https://williambader.com/museum/cubes/cubes.html
Rubik's Cube Solutions plus Puzzles and 8-Balls
Contents:
Introduction #intro
Cube Links #links
Basic Cube Group Theory #basiccubetheory
Rubik's Cube Solution #solution
Cube Solution Supplement for Patterned Faces #patternsupplement
Rubik's Revenge Solution #revengesolution
Advanced Cube Group Theory #grouptheory
My Cube and Puzzle Collection #cubefamily
[animat]
---------------------------------------------------------------------
[684racingcube]
My old racing Rubik's Cube http://www.rubiks.com is well-worn and
barely holds together. I gave it a lot of wear in June 1982 during
the week between high school finals and graduation ../../scscards/
liberty.html. A cube has 43,252,003,274,489,856,000 (over 4 X 10^19)
different positions http://www-gap.dcs.st-and.ac.uk/~gap/Intro/
rubik.html. This number is the product of 8! corner position
permutations X 12!/2 edge position permutations X 3^8/3 corner twirls
X 2^12/2 edge flips and factors as 2^27 X 3^14 X 5^3 X 7^2 X 11. If
you take a cube apart, there are 12 different ways that you can put
it together such no set of moves will take the cube from one of the
12 assemblies to the other. When I was in practice, I could solve a
scrambled cube in about a minute using about 65 moves (counting
slices as a single move). I usually do the top corners, three of the
top edges, the bottom corners, the bottom edges, the last top edge,
and then the center edges.
[685rubiksrevenge]
A Rubik's Revenge Cube http://www.rubiks.com from 1982. A Rubik's
Revenge Cube has
7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 (over 7
X 10^45) different positions http://www.geocities.com/jaapsch/puzzles
/cube4.htm. When I was in practice, it took about twice as long and
about twice as many moves to solve as the original cube.
[solarsystem]
For an idea of the size of the number of possible positions on a
Rubik's Revenge cube, note that a Rubik's Revenge cube is 2.6 inches
(6.5 cm) per side and the earth's orbit around the sun has an average
radius of 93 million miles (150 million Km) or about 2.3 X 10^12
cubes. If the sun expanded to the size of the earth's orbit, using V
= 4/3 Pi r^3, it could hold about 50 X 10^36 cubes. A giant pile with
one cube for every different position would have a radius about 500
times the radius of the earth's orbit (((7.4 X 10^45)/(50 X 10^36))^
(1/3)). For comparison, the radius of Neptune's orbit is about 30
times radius of the earth's orbit, and the radius of Pluto's orbit is
about 40 times the radius of the earth's orbit.
[001revengecubeinside]
A Rubik's Revenge Cube has a round center.
[002revengecubeparts]
Rubik's Revenge parts are fragile, especially the center pieces (on
the left). The two pieces in the middle are edges, and the two pieces
on the right are corners.
---------------------------------------------------------------------
Some cube links:
Version of this page in Belorussian http://onlinecasinospotlight.com/
web/rubiks-cube-solutions-be
Jessica Fridrich's 17 second 56 move solution http://
www.ws.binghamton.edu/fridrich/
Dan Knight's 17.3 second solution http://www.knightslab.org/#!
speedcubing/cy62
Lars Petrus's 24 second 60 move solution http://lar5.com/cube/
Lego Robot Solves Rubik's Cube Puzzle In 3.253 Seconds http://
tech.slashdot.org/story/14/03/19/0333218/
lego-robot-solves-rubiks-cube-puzzle-in-3253-seconds
How to build a LEGO Speedcuber in under 5 minutes using ARM Powered
devices https://www.youtube.com/watch?v=u7h5KSOFXSc
Robot solves cube in 12 seconds http://singularityhub.com/2010/02/17/
lego-robot-solves-any-rubiks-cube-in-less-than-12-seconds-video/
Robot solves larger cubes http://singularityhub.com/2010/04/28/
lego-robot-solves-bigger-and-harder-rubiks-cubes-video/
Android phone solves Rubik's cube in 12.5 seconds http://
deviceguru.com/android-phone-controlled-robot-solves-rubiks-cube/
This robot can solve a Rubik's Cube in 1 second http://thenextweb.com
/shareables/2016/01/25/this-robot-can-solve-a-rubiks-cube-in-1-second
/
Robot sets new Rubik's Cube record of 0.637 seconds http://
www.bbc.com/news/technology-37925028
Ben Katz and Jared Di Carlo 0.38 Second Rubik's Cube Solve http://
build-its-inprogress.blogspot.co.uk/2018/03/
the-rubiks-contraption.html
OpenAI's AI-powered robot learned how to solve a Rubik's cube
one-handed https://www.theverge.com/2019/10/15/20914575/
openai-dactyl-robotic-hand-rubiks-cube-one-handed-solve-dexterity-ai
Teenager Lucas Etter sets new fastest time to solve a Rubik's Cube
world record of 4.904 seconds http://www.guinnessworldrecords.com/
news/2015/11/
confirmed-teenager-lucas-etter-sets-new-fastest-time-to-solve-a-rubiks-cube-wor
Philip Marshall's 65 move solution http://helm.lu/cube/
MarshallPhilipp/
Christophe Goudey's CubeLand with solutions, games, contests and
links http://cubeland.free.fr/
Michael Reid's links, patterns, and optimal solver http://
www.cflmath.com/Rubik/index.html
Morley Davidson, John Dethridge, Herbert Kociemba, and Tomas Rokicki
prove that the best solution is 20 moves http://www.cube20.org/
Herbert Kociemba's CubeExplorer finds solutions of 19 moves on
average http://kociemba.org/cube.htm
DeepCubeA finds the best solution 60% of the time http://
deepcube.igb.uci.edu/
Rubik's Cube solver https://rubiks-cube-solver.com/
How hard is it to scramble Rubik's Cube? https://phys.org/news/
2020-01-hard-scramble-rubik-cube.html
Cooperman and Kunkle's 26 move solution (May 2007) http://www.ddj.com
/blog/portal/archives/2007/05/rubiks_cube_bus.html and http://
www.sciencedaily.com/releases/2007/05/070531131326.htm and http://
www.yubanet.com/artman/publish/article_58110.shtml
Jaap's Puzzle Page http://www.jaapsch.net/puzzles/compcube.htm
Tomas Rokicki's 25 move solution (March 2008) http://arxiv.org/abs/
0803.3435 http://arxivblog.com/?p=332
Demaine, Demaine, Eisenstat, Lubiw, and Winslow's Algorithms for
Solving Rubik's Cubes (June 2011) http://arxiv.org/abs/1106.5736
A deep learning technique to solve Rubik's cube (Feb 2021) https://
techxplore.com/news/2021-02-deep-technique-rubik-cube-problems.html
Matthew Monroe's page http://jjorg.chem.unc.edu/personal/monroe/cube/
rubik.html
Josef "Gloom" Jelinek's and Dr. Hana M. Bizek's solution page http://
cube.misto.cz/
Mark Jeays' solution page http://jeays.net/rubiks.htm
Diego de Pereda's solution page http://thecube.guru/
rubiks-cube-solution/
R. A. Hoffman's graphic solution http://www.mts.net/~rhoffman/
index.htm
Peter's Cube Page http://www.speedcubing.com/peter/
Chris Hardwick's Cube Page http://www.speedcubing.com/chris/
Cubeman's page http://cubeman.vg-network.com/ http://cubeman.org/
Ryan Heise's Cube Page http://www.ryanheise.com/cube/
PuzzleSolver.com http://www.puzzlesolver.com/
Solution Guide for Beginners http://www.rubiksplace.com/
