still some thinking to do.

3W5B - ?? (by symmetry)
2W6B - 3 (by symmetry)
1W7B - 1 (by symmetry)
0W8B - 1 (by symmetry)

What say you of 5W3B?
In addition to the 10 that are counted here, I figured 6 more distinct patterns (3 white and 3 black for the 3/5 pattern) as well as 8 distinct patterns (4 black and 4 white for the 4/4 pattern) for a total of 24 distinct patterns. Does that look right?

9. Any guesses for the total number of distinct patterns for this shape?

10. Originally Posted by tkhunny