A very interesting problem (1 Viewer)

[_ShiFTy_]

So whats the answer??

 

[KeypadSDM]

6x9 = (42)₁₃
72 is just ... who cares, I'm drunk.

 

[fishy89sg]

obviously the answer is 0 because you cant put 12 marbles in a circle

 

[STx]

obviously the answer is 0 because you cant put 12 marbles in a circle

lols

 

[KFunk]

This kind of question is nasty, I don't think you can solve it through methods other than sheer force. It's not too disimilar to asking how many distinct ways there are to arrange n identical objects, e.g. for 4 objects you have groups of 4, 3+1, 2+2, 2+1+1, 1+1+1+1... which is much the same same deal as the partition function which you will see, if you click the link, isn't the simplest of things.

 

[Affinity]

80 should be the answer, you don't need brute force for this... think burnside+frobenius+cauchy

 

[hyparzero]

80 should be the answer, you don't need brute force for this... think burnside+frobenius+cauchy

opps..
wait 80
sorry

I don't think the question is based number theory as Burnside Lemma implies. the BFC is used if we assume the circle to be a 3D object, (one where we can walk around)

But in this case, its not.

 

[Affinity]

80 should be the answer, you don't need brute force for this... think burnside+frobenius+cauchy

EDIT: I mistakenly place 157 there.. It's 80

Here's how you do it:

You consider the spots aroud the circle as distinct, and look at how it behaves under rotations

Burnside's lemma states that the number of orbits in a permutation group is equal to the average number of fixed points under each permutation...
applied here, it would mean that:

The number o different placements(orbits) is equal to the average number of placements which does not change(these are fixed points) under different amounts of rotation(there are 12 possibilities)


For rotation of 0 degrees, every placement is a fixed point: there are 12C = 924 of them
For rotation of 30 and 330 degrees there are no configurations which can be fixed
for rotation of 60 and 300 degrees, there are 2 configurations which are fixed for each (4 total) (both alternating white and black, one starting with white, the otehr starting with black)
and the others:
90,270: none
120:240: 4C2 = 6 each
150,210 : none
180: 6C3 = 20

And then just apply the lemma:

number of distinct placements = (1/12)*(924 + 2 + 6 + 20 + 6 + 2) = 80

(I said 157 first time coz I made a mistake and added another 924 there :S)

 

[Affinity]

I don't think the question is based number theory as Burnside Lemma implies. the BFC is used if we assume the circle to be a 3D object, (one where we can walk around)

But in this case, its not.

all you need is a permutation group G (here, rotation) and a G-set X (here, the different arrangements, where each spot on the circle is considered distinct)

an elementary application:

how many ways can you arrange 12 people in a circle?

Rotation of 0 degrees: all 12! configurations are fixed
Other roations: none

So there are 12!/12 = 11! ways agreeing with folklore

 

[Affinity]

If anyone cares, for general n.

(D is the set of divisors of n greater than 2. The first formlua applies if n is even, the second when n is odd)