Solving the Rubik's Cube Without Human Knowledge https://arxiv.org/
abs/1805.07470
Jim Cranwell's NxNxN cube solution page http://www.geocities.com/
xulfrepus/rubik/cube.html
4D cube http://www.superliminal.com/cube/cube.htm
Werner Randelshofer's disassembled cube and pattern collection http:/
/www.randelshofer.ch/
Joshua's Virtual Puzzle Museum http://www.twistypuzzles.com/
W. D. Joyner's cube theory lecture notes http://web.usna.navy.mil/
~wdj/rubik_nts.htm
Jim Mahoney's Group Theory & Rubik's Cube lecture notes http://
www.marlboro.edu/~mahoney/courses/Spr00/rubik.html
Speed Cubing page http://www.speedcubing.com/
Gilles and his cubes http://grrroux.free.fr/
Jasmine's Speed Cuber Blog http://speedcuber.blogspot.com/
Georges Helm's page with a 600-item bibliography http://
webplaza.pt.lu/geohelm/myweb/cubeold.htm
Official Rubik's Cube web site http://www.rubikscube.com/
Meffert's Puzzles sells various type of cubes and puzzles http://
www.mefferts.com/
Magic Cube 4D http://www.superliminal.com/cube/cube.htm
Wikipedia has a cube overview http://en.wikipedia.org/wiki/
Rubik%27s_Cube and optimal solutions http://en.wikipedia.org/wiki/
Optimal_solutions_for_Rubik%27s_Cube
Yahoo Speed Solving Group http://games.groups.yahoo.com/group/
speedsolvingrubikscube/
Discussions on the mathematics of the cube http://
cubezzz.homelinux.org/drupal/
Solving the Rubik's Cube Optimally is NP-complete http://xxx.lanl.gov
/abs/1706.06708
The most complicated yet elegant artifact http://
shl.stanford.edu:3455/TenThings/417
The World's Largest Free Standing Rubik's Cube https://
www.youtube.com/watch?v=1EUDKvShiT8
Children of the Cube (Aug 2018) https://www.nytimes.com/2018/08/15/
sports/cubing-usa-nationals-max-park.html
The Amazing Math Inside the Rubik's Cube (Dec 2019) https://
www.popularmechanics.com/science/math/a30244043/solve-rubiks-cube/
A Rubik's microfluidic cube (Jun 2020) https://phys.org/news/
2020-06-rubik-microfluidic-cube.html
J Perm Attempts to Solve the Biggest Rubik's Cube in the World
21x21x21 (Oct 2021) https://www.youtube.com/watch?v=CoAeCy1R6Qo
A.I. Solves a 55 x 55 x 55 Rubik's Cube (Dec 2018) https://
www.youtube.com/watch?v=f9smvQ5fc7Q
Speedcuber solves Rubik's Cube in 3 seconds, sets new world record
(Jun 2023) https://www.yahoo.com/lifestyle/
watch-speedcuber-solves-rubik-cube-220842202.html
---------------------------------------------------------------------
Some basic cube theory:
A 3x3x3 cube has 27 pieces: 8 corners with three external sides, 12
edges with two external sides, 6 centers with one external side, and
the middle with no external sides.
Cube moves work differently than arithmetic. On a calculator, if you
add 1, then add 2, then subtract 1, and then subtract 2, you go back
to 0 where you started. On a solved cube, if you make a move, then
make a second move, then do the first move in reverse, and then do
the second move in reverse, the cube will usually end up messed up.
Order matters on a cube so that 1 plus 2 is often not the same as 2
plus 1. In group theory terms, two elements commute if they produce
the same result when applied in either order. A group is Abelian if
every pair of elements in the group commutes. The group formed by
addition on the set of integers is Abelian, while the group formed by
moves on a cube is non-Abelian.
What helps in finding solutions is that 1 plus 2 is often close
enough to 2 plus 1 that the result of 1 plus 2 minus 1 minus 2 leaves
only a few pieces out of place. Sequences of moves that leave only a
few pieces out of places are building blocks for solving the cube. In
group theory, moves of the format X Y X-in-reverse Y-in-reverse are
called commutators. I will use the non-standard notation X * Y for X
Y X' Y'. The usual group theory notation is (X,Y) = X' Y' X Y.
If you repeat any sequence of moves, you will eventually return to
your starting position. Any move affects only a fixed number of
pieces, and those pieces have a fixed number of different
permutations. Once you repeat the move more times than the number of
permutations, you must return to a permutation that you saw before.
Repeating a sequence of moves often produces cube patterns with only
a few cubes out of place, especially at repeat counts that evenly
divide the number of times that you can repeat the move before
returning to the initial position. These sequences are not good for
fast solutions because they can be long and because there are often
more direct ways to reach the positions, but they are good for
exploring when you are starting out. In group theory, the sequence of
moves that you repeated is a generator for the subgroup of the moves
reachable by repeating that sequence of moves. The subgroup is also
called the orbit of the element that you used as the generator. The
number of times that you can repeat the move until you reach the
starting position is called the order of the subgroup.
Once you find a move sequence that shifts a piece from one position
to another position, you can use it to shift other pieces. You first
do a setup move to place the piece that you want to move into the
position that your move touches, then you do your move, and then you
undo the setup move. It does not matter how much your setup move or
your shifting move scrambles the cube, except that your setup move
should not leave any solved pieces in positions that your shifting
move will scramble. Moves of the format X Y X-in-reverse are called
conjugate moves. I will use the non-standard notation X + Y for X Y
X'.
To continue to more advanced cube group theory, see #grouptheory.
For an introduction to cube group theory, see http://
members.tripod.com/~dogschool/.
For some basic hints, see http://remus.rutgers.edu/~rhoads/Fun/
Rubik.hints.
When you are starting, if you are lost, it is OK to start over by
taking the cube apart as long as you are careful, but keep in mind
that each time you take the cube apart, it will get a little looser
until it no longer stays together. If you feel the need to play a
trick on someone, you can put a cube together in the wrong way if you
must, but never peel off a sticker -- they are glued on and not
sticky underneath.
No one has yet found the shortest solution, but it is probably in the
low 20's of moves. This means in reverse, that it takes at least 20
moves to mess up a cube completely. When someone who hasn't played
with cubes much messes up a cube for me to solve, they often do not
make enough moves, or enough of the right moves, to scramble the cube
completely. If you are careful when solving the first face, some of
other faces will sometimes solve themselves.
The original cube had the pairs of opposite faces as white and blue,
green and yellow, and red and orange. You should note the layout of
your cube before trying to solve it.
