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How many distinct 3-colorings of the edges of square?
I'm trying to use Burnside's Lemma but I'm not getting a natural number.
I'm getting [math]\frac{1}{8}\left ( 3^4+3+3+3+3^3+3^3+3^2+3^2 \right )=20.25[/math]
where the numbers in the parentheses (in order) are the number of configurations fixed under
1: 0 degree rotation
2. 90 degree rotation
3. 180 degree rotation
4. 270 degree rotation
5. reflection across vertical axis
6. reflection across horizontal axis
7. reflection through one diagonal
8. reflection through the other diagonal
What am I doing wrong?
>>
Help me out guys. I feel like I'm overlooking something really simple.
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There's gotta be at least one person on /sci/ right now that knows combinatorics.
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>>8514290
lol no were too busy shitposting
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>>8514290
If you really want help with this problem why don't you explain it in more detail what it is you're trying to calculate? I'm happy to help but I have no idea what "3-colorings of the edges of square" means. Also, why are you dividing by 8?
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The 180 degree rotation has 2 orbits.
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>>8514317
I thought it would take too long to explain the problem and Burnside's Lemma than to simply have someone familiar with the problem recognize the mistake I was making.
Basically, we're counting many distinct ways there are to color the edges of a square using three colors, keeping in mind that certain configurations may be rotationally or reflectionally equivalent.
>>8514318
Thanks a lot!
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>>8514322
Oh I see. The picture threw me off, I thought you were trying to distinctly color in the 8 sections divided by the dotted lines. And if it's not too much to ask why did you have to divide by 8? To avoid double counting?
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