Addendum

Discussion in 'Physics & Math' started by noodler, Oct 10, 2009.

  1. noodler Banned Banned

    Messages:
    751
    I've been itching to post this table of permutations for the 2x2x2 Rubik's "toy'. This P-space is known to be complete (it has a small algebraic height), however no table is known to have been composed for the 3x3x3 or higher-N puzzles.

    This is indirectly related to the notion of an intersection "identity"; I consider this table to show the "carry/lookahead" function in the QTM and FTM metrics. F "borrows" from Q, and Q "carries" F (across to the right-handed total). Note also that the number of intersections per carry/borrow, increases as n+1, up to n = 4 in the FTM.

    Code:
    		FTM	
     	0	1	2	3	4	5	6	7	8	9	10	11	Total
    QTM 0	1												1
        1		6											6
        2		3	24										27
        3			24	96									120
        4			6	144	384								534
        5				72	768	1,416							2,256
        6				9	564	3,608	4,788						8,969
        7					126	3,600	15,072	14,260					33,058
        8					5	1,248	18,996	57,120	36,780				114,149
        9						120	9,480	85,548	195,400	69,960			360,508
        10							1,728	55,548	341,604	487,724	43,984		930,588
        11							72	14,016	238,920	841,500	256,248	96	1,350,852
        12								1,032	56,440	455,588	268,532	944	782,536
        13								12	928	32,968	54,876	1,496	90,280
        14										8	160	108	276
    
    Total	1      9      54     321     1,847    9,992   50,136  227,536 870,072 1,887,748 623,800 2,644   3,674,160
    
    
    That is, the table says something about the symmetry.
     
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  3. noodler Banned Banned

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    751
    Well, the array isn't sparse, it's compact. (duh!)

    Is there a straightforward formula that gives the width or depth of a particular intersection, for a row or column respectively, that only selects nonzero values? Can it be written in terms of m,n the row,column metrics?
    How to account for the apparent nonlinearities (for instance the column depth is 5 in three places and the row width is 5 in five places) in what must be a multilinear set of relations?

    Ed: apologies for vague terminology like "number of intersections", which should be "number of groups intersecting the QTM, for n" as the def. for depth in the graph G, or matrix M or array A, ...whatever.
     
    Last edited: Oct 12, 2009
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  5. noodler Banned Banned

    Messages:
    751
    There is a solution to the problem of building a table for the 3x3x3 which is 6th order in polynomial time.
    http://www.math.toronto.edu/~drorbn/Talks/Mathcamp-0907/NCGE.pdf


    It uses a table of permutations as generators, so the inductive step is that a permutation table exists for the 2x2x2.
    So that Gaussian elimination and noncommutative relations in G will reveal the widths and depths.
    The rule is that the quarter turn metric is "squared" by a full turn. So the intersection at (1,1) = 6 means there are 6xQ and 6xF isometries. You can't make a single quarter turn into a squared full turn so they must be separate automorphisms. When m = 2 there are 3 isometries for a single f (at n = 1).

    Which values of n and m have intersections that include identical states, since both metrics map the entire distance |G| or have the same outer extreme?
     
    Last edited: Oct 12, 2009
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