First page . . . Forward to page 3 . . . Last page (page 6)
Updated Table of Algorithms
This table brings together all the algorithms I have ever learned used for solving the cube. They are grouped by function in the following order: Corners, then Edges, then Centers; posiioning before orienting; "dirty" algorithms before "pure"; small cubes then larger cubes. I also list a few that I have not actually learned but believe that I should.
This table adds one extension to the notation:
Rw = Turn the whole cube 90o along the same axis as an R move.
|
2009 Aug 30
Using a full search, still without symmetry, rubik3 finds that 10s is the minimum for a pure algorithm to move 2 pairs of centers:
R2 D2 L2 B2 L2 D2 R2 F2 (8s) (moves the UF pair to R and the RF pair to U)
This is a great algorithm because it can be executed easily in the following way:
((R2Uw)2 R2Rw' F2Rw')2 (8s+8w)
This moves the FD pair onto R and the RD pair onto F, and is optimized for easy turning by a right-handed cubist.
Emulating Cuboid Puzzles
Any standard cube can be used to emulate one or more smaller cubes and/or cube-like puzzles with non-cube rectangular dimensions. For example, if you have a 4×4×4 cube and limit your motions to those that turn an outer and inner layer at the same time (R12, F12, etc) you are "emulating" a 2×2×2 cube, and all patterns you get will be identical to patterns on the 2×2×2 (assuming you started with your 4×4×4 cube solved).
Using an odd-number sized cube (which includes the 3×3×3 and 5×5×5) you can emulate smaller puzzles that have an odd number of slices in each dimension. Using an even-number sized cube (like the 4×4×4) you can emulate any equal or smaller size in each dimension. For example, the 4×4×4 can emulate a 2×3×4 puzzle.
These are discussed in order with the smallest numbers at the beginning. For example, the 1×2×2 comes before the 2×2×2, then the 2×2×3. To ensure a specific ordering, each puzzle is named with smallest numbers first, so the 2×2×3 is not called a "2×3×2" or "3x2x2" even though those names really mean the same thing.
1×1×N Puzzles
Any 1×1×N puzzle is trivial to "solve" because the individual layers do not affect each other. Some 1×1×2 and 1×1×3 puzzles have been built by avid hobbyists:
MeMyselfAndPi's 1×1×2: watch
Kenneth Brandon's 1×1×3: watch
1×2×2
This size was the smallest "two dimensional" case discussed by Singmaster([1] pp 51-52), who was considering an imaginary flat (two-dimensional) "cube" with individually-reversible rows and columns and either one side (no distinction between front and back) or two.
Here is my 1×1×2, which is simply a Rubik's Ice Cube bandaged with sticky notes:
2×2×2 emulating a 1×2×2
A paper model is easy to build: it is just two pairs of cubes with some thread; the threads are wound around each other a couple times. As you can see in this video, if you turn it far enough in one direction the pieces come back apart.
Solution: The 1×2×2 is a little less trivial than the 1×1×N cases, but still extremely easy to solve. Because there are no fixed centers, the back-left cubie can be assumed to be in the correct position. There are 6 possible combinations and two valid moves: R2 and F2. By repeating these two moves in alternation you go through all 6 combinations. Thus, solving the puzzle takes no more than 5 moves, or 3 if you know which move to start with.
1×2×3
Here are two ways to emulate the 1×2×3: Using a normal 3×3×3 cube, cover a center vertical slice with black to "ignore" it, and cover the remainder of the four sides (not the top and bottom) with 3x1 strips. To emulate the 1×2×3 without the ignored black regions, you need a 4×4×4. Here are both bandaged cubes, in an equivalent scrambled state:
two ways to emulate a 1×2×3
The picture illustrates a few different types of bandaging used to emulate these small puzzles on real cubes. The small cube uses 3→1 bandaging in the vertical axis. The large cube uses 4→1 bandaging in the vertical axis, 4→2+2→2 on the left-right axis, and 4→1+2+1→3 along the front-back axis. Notice in particular that these divisions of 4 all divide at different points, the result is that if you turn anything by 90o, you then cannot turn on another axis. The bandaged 2×2×3 behaves similarly, but see the 3×3×5 discussion for other possibilities.
Notation: Pictured are the U, F and R faces. Listed clockwise from the very back, the six pieces are called LB, RB, R, RF, LF, and L. All turns are 180o and are indicated with the superscript 2 to retain consistency with other puzzles.
The 1×2×3 has six pieces and at any point you have three choices of what to turn. There are two center or "center-edge" pieces and four corner pieces. One of the centers can be assumed to be in the correct position (and therefore, the other one is too). All 4!=24 permutations of the corners are possible, and each corner's orientation is determined by its position. In addition one of the centers can be flipped with respect to the other, giving 24×2 = 48 combinations. No position is more than 6 moves from the start.
