Discussions on the mathematics of the cube

FTM Antipodes of the Edge Group

I have done my own independent breadth-first search of the edge group using the face-turn metric. I used symmetry/antisymmetry equivalence classes to reduce the number of elements in the search space. I confirm the "Unique mod M+inv" values for this group/metric that Rokicki reported in 2004.

I reduced the "coordinate space" for the search to 5022205*2048=10285475840 elements by using symmetry/antisymmetry equivalence classes of the edge permutation group. (This gives a much more compact overall coordinate space than using an edge orientation sym-coordinate, at a cost of more time required to calculate representative elements. This allowed me to keep track of reached equivalence classes with a ~1.3 GB bitvector in RAM and 5022205 KB disk files to keep track of distances.)

God's Algorithm out to 13f*

Just finished running out to a distance of 13 in the face turn metric.

First, the positions at exactly that distance:

 d   mod M + inv          mod M       positions
-- ------------- -------------- ---------------
 0             1              1               1
 1             2              2              18
 2             8              9             243
 3            48             75            3240
 4           509            934           43239
 5          6198          12077          574908
 6         80178         159131         7618438
 7       1053077        2101575       100803036

God's Algorithm out to 14q*

I've computed the count of positions out to 14 quarter turns.

First, positions at exactly the given distance:

 d  mod M + inv         mod M      positions
-- ------------ ------------- --------------
 0            1             1              1
 1            1             1             12
 2            5             5            114
 3           17            25           1068
 4          130           219          10011
 5         1031          1978          93840
 6         9393         18395         878880
 7        86183        171529        8221632
 8       802788       1601725       76843595

God's Algorithm out to 12f*

I just completed exploring all positions of the cube out to depth 12 in the face turn metric.

The first table is the count of positions with exactly the given depth.

 d  mod M + inv        mod M      positions
-- ------------ ------------ --------------
 0            1            1              1
 1            2            2             18
 2            8            9            243
 3           48           75           3240
 4          509          934          43239
 5         6198        12077         574908
 6        80178       159131        7618438
 7      1053077      2101575      100803036

Twenty-Nine QTM Moves Suffice

With 25,000 QTM cosets proved to have a distance of 25 or less,
we have shown that there are no positions that require 30 or more
quarter turns to solve. All these sets were run on my personal
machines, mostly on a new single i7 920 box.

These sets cover more than 4e16 of the total 4e19 cube positions,
when inverses and symmetries are taken into account, and no new
distance-26 position was found. This indicates that distance-26
positions are extremely rare; I conjecture the known one is the
only distance-26 position.

In order to take the step to a proof of 28, I would need a couple

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Interesting Problem/Puzzle/Game

ok here is the game:
your opponent has a secret number that is 4 digits long. the digits are 0-9 and no digit can be repeated in the number (in other words all 4 numbers are different)
examples: 1234, 1948, 4950
you have to guess the number
when you guess a number: your opponent gives you back 2 numbers (x,y)
number x is the amount of numbers in the right place
number y is the amount of numbers right but in the wrong place
anyone have an algorithm to solve this problem using the 2 numbers given back?

subgroup enumeration

I've been playing around with the Rubik's cube subgroup generated by the turns: (U U' D D' L2 R2 F2 B2) which I refer to as the D4h cube subgroup after the symmetry invariance of the generator set. This, I believe, is the subgroup employed by Kociemba in his two phase algorithm. Anyway, I have performed a partial enumeration of the subgroup and its coset space. I was wondering if anyone might be able to confirm these numbers as a check on my methodology.

Six Face/D4h Coset Enumeration (q turns)
Depth  Cosets     Total
 0         1         1
 1         4         5

Twenty-Five Random Cosets in the Quarter Turn Metric

As the next step in my exploration of the quarter turn metric, after finishing a proof of an upper bound of 30, I decided to run 25 cosets all the way (until I have optimal solutions for all positions). Unlike my runs in the half turn metric in November of 2008, I went ahead and made the search phase run in parallel. In addition, I acquired a newer, faster machine (a Dell Studio XPS with an Intel i7 920 processor).

I decided to run the same 25 cosets I ran for the half turn metric. The results are summarized in the table below.

Sequence             9 10  11  12   13    14     15      16       17        18         19         20         21         22       23

Thirty QTM Moves Suffice

With 10,114 cosets solved in the quarter turn metric, I have shown
that 30 or fewer quarter turns suffice for every Rubik's cube
position. Every coset was shown to have a bound of 25 or less,
except the single coset containing the known distance-26 position.

I also solved every coset exhibiting 4-way, 8-way, and 16-way
symmetry, and each of these also were found to have a bound of
25 or less. Thus, if there is an additional distance-26 or
greater position, it must have symmetry of only 2, 3, or 6, or
no symmetry at all. I believe, based on this, that it is likely
that on other distance-26 positions exist.