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 Post subject: Number of solved states 3x3
PostPosted: Tue Mar 05, 2013 6:47 pm 
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I was just wondering, bearing in mind a 3x3 cube has rotatable centres, how many states actually count as "solved"?

At first I thought it was 4^6, then I thought it might be half that since they always rotate in pairs of 90 degrees...

Anyone better at maths who can explain it?

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 Post subject: Re: Number of solved states 3x3
PostPosted: Tue Mar 05, 2013 7:08 pm 
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The first 5 faces can be in any 4 orientations. The last one has only 2 options. I don't understand the math behind it but I know that these are the restrictions.
So it's 4^5 * 2 = 2048

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 Post subject: Re: Number of solved states 3x3
PostPosted: Tue Mar 05, 2013 7:21 pm 
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GuiltyBystander wrote:
The first 5 faces can be in any 4 orientations. The last one has only 2 options. I don't understand the math behind it but I know that these are the restrictions.
So it's 4^5 * 2 = 2048
It also depend on rather you consider global rotations of the entire puzzle as different states. If you do then you can multiply this by 24. I remember that figure as 6*4. You can pick 1 of the 6 faces to be up. You then can pick only 1 of 4 faces to be forward.

Other then that I agree with GuiltyBystander.

Carl

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 Post subject: Re: Number of solved states 3x3
PostPosted: Tue Mar 05, 2013 8:06 pm 
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Surely there are an infinite number of solved states: solved puzzle on a table, on a chair, under this book, or that book, 2mm this way, 4cm that way, rotated by 12 degrees to the left, raised by 3 meters, 8 degrees warmer, moving at 12m/s this way, accelerating 5mm/hour that way, spinning at 6 rpm clockwise about this axis, or 5.7625548 million rpm anticlockwise about that axis, etc. 8-)

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 Post subject: Re: Number of solved states 3x3
PostPosted: Tue Mar 05, 2013 10:12 pm 
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KelvinS wrote:
Surely there are an infinite number of solved states: solved puzzle on a table, on a chair, under this book, or that book, 2mm this way, 4cm that way, rotated by 12 degrees to the left, raised by 3 meters, 8 degrees warmer, moving at 12m/s this way, accelerating 5mm/hour that way, spinning at 6 rpm clockwise about this axis, or 5.7625548 million rpm anticlockwise about that axis, etc. 8-)
Come on... do you really think Monk would consider those solved stated? To be solved it must be at rest in the dead center of a perfectly white square room. Preferably a class 1 clean room that is temperature and humidity controlled. With all the side of the cube parallel to the walls of the room. Arg!!!! 24 is not a multiple of 10... take back everything I just said. Monk would clearly just take all the stickers off and replace them with all white stickers. There... just 1 solved state. Monk is happy now.

Carl

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 Post subject: Re: Number of solved states 3x3
PostPosted: Wed Mar 06, 2013 1:40 am 
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KelvinS wrote:
Surely there are an infinite number of solved states: solved puzzle on a table, on a chair, under this book, or that book, 2mm this way, 4cm that way, rotated by 12 degrees to the left, raised by 3 meters, 8 degrees warmer, moving at 12m/s this way, accelerating 5mm/hour that way, spinning at 6 rpm clockwise about this axis, or 5.7625548 million rpm anticlockwise about that axis, etc. 8-)

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