Page 1 of 2 12 LastLast
Results 1 to 10 of 11

Thread: How Many Combinations are Possible for Octahedron?

  1. #1
    New Member
    Join Date
    Jan 2018
    Posts
    46

    How Many Combinations are Possible for Octahedron?

    Tough question.
    Last edited by bennyJ; 04-23-2018 at 01:51 PM.

  2. #2
    Elite Member
    Join Date
    Apr 2005
    Location
    USA
    Posts
    9,390
    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?
    "Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.

  3. #3
    New Member
    Join Date
    Jan 2018
    Posts
    46
    Quote Originally Posted by tkhunny View Post
    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. #4
    Elite Member
    Join Date
    Apr 2005
    Location
    USA
    Posts
    9,390
    Quote Originally Posted by bennyJ View Post
    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:
    1) Adjacent horizontally
    2) Adjacent vertically
    3) No Adjacent

    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?
    "Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.

  5. #5
    New Member
    Join Date
    Jan 2018
    Posts
    46

    Red face

    Quote Originally Posted by tkhunny View Post
    I have to agree with this. We have:

    With reference to the provided drawing:
    1) Adjacent horizontally
    2) Adjacent vertically
    3) No Adjacent

    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.
    Last edited by bennyJ; 01-12-2018 at 04:22 PM.

  6. #6
    New Member
    Join Date
    Jan 2018
    Posts
    46
    Quote Originally Posted by bennyJ View Post
    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?

  7. #7
    Elite Member
    Join Date
    Apr 2005
    Location
    USA
    Posts
    9,390
    Start with the three 2-6 patterns and explore the 3-5s.
    "Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.

  8. #8
    New Member
    Join Date
    Jan 2018
    Posts
    46
    Quote Originally Posted by tkhunny View Post
    I have to agree with this. We have:

    With reference to the provided drawing:
    1) Adjacent horizontally
    2) Adjacent vertically
    3) No Adjacent

    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. #9
    New Member
    Join Date
    Jan 2018
    Posts
    46
    Any guesses for the total number of distinct patterns for this shape?

  10. #10
    New Member
    Join Date
    Jan 2018
    Posts
    46
    Quote Originally Posted by tkhunny View Post
    Start with the three 2-6 patterns and explore the 3-5s.
    I think I may have double counted the 4-4 patterns. Since each of the four 4-4 patterns I counted appears to work as either black or white, there's no need to count them twice, as I did before. That brings my total to 20. Is that different from what you got?

Bookmarks

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •