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Combinations with Symmetry Question

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How many ways are there to put 9 differently colored beads on a grid if the purple bead and the green bead cannot be adjacent (either horizontally, vertically, or diagonally), and rotations and reflections of the grid are considered the same?

I've had quite a bit of trouble with this one. Does anyone know how to do it?

Dec 17, 2019

#1
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I did it in a circle and I realize now that this is not right. You want it in a grid.

But you have not said what shape the grid is. I suppose it is supposed to be a 3 by 3 grid but this has not been stated.

So I might think about it again......

Dec 17, 2019
#2
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I assume you do not know how to do it but do you know what the answer is?

Or do you have any information that could help answerers.

Dec 17, 2019
#3
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How many ways are there to put 9 differently colored beads on a grid if the purple bead and the green bead cannot be adjacent (either horizontally, vertically, or diagonally), and rotations and reflections of the grid are considered the same?

A grid of 3 dots by 3 dots.      9 colours but pink and green cannot be next to each other.

I really do not know but here is my 'guess'

So there are 7 colours that can go into the middle.

Rotations are the same just put a green in a corner

You cannot have a pink next to it but you cannot have a pink on the adjacent corners either because it it is rotated they will become diagonally adjacent,

So

I get   7*1*6*5*4*3 *3*2*1 = 15120

There are 4 axes of symmetry so divide by 2^4=16

15120/16 = 945

I doubt that it is correct though.

If you know what the number answer is then please share that knowledge with the forum.

Dec 17, 2019
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I'm getting an answer of 5*6! = 3600.

Dec 17, 2019
#5
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This question is actually very ambiguous.

Sometimes an outcome appears to be valid but if it is rotated it becames invalid.

Since rotations are considered euqivalent then that should mean that the original outcome is also invalid.

I tried to take this into acoount with my answer.  I have said both must be invalid.

It is the diagonal condition that causes this problem.

---------------

Anyway guest, you have not attempted to explain how you got your answer. So it is impossible for anyone to consider your logic.

Melody  Dec 17, 2019
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Attn: chenxander

I am still waiting for your response.

Dec 17, 2019
#7
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Hello Melody,

I understand your confusion; I believe the grid is a three-by-three grid.

The answer is 20160, but I'm not sure why.

Thanks.

chenxander  Dec 18, 2019
edited by chenxander  Dec 18, 2019
edited by chenxander  Dec 18, 2019
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Thanks Chenxander,

I do not know either BUT

If rotations just mean the corner colours must stay in corners

THEN

I get two different initial positons that can be rotated.

The green, which is my fixed one, can be in a corner, or in the middle of a side.

If ithe green is in a corner then I get this

G 6 1

5 7 2

5 4 3

7!*5 =   25200  permutations

If the green is in the middle of a side then I get this

6 G 5

4 7 3

3 2 1

7!*3 = 15120 permutation

Now       25200+15120 = 40320

[Your given answer  of 20160 is exactly half of this which appears to me to be more than just a coincidence  ]

According to my logic 40320 is the number of permutations if rotations of whole sides is considered the same.

However I have not considered reflections yet.There are 4 axes of symmetry so I want to divide by 2^4=16.

I do not know why the given answer is only divided by 2.

Melody  Dec 18, 2019
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If you get a full answer can you explain it here please.

You can send me a private message to let me know you have added it.

Otherwise i might not notice.

Thanks :)

Melody  Dec 18, 2019