There is no magic secret to solving the cube quickly. Whatever system
you use, the more moves that you memorize, the faster you can work.
The fastest systems require memorizing several hundred moves, plus
their inverses and reflections.
You also need fast fingers. A friend in high school who could solve
cubes in about 24 seconds had a move where he could flick the right
and upper faces in a single motion, although he would regularly
explode his cube.
Avoid cubes with hollowed-out corner pieces. The edges around the
holes can catch on other pieces as you make moves. Better cubes have
smooth corners without large mold marks. You can sometimes file down
excessive mold marks.
I have never found a good cube lubricant. Avoid petroleum products --
they dissolve and soften plastic. Graphite powder works, but it
leaves messy black dust, does not last long, and makes a gummy goo if
it mixes with any liquid lubricant. Some people say that
silicone-based lubricants work well. The cube itself makes a fine
dust of plastic powder as the pieces rub against each other when you
use the cube. You can see some cube dust on the picture of the broken
Rubik's Revenge Cube above. This cube dust acts as a lubricant when
it is dry, but it can get gummy if it mixes with a liquid lubricant.
My Rubik's Cube moves:
My notation is the standard U/D/R/L/F/B for a clockwise quarter turn
of the outer slice of the face, U', etc., for a counter-clockwise
quarter turn, and Us, etc., for a clockwise quarter turn of the
middle slice with respect to the face.
An anti-slice is combination move of a face and the face opposite to
it both clockwise or both counter-clockwise. The anti-slice Ua is UD,
Ra is RL, Fa is FB, etc.
Read the moves from left to right, so that X Y means to do X and then
Y. I sometimes add parentheses or spaces for readability or to help
with memorization. The parentheses and spaces do not correspond to
moves.
I also use some additional notation:
X * Y = X Y X' Y'
X + Y = X Y X'
X ^ 2 = X2 = X X
X ^ 3 = X3 = X X X
X ^ 4 = X4 = X X X X
X ^ n = repeat X an arbitrary number of times
This more compact notation shows the beauty and simplicity of the
moves. It also helps to show how the moves work.
Repeat counts and inverses (') bind tighter than * and + so (X+Y') =
(X+(Y')) = X Y' X'.
I sometimes temporarily shift the position of the cube in my hands as
I make moves, especially for moves containing Fs. Some people have a
notation for this (for example, x, y, and z for clockwise rotation
from R, U, and F), but adding the shifting notation would make the
move sequences longer without helping people who hold the cube
differently than I hold it.
If you actually read these moves and find a mistake, please let me
know. My email address is at the bottom of my home page ../../
index.html.
Getting started:
I do the top corners and three of the four top edges first. I do not
spend too much time deciding the first face to solve unless I am
counting moves. To avoid wasting time, I usually pick the white or
blue face, whichever is closest to the top when I first pick up the
cube, unless I happen to notice that one face happens to have two or
more correct pieces than the face that I would have picked.
Most of the web sites listed above have detailed instructions for
solving the top layer.
Top corners:
Hold the cube so that the upper face has an unsolved corner in the
UFR position. Rotate the bottom face so that the corner that belongs
at UFR is at DFR. Find the position of the color of the upper face on
the corner at DFR.
If it is at D, use R'D2R DR'D'R = (R'+D2)(D*R').
If it is at F, use FDF' = F+D.
If it is at R, use R'D'R = R'+D'.
If all of the remaining corners for the upper face are on the upper
face, but in the wrong position, use one of these moves to shift one
of the incorrect corners from the upper face to the bottom face. When
you shift the corner to the bottom, try to avoid leaving the side
with the color of the upper face on the bottom face.
You can sometimes place the first few corners in a single move with R
or R2, etc., if the adjacent edges and corners are not yet
positioned.
Top edges:
Hold the cube so that the upper face has the unsolved edge in the UF
position.
If the edge that belongs at UF is on the middle slice (between the
upper and bottom faces), rotate the middle slice to place the edge in
the FR position and find the position of the color of the upper face
on the FR edge.
If it is at F, use UL'Us2L = U(L'+Us2).
If it is at R, use ULUsL' = U(L+Us).
If the edge that belongs at UF is on the bottom face, rotate the
bottom face to place the edge in the F position and find the position
of the color of the upper face on the FD edge.
If it is at F, use D'Rs'DRs = D'*Rs'.
If it is at D, use Rs' D2 Rs = Rs'+D2.
If the edge that belongs at UF is on the upper face in the UF
position with the color of the upper face on F, use F'UsF2Us2F' =
(F'+Us)(F+Us2).
If the edge that belongs at UF is on the upper face in the UB
position with the color of the upper face on U, use Rs2 D2 Rs2 =
Rs2+D2.
If all of the remaining edges for the upper face are on the upper
face, but in the wrong position, move the edge that belongs in UF to
RB or LB with R if it is at UR, B or B' if it is at UB, or L' if it
is at UL. Then use F or F', depending on the alignment of the edge,
and then Us, Us2, or Us', depending on the position of the edge, to
slide it into the slot on the F face, and then restore F and the
other face.
Remember that you need to place only three of the top four edges. The
fourth edge will be done in a later step.
Positioning the corners:
Next I turn the cube over, so that the almost-solved face is at the
bottom and the unsolved face is at the top. Check the positions of
the corners. You should be able to turn the top face so that either
all of the corners are in place, only the two corners at the back are
exchanged, or both the two corners at the front and the two corners
at the back are exchanged.
If the two corners in the back are exchanged (4 in 6 chance), use
FU'B'UF'U'B U2.
If both pairs of corners are exchanged (1 in 6 chance), use F URU'R'
F' = F+(U*R).
Orienting the corners:
Next check the orientation of the corners. Either all of the corners
will be correct, or two corners will be twirled with one corner
clockwise and the other corner counter-clockwise, or three corners
will be twirled either all clockwise or all counter-clockwise, or all
four corners will be twirled with two corners clockwise and the other
two corners counter-clockwise.
If only two corners are not oriented correctly (12 in 27 chance),
rotate the upper face so that a bad corner is in the FR position and
the color of its F face should be on the top. Use the move R'DR FDF'
then rotate the upper face so that the second corner is in the FR
position and finish with the move FD'F' R'D'R. The second half is the
inverse of the first half. The entire move is ((R'+D)(F+D))*(U^n).
If three corners are not oriented correctly (8 in 27 chance), place
the correctly oriented corner in the FL position and look at the
corner in the FR position.
If the R face of the FR corner has the color that should be on top,
use the move U2 F'U'FU'F' U2 F.
If the F face of the FR corner has the color that should be on top,
use the move F' U2 FUF'UF U2. This is just the previous move in
reverse.
If all four corners are not oriented correctly, rotate the top face
so that the F face of the FL corner and the B face of the BL corner
have the color that should go on top. Then look at the FR corner.
If the R face of the FR corner has the color that should be on top (4
in 27 chance), use the move F URU'R' URU'R' F' = F+((U*R)^2).
If the F face of the FR corner has the color that should be on top (2
in 27 chance), use the move R2U2 R U2R2 U = (R2U2+R)U.
If any other combinations appear, someone took your cube apart and
did not put it back together correctly. You will need to pop it open
to orient the corners.
Positioning the top and bottom edges:
Orient the cube so that the first face with all but one solved edge
is at the left and the second face with only the four solved corners
is at the right. From here on, the middle slice refers to the slice
between the left and right faces.
Find an edge in the middle slice that belongs on the right face. Turn
the middle slice so that the edge is on the bottom face, turn the
left face so that the single unsolved edge is at the top, and turn
the right face so that the position where the edge on the middle
slice belongs is at the top. Turn the upper face a quarter-turn
clockwise or counter-clockwise, depending on the orientation of the
edge, rotate the middle slice to place the edge in position, and then
turn the upper face back.
If the remaining edges for the right face are all in incorrect
positions on the right face, rotate the right face to place one of
the incorrect edges at the top and then use U Rs' U' to push it into
the middle slice (or U Rs U' to push it into the left face if it has
the correct orientation).
Very rarely, if all of the R face edges are on the R face but in
incorrect positions, you can temporarily turn the cube to make the R
face into the U face and use moves as if you were solving the cube
top-middle-bottom. You can do a three-face cycle with F2 D (R2 Fs U2
Fs') D' F2 (cycles UF to UR, UR to UL, UL to UF), a three-face cycle
with a flip with Rs' D (R2 Fs U2 Fs') D' Rs (same cycle but flips UR
and UF), an adjacent edge-pair exchange with Fs2 D' (F2 Rs2 F2 Rs2) D
Fs2 (exchanges UF with UR, UL with UB), an opposite edge-pair
exchange with Fs2 D Rs2 D2 Rs2 D Fs2 (exchanges UF with UB, UL with
UR), etc.
To place the last edge in the left face, instead of using a middle
slice, use a three-edge cycle like F2 Rs F2 Rs' to avoid disturbing
the right face. For example, to place DF in UL, use U' F2 Rs F2 Rs' U
= U'+(F2*Rs).
If the last edge of the left face ends up in the last position of the
right edge, you can place both edges at once if you line up the
middle slice so that you need only a quarter turn. It is probably
faster to do this even when one of the edges ends up with the wrong
orientation. You can fix it later.
If the last left edge ends up in the last right position, and the
last right edge ends up in the last left position, you can exchange
them with U Rs2 U2 Rs2 U = U+(Rs2*U2).
Positioning the middle slice edges:
Now, only the middle slice is left. Orient the cube so that the two
solved faces are at the left and right.
Place the edges in the correct position.
If three edges are out of position (8 in 12 chance), hold the cube so
that the FD edge belongs in UB, the UB edge belongs in UF, and the UF
edge belongs in FD, and then use F2 Rs' F2 Rs = F2*Rs'.
If four edges are out of position (3 in 12 chance), hold the cube so
that UF and UB need to be exchanged, and DF and DB need to be
exchanged, and then use D2 Rs2 D2 Rs2 = D2*Rs2.
If two edges are out of position, someone took your cube apart and
put it together incorrectly. You might as well pop the cube apart
now.
If one edge is out of position, someone exchanged a piece with
another cube. You will need to find another cube and swap pieces.
Orienting the middle slice edges:
Now, either the cube will solved, or two edges will be flipped, or
all four slice edges will be flipped.
If only two edges are flipped (6 in 8 chance), temporarily reorient
the cube so that the solved faces are at the front and back and one
of the flipped edges is at UR and the other is at UL, then use the
move R'UsR2Us2R' U2 RUs2R2Us'R U2 = ((R'+Us)(R+Us2))*(U2). If the
second flipped edge is at DL instead of UL, use L2 before and after
the move.
If all four edges are flipped (1 in 8 chance), use the move F'L'F
(RsU RsU RsU RsU) F'LF = (F'+L)+((RsU)^4).
If only one edge is flipped, someone has messed with your cube, and
you will need to take your cube apart to fix it.
When I am solving the final edge of the top face, if it happens to be
in position but flipped, I leave it and then orient it in this step
along with the middle slice. If I end up with three flipped edges in
the middle slice, I place the flipped edge from the previous step at
UL, and I rotate the middle slice so that the single correctly
flipped edge is at UF, and then I use the move RsU RsU RsU RsU.
Faster solutions:
One of the faster methods is to solve the top edges, then the top
corners and middle edges at the same time, and then the bottom. The
moves to position a top corner in combination with the middle edge
below it are not much more complicated than just positioning the top
corner. Finishing the bottom requires four steps: positioning and
orienting the edges and positioning and orienting the corners. Faster
solutions pair these into two steps that either position and then
orient everything at once or that solve the edges and then solve the
corners at once. In the easier, but less efficient, procedure that I
use, my moves to position the bottom corners also twist them, and
most of my moves for positioning and orienting corners also move the
edges. You can take advantage of this by tracking the other pieces
that the moves affect and coming up with similar moves for all of the
cases.
Centers on Rubik's Cubes with patterned faces:
To solve a cube with pictures on the faces, first solve the cube
without worrying about the pictures. After you follow the solution
for a plain cube, everything should be in place, except that some
centers might be rotated.
Find two centers that need opposite rotations. If all of the centers
are rotated in the same direction, pick any two centers. Make the
pattern called "dots" or "boxes" to place one center inside the face
of the other center, then rotate that face to solve the center (if
the center was rotated clockwise initially, rotate the face
counter-clockwise), then undo the "dots" and unrotate the face. I
usually start by placing the face with the center under consideration
at the front of the cube and make "dots" by slicing that center
through the top position.
Use 6 dots for adjacent centers on F and U, (Rs Bs Ls Fs) U^n (Fs'
Ls' Bs' Rs') U^n' = (Rs Bs Ls Fs)*(U^n).
Use 4 dots for opposite centers on F and B, (Rs Bs2 Ls Fs2) B^n (Fs2
Ls' Bs2 Rs') B^n' = (Rs Bs2 Ls Fs2)*(B^n).
After repeating this procedure, you will either solve the cube or end
up with only one rotated center. The rotated center will always be
upside down. To solve it, hold the cube so that the face with the
rotated center is on the top, and then make the move (RL U2 R'L') U
(RL U2 R'L') U = ((Ra+U2)U)^2 (where U/R/L means rotating the upper/
right/left face one-quarter turn clockwise, U2 means rotating the
upper face one-half turn, R' means rotating the right face
counter-clockwise, and Ra means the right anti-slice RL).
---------------------------------------------------------------------
My Rubik's Revenge moves:
My notation is the standard U/D/R/L/F/B for a clockwise quarter turn
of the outer slice of the face, Us, etc., for a clockwise quarter
turn of the inner slice of the face, and U', etc., for a
counter-clockwise quarter turn.
Note that slices on opposite faces like Us and Ds' refer to the same
slice on a Rubik's Cube but to different slices on a Rubik's Revenge.
Some people use u for Us, etc., or U3 or U-1 or U- for U', etc.
I sometimes add parentheses or spaces for readability or to help with
memorization. The parentheses and spaces do not correspond to moves.
Be careful moving the cube. When I try to go fast, I often end up
jamming the cube or knocking out a piece. The movements are in my
fingers, not in my head, and require no conscious thought. When the
cube jams, it is hard to restart a sequence of moves from the middle.
After each move, I square-off the cube to keep the faces in
alignment. This helps to keep the cube from jamming.
To reassemble an exploded cube, first do the centers, then build out
from a corner, and do the final edges last. The centers have a thin
leg that is easy to break. To pop in the last edge, it helps to turn
the face 45 degrees. I have gone through a few different Rubik's
Revenge cubes. The centers often break on the cubes with stiffer
plastic, while the cubes with softer plastic often become too loose
and fall apart.
I do the same top, bottom, middle routine as on a 3x3 cube, except
that I also do the top and bottom centers along with the top and
bottom corners and edges.
Unlike a 3x3 cube, you can not flip edges. The position of an edge
determines its orientation. If you take a cube apart, you can see
that the edges are rounded so that it is physically not possible to
pop out a single edge, flip it upside down, and reinsert it. If an
edge appears to be flipped, what really happened is that you are
looking at the edge's pair that has the same colors but the opposite
orientation. As a side note, unlike a 3x3 cube, it is possible to
exchange any two edges. (In fact, I have the move sequence below as
part of the solution.) If you exchange two adjacent edges, you can
make a position that resembles the impossible single flipped edge on
a 3x3 cube.
Top and bottom centers:
I first do the top and bottom centers. You can push the first few
centers into position easily if you do two adjacent top centers, then
do the two adjacent bottom centers across from the top centers (for
example, if the top centers are at LF and LB, do the bottom centers
at RF and RB), then turn the top or bottom face a half-turn, and do
two more centers. Usually some centers are on diagonals instead of
adjacent, and you won't get lucky enough to place all of the centers
easily.
For the last few centers, find a center on the middle slice (i.e. Us
or Ds) and a hole on the U face where the center belongs. Rotate Us
or Ds to bring the center onto the F face, and then rotate the F face
to bring the center to LD. Rotate the U face to bring the hole to RB.
Then position the center with Rs' F' Rs.
You can reduce the number of setup moves by using reflections and
inverses of this move:
(The center is always on the F face, and the hole is always on the U
face.)
If the center is at LD, place the hole at RB, and use Rs' F' Rs.
If the center is at LU, place the hole at RF, and use Rs' F Rs.
If the center is at RU, place the hole at LF, and use Ls F' Ls'.
If the center is at RD, place the hole at LB, and use Ls F Ls'.
If two centers are at LU and LD, and you can place two holes at RF
and RB, use Rs' F2 Rs.
If two centers are at RU and RD, and you can place two holes at LF
and LB, use Ls F2 Ls'.
After you finish the top center, or if you ran out of top center
pieces because some centers are on the bottom, flip the cube to bring
the bottom to the top and use the same procedure for positioning
centers.
You should be one with your cube and know all of the pairs of
opposite colors so that you can place the correct color on the
bottom. If you are not sure, check the colors on the corners. If you
do the corners first, the centers take an extra move to position
because you must follow Rs' F' Rs with F, and the same for the other
moves.
Corners:
Do the top and bottom corners the same as for a normal cube.
Top and bottom edges:
Do the top and bottom edges the same as for a normal cube. There are
twice as many edges, but the moves are the same. When you finish the
bottom edges, you can sometimes slide in both edges at the same time.
While you are sliding in edges, if you are only sliding in one edge
into the target face, but the adjacent position in the target face
has an incorrectly positioned edge for the face, you can do a double
slice to shift the incorrect edge to a center slice with the same
move that you position the first edge.
It sometimes helps to do the last top and bottom edges together.
Place one of the top edges in the bottom, and place the bottom edge
in a middle slice, so that when you slice the middle edge into the
bottom, the edge in the bottom goes into position on the top.
Left slice:
After I finish the top and bottom, I reorient the cube so that the
solved faces are at the right and left. I work on the left slice
first and pick the orientation that gives me the most pieces in
correct positions. To do the first three left slice edges, I orient
the cube so that the first good edge piece (if present) is at the
upper front of the Ls and the second good edge (if present) is at the
upper back of the Ls. I then rotate the Rs so that the edge piece
that belongs at the lower front of the Ls is at the upper back
position of the Rs with the color of the front face showing on its
upper face. The move is then D2 Rs D2. If the edge that you need is
on the Rs, depending on how far you are, you can either do this move
in reverse to kick an edge from the Ls to the Rs, or you can rotate
the Ls for a better starting position, or you can reorient the left
and right sides of the cube in your hands.
To place the final Ls edge, orient the cube so that the open position
on the Ls is at the upper front, and then rotate the Rs so that the
piece for that position is at the upper back. The move is then (D2 Rs
D2 Rs') U2 (Rs D2 Rs' D2) U2 = (D2*Rs)*(U2). To make the move, hold
the cube with your left hand high on the F and B faces, and do the
moves with your right hand. To help remember the move, note that it
starts D2 Rs the same as move for the other edges and has a simple
formula in commutator notation. I must admit here that I have
forgotten and rederived this move several times because in 1984, my
boss threatened to glue my cube together if he saw it one more time,
and as I write this in 2002, I've played with cubes less than once a
year since. Hopefully now, I will be able to find the move on this
page whenever I need it.
Right slice:
To do the Rs, I place the two pieces that I want to exchange on the
upper face and use the move F2 (D2 Rs D2 Rs') U2 (Rs D2 Rs' D2) U2 F2
Rs. This move is just the previous move for the final Ls edge wrapped
by F2.
If the two pieces are at diagonals, you can either do the move above
twice or place the two edges at UB and FD and use a setup sequence
like (D' Rs' D Rs) before the move above and (Rs' D' Rs D) after, or
(D'*Rs')+((F2+((D2*Rs)*(U2)))Rs) in total.
Centers:
Now only the centers are out of place. The setup moves will usually
require rotating faces to place pieces in position for the moves.
If two opposite faces have two adjacent centers that it would
correctly position at least two centers to exchange, place the
centers on the Ls of the U and D faces and use Ls2 Fs2 Ls2 Fs2 =
Ls2*Fs2.
If two adjacent faces have centers that need to be exchanged, orient
the cube with the faces at the L and B. Rotate the faces so that the
L center is nearest U and F faces, and the B center is nearest the U
and R faces, and then make the move B2 (Rs' Fs Rs) B2 (Rs' Fs' Rs) =
B2*(Rs'+Fs). You can do four pieces at a time by exchanging the Fs
with FsBs' and Fs' with Fs'Bs.
If three faces have centers that need to be cycled, orient the cube
so that the center in the F face needs to go to the L face, the
center on the L face needs to go to the B face, and the center on the
B face needs to go to the F face. Rotate the faces so that the F
center is nearest the R and D faces, the L center is nearest the F
and U faces, and the B center is nearest the D and L faces, and then
make the setup move Rs2 and then the previous move for two faces. You
can see now why it helps to start the previous move with an extra B2.
As with the previous move, you can do six pieces at a time by
exchanging Fs with FsBs'. When I make the setup moves, I usually set
up the F and B faces and then the L face. After doing the exchange, I
let the colors of the faces guide me in undoing the setup moves.
If you end up with the last two centers on opposite faces, do a three
center exchange to move the last two centers to adjacent faces, and
then do a two center exchange. To reduce the chances of ending up
this way, you should work on centers that are on the face opposite to
where they belong by using the three center exchange before working
on centers that are on the face adjacent to where they belong. If the
centers are positioned so that a three center exchange will solve at
least two centers without leaving a center opposite to its face, the
exchange is probably worth doing.
Other solutions:
It is possible to solve a Rubik's Revenge by positioning the centers
and pairing the edges and then treating it like a Rubik's Cube. The
tricky part about pairing the edges is that you can end up with the
equivalent of the impossible single flipped edge on a Rubik's Cube,
and then you have to start over again.
It is also possible to solve a Rubik's Revenge by layers as
top-middle-bottom, but there are more middle edges than on a Rubik's
Cube, and it is hard to move edges without scrambling centers. Also,
the final layer has more possible edge permutations than a Rubik's
Cube.
Notice that the last few steps of both cubes require only a few move
sequences, although the move sequences are relatively long. Since you
need to look at the cube for just a second to decide what move to
apply next, it is possible to solve a cube while having a
conversation with someone and making eye contact with them and not
the cube. This often surprises non-cubists, although I guess after
writing this, I can't claim to just have been lucky when I find a
scrambled, dusty cube at a yard sale or on someone's bookshelf and
accidentally solve it.
---------------------------------------------------------------------
More group theory:
For some optional additional group theory (which is OK to skip now if
you promise to come back later), define a group G as a set with an
associative binary operator, with an identity element, and with all
of its elements invertible. Groups are usually written in uppercase
letters like "G" and "H", while elements of groups are usually
written in lowercase letters like "g" and "h". The operator is often
written in a product format (and assumed between adjacent elements),
and the identity element is often written as '1' (but sometimes as
'0' for groups that are written in additive format, and sometimes as
'I' for groups that are transforms). Associative means that x*(y*z)=
(x*y)*z, for example, in multiplication over the set of integers, 2*
(3*4)=(2*3)*4=24. Lie and Jordan algebras have non-associative
multiplication and are sometimes used in topology. An element g in G
is invertible if there is a g' in G such that g*(g')=1. I will
usually write g' for the inverse of g, but the inverse can also be
written as g^-1 or -g. A ' on a group instead of on an element as in
G' means another group, often somehow derived from G, according to
context.
In the set of integers over addition, the identity element is 0, and
the inverse of every integer is its negation, for example, 3+(-3)=0.
A group is Abelian (or commutative) if a*b=b*a for all elements a and
b in the group. Addition over the set of integers is commutative, for
example, 3+5=5+3=8.
For a group G, define the commutator subgroup G' as the subgroup of G
generated by all of the commutators (g,h) = g'h'gh where g and h are
in G.
A subgroup K is defined as a normal subgroup of G if g'kg is in K for
every g in G and k in K.
If K is a normal subgroup of G, then K' is also a normal subgroup of
G. For example, if f is a homomorphism from G to K, f(g,h) = f
(g'h'gh) = f(g')f(h')f(g)f(h) = (f(g),f(h)), so f(G') is contained in
K'. If f covers all of K, every commutator (f(g),f(h)) is in f(G')
and f(G') = K'. For a in G and k = (g,h) in K, a'f(k)a = a'f(g'h'gh)a
= a'f(g')f(h')f(g)f(h)a = (a'f(g')a)(a'f(h')a)(a'f(g)a)(a'f(h)a) =
(a'f(g')a,a'f(h)a) which is in K', so K' is normal.
G/K is defined as the quotient group of G relative to the normal
subgroup K. Elements in G/K are equivalence classes of G. Each
element g of G maps to the element gK in G/K where gK is the set of
all elements of the form gk where g is in G and k is in K. For
example, the group of addition of hours on a 12-hour clock has the
form Z/12Z, where Z is the set of integers or 0, 1, -1, 2, -2, etc.,
and 12Z is the set of all integral multiples of 12. Z/nZ is sometimes
written as Zn. For example, Z12 is short for Z/12Z.
| S | is defined as the number of elements in the set S. For example,
|Z12| is 12.
For the elements g1, g2, ..., gn in G, defines the
subgroup generated by all of the products of powers of g1, g2, ...,
gn. Each of the gi's are called generators. For example,
, the set of moves generated by quarter-turns on the six
faces of a cube, generates all of the possible moves on a cube.
If a group of n items can be generated by powers of a single
generator, the group is Abelian, and it is isomorphic to the cyclic
group of n items. For example, Z12 can be generated by 1. Z12 is
Abelian, and it is isomorphic to the cyclic group of 12 items.
Two numbers that do not share any common factors are called
relatively prime or coprime. The probability that two numbers are
coprime is related to the Riemann zeta function http://
www.cacr.caltech.edu/~roy/upi/coprime.html.
If a finite cyclic group of n items has a generator, it has phi(n)
elements that are generators, where phi(n) is the Euler phi-function,
which is the number of positive integers between 1 and n that do not
share any factors with n. If n is written as a unique product of
prime numbers p1^i1 X p2^i2 X ... X pN^iN, phi(n) is (p1^i1 - p1^
(i1-1)) X (p2^i2 - p2^(i2-1)) X ... X (pN^iN - pN^(iN-1)). For
example, 12 = 4 * 3 = 2^2 * 3^1, so phi(12) = (2^2 - 2^1) * (3^1 - 3^
0) = (4-2)*(3-1) = 2*2 = 4. This means that Z12 has exactly four
elements that are generators. The generators are 1, 5, 7, and 11. The
group generated by clockwise turns of the right face of a cube, ,
is cyclic with four elements, R, R2, R3(=R'), and R4(=I). phi(4) = 2^
2-2^1 = 4-2 = 2, so {R,R2,R3,I} has two elements that are generators.
One is R, and the other is R3. Someone who is left-handed could
express cube moves in terms of instead of
. R2 has order 2 because has only two elements, R2
and I. is a subgroup of . The number of elements in a
subgroup always divides evenly into the number of elements of its
group. The cube group contains some larger cyclic subgroups. For
example, has order 105. Each RU move cycles 7 edges, 5 corners,
and twists 1 corner, so the period of the entire move is the least
common multiple, 7 * 5 * 3 = 105. has order 42 because each RU'
move cycles 7 edges and 6 corners.
A function f that maps elements from group G to group H such that for
all a and b in G, f(ab) = f(a) * f(b) is called a homomorphism. If f
maps from G into itself, it is called an endomorphism. If f maps
1-to-1, it is called an isomorphism. If f maps 1-to-1 from G into
itself, it is called an automorphism.
