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Date: Tue, 03 Jun 1997 23:57:22 -0400 (EDT)
From: Jerry Bryan
Subject: Re: Number of States for 2x2x2 cube
In-reply-to: <199706031013.GAA17406@life.ai.mit.edu>
To: Richard M Morton RMM - ICOMSOLS
Cc: cube-lovers@ai.mit.edu
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On Tue, 3 Jun 1997, Richard M Morton RMM - ICOMSOLS wrote:
>
> Reading about the number of possible states (88million) for the corners
> of the 3x3x3 cube (equiv. to 2x2x2) made me try working this out for
> myself. My logic was :
Actually, the corners of the 3x3x3 are not usually considered to be
equivalent to the 2x2x2. There are 24 times fewer states in the 2x2x2
because any of the 24 rotations in space of the 2x2x2 are considered to be
equivalent. Another way (and the typical computer way) to model this
effect is to fix one of the corners of the 2x2x2.
>
> Cube has 8 corners, each of which can have 3 orientations. Number of
> possible states is
>
> (8*3)*(7*3)*(6*3)*(5*3)*(4*3)*(3*3)*(2*3)*(1*3) = 8!*3**8 = 264,539,520
>
> This figure of course includes some states only possible by disassembling
> the cube (or maybe by twisting it in a fourth dimension ?). Without this
> the last corner can only have one orientation so the number of states
> achievable by twisting only in 3d is 8!*3**7 = 88179840
>
> I assume that this is the correct figure but what I would like to know is
> whether my logic is correct ie is the assumption about the last corner
> being fixed in orientation the only requirement (I am not a mathematician
> so please excuse me if this is obvious).
>
Your logic is correct. The larger size of 264,539,520 is the number of
corner configurations in the so-called constructable group. The
constraint on the twist of the last corner cubie gets you down to
88179840, and is the only other constraint on the corners alone. When you
add in the edge cubies, there are two additional constraints -- one for
the edge cubies alone, and one for the corner and edge cubies combined
(they have to have the same parity).
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990