Discussions on the mathematics of the cube

Lower Bounds for n x n x n Rubik's Cubes (through n=20) in Six Metrics

In January 1981, Dan Hoey posted to cube-lovers a description of a
technique to compute a lower bound on God's Number. This technique
considered the maximum number of positions that can be reached by what
is called "syllables"---consecutive moves on the same axis, possibly
turning distinct faces. Since all the moves that make up a syllable
commute, we can select a single canonical move sequence to represent
every syllable, and then determine how many move sequences of a given
total length can be made out of only these syllables. This gives a
more accurate bound on God's Number because it eliminates many

Number of 4x4x4 positions for up to 5 moves

After hearing about this paper which talks about the size of God's number for nxnxn cubes, I was thinking about what we currently know about God's algorithm for the 4x4x4 and realized that I couldn't seem to find any partial distance distributions on the 4x4x4 on the web.

So I've run my own analyses to get the number of positions up to 5 moves from the solved state. I have done this for six metrics. First, I have "single-slice" metrics where a move is only allowed to turn a single layer. Second, I have twist metrics where a move is only allowed to twist the cube along one plane. This is sometimes called face-turn metric because a face layer (possibly along with additional adjacent layers) is (are) turned with respect to the rest of the cube. Finally I also used what I termed "block turns" where some block of one or more adjacent layers (not necessarily including a face layer) are turned with respect to the rest of the cube. For each of these, I also considered whether or not a move must be restricted to quarter-turns only.

An Alternate Universe of the Cube

I've been meaning to explore new variations on the 3x3x3 cube for a while and I think I've come up with something new.

If we consider the rotation of a 2x2x2 block as one move, say the UFR block, and call it the z-move for lack of a better name. Now all sorts of weird stuff becomes possible, e.g. (z, C_U2)^6 will generate a 5-spot pattern! The centres cycle (U,L,R,B,F) in befuddler notation.

So in this particular universe of the 3x3x3 cube we are considering 8 corner moves which move a 2x2x2 block rather than the usual 6 face moves. I won't spoil all the fun, but a 3-spot, 4-spot and 6-spot are possible and these spots patterns are quite different from what is possible in the 'normal' universe of the cube.

A Curiosity

Playing around with an optimal slice turn solver of mine, I found two turn sequences which are identical except for two turns which are swapped:

R U2 R MU2 MF L' MU' L2 B2 R MU2 MF L U2 R MU2
R U2 R MU2 MF L' MU' L2 B2 R MF MU2 L U2 R MU2

The first gives superflip and the other gives the 26 q-turn hermit position—superflip composed with a four spot pattern. I don't know what if any significance this has, but I find it remarkable.

Three Million Random Positions in the Quarter Turn and Half Turn Metric

Last year I solved one million random cubes in both the half-turn and the quarter-turn metric. Unformtately, the random number generator I used was good old |drand48()|, which is not of the highest quality. This time, I generated 3,000,000 positions using the Mersenne Twister random number generator and solved all of these with a new faster optimal solver. This is the result:
    12h 13h 14h  15h   16h    17h     18h    19h     sum
14q   -   1   2    -     -      -       -      -       3
15q   -   4  19   13     -      -       -      -      36
16q   1  11  47  124   126      -       -      -     309

NxNxN cubes in GAP

Since defining GAP definitions for large cube sizes can be very tedious, I have implemented some GAP code for defining NxNxN cubes. The main function is called GenCube and returns a group representing a cube of the size specified by the parameter n. This function has a 2nd parameter (center_ori) used for odd cubes that allows specifying whether or not you wish to have the orientation of the most central pieces on each face to be considered significant.

The code uses a face-based numbering system. The facelets on the U face are numbered 1 to n2, the L face uses numbers n2 + 1 to 2n2, and the remaining faces are similarly numbered in the order F, R, B, D. For handling orientation of the most central pieces on each face, 18 additional numbers are used, starting at 6n2 + 1.

Optimal parity fix maneuvers on the 4x4x4 cube

I recently investigated optimal reduction parity fixes on the 4x4x4 cube.

First some explanation of terms. A common strategy used in solving the 4x4x4 cube is to solve the center pieces, and then pair up the 12 pairs of edge pieces. The puzzle can then be solved like a 3x3x3 cube by turning only the face layers, except for two possible types of parity conditions that can't normally occur on the 3x3x3. These parity conditions are often called OLL parity and PLL parity (since whether or not these parity conditions are present typically isn't recognized until attempting to solve the last layer). Since the 4x4x4 is "reduced" to a pseudo-3x3x3 cube, this strategy is generally referred to as reduction.

The Void Cube in GAP

I have been amusing myself messing around with GAP and have modeled the void cube. The void cube is a standard cube with indistinguishable center cubie facelets. The void cube may be modeled by the group: < R , U , F , TR , RU , TF > , where the latter three generators are "Tier" or "Tandem" moves of a face and the adjacent middle slice. Note that the generators do not move the DBL cubie. As such, this is a fixed corner cubie model. The DBL cubie provides the necessary frame of reference which defines which face is Up, which face is Right and so forth. The tandem moves are the fixed corner cubie model counterparts of the L , D , B moves in the standard fixed center facelet model--they perform the same rearrangement of the cubies relative to one another.

PSL(2,7) embeds in the 2x2x2 cube group

I don't see any explicit reference to this but I've noticed
that the simple group PSL(2,7) occurs naturally as a subgroup
of the 2x2x2 cube group of order 3674160 (well - with a slight
amount of wilful tinkering!). This is the model in which one
of the the 8 cubelets stays fixed.

One way of seeing how it is realised is to view to view the
corner cubelets as a single block, i.e. suppose all three
elements of each corner cubelet have the same colour.

Then taking the following labellings where all of 1 could be
coloured red, all of 2 yellow, etc. (UFR refers to the the
cubelet in the "Up" "Front" "Right" position, etc).

Site URL changed

I was forced to update the URL of the site this morning. http://dyndns.org saw fit to shut down access from the homelinux.org domain and I had to scramble to switch the site over to the allowed free domain name.

The new URL is http://cubezzz.dyndns.org/drupal

Note that the numeric ip address will also work http://204.225.123.154 as I can't guarantee they won't make it necessary to switch to another service in the future as free services tend to disappear.

Let me know at cubexyz at gmail dot com if anything is broken.

Please update your links accordingly.