Consider a function that maps i in Z12 to R^i on a cube. This map
takes 0,1,2,3,4,5,6,7,...,11 to I,R,R2,R3,I,R,R2,R3,...,R3. You can
check by hand that it is a homomorphism. For example, f(5+6) = f(11)
= R^11 = R^3 = R^1 * R^2 = R^5 * R^6 = f(5) * f(6). This mapping
works because Z12 is isomorphic to Z3 X Z4, so it contains Z4 as a
subgroup, and Z4 and are both isomorphic to the cyclic group of 4
elements. The same map f(i) = R^i in Z10 instead of Z12 is not a
homomorphism because f(5+6) = f(11) = f(1) = R, but f(5) * f(6) = R^5
* R^6 = R^1 * R^2 = R^3.
The kernel of the homomorphism f from G to H is the set of elements
in G that map to the identity element in H. In the previous example
mapping Z12 to , the kernel is the set {0,4,8}. The kernel is a
normal subgroup of G, because for g in G and k in the kernel, f(g'kg)
= f(g')*f(k)*f(g) = f(g)'*I*f(g) = I, so g'kg is in the kernel.
Similarly, from a normal subgroup K of a group G, you can construct a
mapping f(g) from g in G into gK in G/K, where gK is the set of
products gk for all g in G and k in K. Each gK is an equivalence
class of elements in g that map to the same element in G/K. For
example, in Z12 with K = {0,4,8}, 0, 4, and 8 map to 0+K or {0,4,8},
1, 5, and 9 map to 1+K or {1,5,9}, 2, 6, and 10 map to 2+K or
{2,6,10}, and 3, 7, and 11 map to 3+K or {3,7,11}. Z12/K is
isomorphic to Z4 with the map i+K in Z12/K to i in Z4.
Consider a group G and its commutator subgroup G'. G/G' is Abelian
and every normal subgroup K such that G/K is Abelian contains G'. G
is Abelian if and only if G' = 1 because for any g and h in an
Abelian group G, (g,h) = g'h'gh = g'gh'h = 1 and if G' = 1 then every
pair of elements in G must commute. For any g and h in G, and K
normal in G, (gK,hK) = (gK)'(hK)'gKhK = (g'K)(h'K)gKhK = (g,h)K, so
that for every g and h, (gK,hK) = K (= 1 in G/K, i.e. G/K is Abelian)
if and only if (g,h) is in K (i.e. K contains G'). Therefore, G/K is
Abelian if and only if K includes all of the commutators of G, which
happens if and only if K includes G'.
I think that the largest Abelian subgroup on a cube is the set of
edge flip moves, plus the set of corner twirl moves, plus the set of
moves for exchanging opposite centers. By the theorem above, none of
those moves can be expressed as commutators.
The center of a group is the set of elements that commute with all
elements in the group. The identity move and the move that flips all
12 edges are the only moves that commute with all other moves. They
form the center of the cube group.
G is defined as solvable if G contains a series of normal subgroups
G, G1, G2, ..., Gn, such that G1 = G, Gn = 1, and Gi+1 is normal in
Gi, and all of the quotient groups Gi/Gi+1 are Abelian. A normal
series constructed from commutator subgroups of G with G, G', G'',
..., G'''...' = 1 satisfies this condition. Sn, the group of
permutations of n items, is solvable only for 4 items or less.
Consider the subgroup of even permutations of n items, called An or
the alternating group of degree n. For n >= 3, An is generated by
3-cycles. An is not Abelian for n >= 4 since the permutations (123)
and (234) do not commute. For n >= 5, if K is a normal subgroup of An
and K is not 1, K must contain a 3-cycle, but if it contains one
3-cycle, it must contain them all, so K = An. Suppose that K contains
the 3-cycle (123), then for any i, j, and k, construct an even
permutation g in An, n >= 5, of the form (1i)(2j)(3k)(4l)(5m)... or
(lm)(1i)(2j)(3k)(4l)(5m)... (taking whichever one is even), and then
g(123)g' = (ijk) is in K, so K = An. Suppose that K contains an
element k that is not a three-cycle. Select the k that affects the
smallest number of elements and is not the identity. Since k is even
and is not a 3-cycle, it must have the form (12345...)... or (12)
(23)... Let g = (345). kgk'g' leaves 1 unchanged without affecting
any more elements than k, but the existence of a permutation that
affects fewer elements than k affects contradicts the assumption, so
K contains a 3-cycle. People discovered the quadratic formula in
ancient times. Del Ferro discovered the formula for some cubic
equations around 1501, Scipione discovered the formula for general
cubic equations in 1515, and Tartaglia later rediscovered the formula
in 1535. Cardano published Tartaglia's procedure in 1545, and
Cardano's student Ferrara discovered the formula for quartic
equations in 1545, but for the next 300 years no one made any further
progress. In 1830, Galois reduced the question of solving an equation
to solving the permutation group of its roots. Since no permutation
group of 5 or more items is solvable, there is no general formula for
solving equations of fifth degree or higher.
Returning to the normal series of finite groups G, G1, ..., Gn, where
G = G1, Gn = 1, and each Gi+1 is normal in Gi, the series is called a
composition series if each Gi+1 is maximal normal in Gi such that
there does not exist a normal subgroup H of Gi that contains Gi+1.
Each Gi/Gi+1 is then called a composition factor, and by the
Jordan-Holder Theorem, every composition series for a group has the
same composition factors (to isomorphism) except that they might be
in a different order. The original group G is solvable if and only if
the factors are all cyclic groups of prime order. This can simplify
proofs on some classes of groups to just examining cyclic groups of
prime order.
If you've read this far and understood everything, either you are a
math major or maybe I shouldn't have joined the dark side and become
an EE... I've also noticed how much I've forgotten since graduating
from college in 1986.
People have tried to apply similar decompositions to the group of
cube moves. For example, David Singmaster's book from 1981 (I have a
picture of the book below) lists an algorithm developed by the
British mathematician Morwen Thistlewaite based on decomposing the
cube group G = into G1 = , G2 =
, G3 = , and G4 = {I}. The total
procedure takes at most 52 moves, but I think that it requires a
computer to follow because it is hard to tell the different groups
apart by eye. I suspect that solutions that are restricted from using
certain moves after a given point can not be optimal. I think that
better solutions have fewer phases because the moves in one phase
often touch pieces positioned by previous phases. Some pieces take
different numbers of moves to exchange depending on the number of
other pieces that must remain in position. Pieces that can be
exchanged only with short moves that affect a lot of other pieces or
with very long moves should be positioned first, while pieces that
can be exchanged with short moves that do not affect other pieces can
be positioned last.
The cube group also decomposes into a direct product of two wreath
product groups. A group H acts on a group K by automorphisms if H
acts on K such that for every h in H, the map of K taking k to hk is
an automorphism. This means that elements of H are permutations of
elements of K that preserve K's structure. The semi-direct product of
K and H is a group on the product set KxH with the binary composition
rule (k1,h1)(k2,h2) = ((h2'k1)k2,h1h2). Now consider a set S, a
transformation group H that acts on S, and a group G. Define the
group G^S (the S-direct power of G) as the set of maps from S into G
with the composition rule (f1f2)(s)=f1(s)f2(s) for fi in G^S and s in
S. For h in H and f in G^S, define (hf)(s)=f(h's). The wreath product
G wr H is the semi-direct product of H and G^S.