Solution: The 1×2×3 is arguably the simplest puzzle that takes any kind of "technique" to solve:
1. If L and R are reversed, turn the whole cube around. If L is now upside down, do L2.
2. Find the piece that belongs at LF. If it is at RF, do F2. If it is at RB, do R2F2. If it is at LB, do B2R2F2.
3. Find the piece that belongs at RF. If it is at RB, do R2. If it is at LB, do B2R2.
4. If the pieces at RB and LB are reversed, do B2.
5. If the piece at R is now upside down, use the single center piece flip algorithm: R2 F2 R2 F2 R2 F2.
Here is a video of someone speedsolving a 1×2×3
1×2×7
Here is a video of a 1×2×7.
1×2×13
Here is a video of a working 1×2×13 puzzle. Photos are here (and you can even order one if you want). There are 26 pieces, most of them with long "feet" that extend all the way into the interior of the central layer.
1×3×3
As of 2009, a we1l-made commercial version is being sold from Japan, and called the "floppy cube". It is also one of the easiest to emulate with a bandaged 3×3×3:
3×3×3 emulating a 1×3×3
This video shows a 1×3×3 made out of 9 loose cubes with magnets. With practice, a magnetic "cube" can be operated as if it were a true mechanical "cube".
This size was one of the "two dimensional" cases discussed by Singmaster, who pointed out that it was equivalent to using only the turns [ L2, R2, F2, B2 ] on a 3×3×3 cube.
The center cubie and the four edge cubies don't move, just like the centers on a 3×3×3 cube, so we can assume they are already in the proper position (although the edges can flip). Each turn swaps a pair of corners and flips an edge over — so we can get any permutation of the corners, but the parity of the permutation is equal to the parity of the flips of the edges. The orientation of the corners is purely determined by their position — every time a corner moves to an adjacent spot, it flips over, so the corner is "right-side up" if it is in its home position or in the opposite corner, and "upside down" if it is in one of the other two corners. So we get 4!=24 combinations for the corners, times 24 for the orientations of the edges, divided by two for the parity constraint, for 24×16/2=192 total combinations.
No position is more than 8 moves from the solved state, and the average number of moves needed is 4.43. The "all alternating" or "checkerboard" pattern L2R2B2F2 (4f) is the closest local maximum.
Notation: We will assume the cube is being held "flat", so the Up and Down faces are the 3×3 sides.
1. Hold the puzzle so the proper Up color shows in the center square of the 3×3 face, and the proper Front color is in the middle edge facing you.
2. Get the corners into place just as you would on the 1×2×3 (see above). Start with one corner and work your way around either clockwise or counter-clockwise so you can put the next corner in place without disturbing the ones already solved.
3. Now either two or four edges are flipped over.
3a. The two edges at F and R can be flipped with R2F2
R2F2 R2F2 (6f). To flip a different pair of edges, the
same moves will work with R and F changed to any two adjacent
sides. (Note that this is an algorithm from the 1×2×3, and also serves
as a useful operator on the 2×3×3 and larger puzzles).
3b. If all four edges happen to be flipped, the 1×3×3
superflip, R2F2L2B2 F2L2 F2L2 (8f)
will set them right in 8 turns. (This happens to be the only position
on a 1×3×3 that needs 8 moves to solve).
Permutation-Only 2×2×2
This is a 2×2×2 that is limited to moves that keep two opposite colors (white and yellow in the pictured examples) on the U and D faces. In other words, L2, F2, R2 and B2 are allowed but L, F, R and B are not. Enforcing this on a normal 2×2×2 requires a complex physical modification, but it can be simulated easily with a bandaged 6×6×6 whose two middle layers are ignored:
The restricted 2×2×2 can be emulated by bandaging a 6×6×6
The permutation-only 2×2×2 is the most important subgroup of the 2×2×2 because its algorithms are applicable to many other cuboids with a similarly restricted move set, including the 2×2×3, the 2×2×4, the 2×3×3, the 3×3×4, and any other cuboid in which two (but not all three) of the dimensions are equal. It also appears as an intermediate step in the 4-stage Thistlethwaite algorithm for solving the 3×3×3.
Computer searches show that the maximum number of moves required to reach any position is 13, and the average is 8.85. Some of the 13-move states are pair swap algorithms identical to the "dedge flip" on the 2×2×4, 3×4×4, 4×4×4 and similar swaps of two edge pieces on larger cubes. Here is an example, which exchanges URF with DFR:
UR2 UF2 U2R2 UR2 D2 F2 U' R2 D (13f)
The same manipulation can be accomplished on the normal (unrestricted) 2×2×2 in 11 moves:
R' F R2 D F' D' F' R2 F' R D (11f)
Number of Combinations : Any single pair swap can be performed (see above), so the 8 pieces can be permuted into any position. There is no fixed orientation, so we can assume that one piece (say the DLB piece) is already solved, or turn the whole puzzle until it is. That leaves 7!=5040 combinations for positioning the other 7 pieces.
First page . . . Forward to page 3 . . . Last page (page 6)
Robert Munafo's home pages on HostMDS © 1996-2012 Robert P. Munafo. about contact
Google+
mrob27
@mrob_27
This work is licensed under a Creative Commons Attribution 2.5
License. Details here
s.13