The group for a Rubik's Cube is contained in (Z3 wr S8) X (Z2 wr
S12), where Z3 is the group of integers mod 3 over addition, S8 is
the group of permutations of 8 items, Z2 is the group of integers mod
2 over addition, and S12 is the group of permutations of 12 items.
This group also covers all of the possible ways that a cube may be
taken apart and re-assembled. The first factor represents the 8
corners, and the second factor represents the 12 edges. The number of
elements of the group is (3^8 X 8!) X (2^12 X 12!). The wreath
product accounts for the way that the positions and orientations of
edges and corners are related as the cube's faces turn. This
factoring shows why it is natural, although not maybe the most
efficient, to solve the cube in phases for corners and edges and in
subphases for positioning and orienting pieces.
Solomon Golomb of the University of Southern California relates
twists on corner pieces to elementary particles made from quarks. See
http://fusionanomaly.net/rubikscube.html and http://
www.puzzlesolver.com/rubik/cube/docs/history.html and http://
www.math.niu.edu/~rusin/known-math/95/rubik.physics and http://
www.lysator.liu.se/~enok/rubics.quark.html and http://
www.chemistrydaily.com/chemistry/Rubik's_Cube and http://
www.jatiyah.freeserve.co.uk/knots.htm.
Quarks have fractional electric charges of 1/3 and 2/3 the same way
that corner pieces on a cube can have twists of 1/3 and 2/3. Positive
charges correspond to clockwise twists, while negative charges
correspond to counter-clockwise twists. Quarks always combine into
particles with an integral charge; a quark and anti-quark pair form a
meson, while three quarks form a baryon. For example, a proton is a
baryon with two up quarks and a down quark, while a neutron is a
baryon with one up quark and two down quarks.
In the same fashion, unless you take a cube apart, the twists on the
corners always add up to an integral value. You can twist one corner
clockwise and another counter-clockwise, or you can twist three
corners in the same direction, but you can not twist a single corner
alone, and you can not twist only two corners in the same direction.
This is the Rubik's Cube analog to quark confinement.
It is possible that someone living in a higher dimensional space
could reach into our universe, extract a lone quark, and return it in
a different state, the same way that you can remove a corner from a
cube, twist it, and then put it back. A person (or, in the case of
some of my cubes, a dust bunny) living in the world of a cube could
calculate the patterns that could be made from a cube with a twisted
corner without ever having the means to construct a cube with a
twisted corner.
For more quark theory, see the slide shows at http://www.phys.ufl.edu
/~acosta/phy3101/quarks/ and http://umaxp1.physics.lsa.umich.edu/
~bloom/smp_L1/.
In August 2000, a group of mathematicians in India discovered the
first deterministic polynomial-time algorithm for determining whether
or not a number is prime or composite; see http://www.cse.iitk.ac.in/
news/primality.html and http://crypto.cs.mcgill.ca/~stiglic/
PRIMES_P_FAQ.html. Mathematicians have been studying prime numbers
since ancient times, and in modern times, fast tests for prime
numbers have important applications in cryptography. The cube-related
group theory above is complete enough that with just a few
definitions below, the prime number paper is understandable.
A ring R is a non-empty set with operators for addition and
multiplication, where the set and the addition operator form an
abelian group, the set and the multiplication operator form a monoid
(similar to a group except that some elements do not have inverses),
and multiplication distributes over addition (i.e., for all a, b, and
c in the ring, a(b+c)=ab+ac and (b+c)a=ba+ca). For example, Z12 forms
a ring with the additive operator as addition mod 12 and the
multiplicative operator as multiplication mod 12.
For a and b in a ring R, a divides b (written a | b) if a is a factor
of b. For example, in Z, the ring of integers, 3|6 but 3 does not
divide 5.
A field F is a ring where the set of non-zero elements forms a
commutative group. For example, Z12 is ring but not a field because
the elements 2, 3, 4, 6, 8, and 9 do not have multiplicative
inverses. Z5 is a ring because multiplication mod 5 is commutative
and 2*3 = 1 and 4*4 = 1 so every non-zero element has a
multiplicative inverse.
A polynomial ring R[x] is a ring of polynomials in the variable x
with coefficients in the ring R. A typical element has the form a0 +
a1*x + a2*x^2 + ... + an*x^n where each ai is in R. For example, in
the polynomial ring Z5[x], (x^3 + 3x^2 + 2x + 4) + (x^2 + 3x + 2) = x
^3 + (3+1)*x^2 + (2+3)*x + (4+2) = x^3 + 4x^2 + 1, and (x + 2)*(2x +
3) = 2x^2 + (4+3)*x + (2*3) = 2x^2 + 2x + 1.
The degree of a polynomial in x is the exponent of the highest power
of x. For example, the degree of 3x^2 + x + 1 is 2.
For polynomials f(x) and g(x) in the polynomial field F[x] and with g
(x) not zero, there exists a unique q(x) and r(x) in F[x] such that f
(x) = q(x)*g(x) + r(x) and the degree of r(x) is less than the degree
of q(x). The division ring F[x]/g(x) contains equivalence classes of
elements of F[x] based on their remainders, r(x), when they are
divided by g(x). If g(x) is unfactorable in F[x], then F[x]/g(x) is a
field. For example, in the field of real numbers, R, the polynomial x
^2 + 1 has no factors, so R[x]/(x^2+1) is a field (and happens to
correspond to the field of complex numbers where x maps to i (the
square root of -1) because x^2+1=0 so x^2=-1).
This should provide enough background to read the paper on prime
numbers.
Cyclic Redundancy Check (CRC) algorithms work in the polynomial ring
Z2[x]. To check a block of bits with a CRC, you append a CRC checksum
such that if you treat the block plus the checksum as coefficients of
a polynomial in Z2[x] and then divide that polynomial by the CRC
polynomial, the result is 0. If the degree of the CRC polynomial is n
and you treat the block of bits as coefficients of a polynomial in Z2
[x], then the checksum is the n bits of the coefficients of the
remainder when you divide the polynomial by the CRC polynomial. CRCs
are good for detecting small numbers of random bit errors in short
data blocks, but they can fail to detect large bursts of errors. CRCs
are easy to calculate which makes them good for detecting random
noise in data connections but insecure for cryptography. For more on
CRCs, see http://www.rad.com/networks/1994/err_con/crc.htm or http://
www.ciphersbyritter.com/ARTS/CRCMYST.HTM. The DEC VAX, a classic CISC
design, has a CRC instruction http://rtfm.net/vax/vax-crc.html.
For some math quotes, see ../../aksnyderquotes.html.
[002cubenotes]
My old notes from 1981 and 1982 with moves calculated from
permutations. Check out the old fan-fold green-bar computer paper.
[001singmasterbook]
My brother Joel gave me this classic book NOTES ON RUBIK'S MAGIC CUBE
by David Singmaster, 1981, ISBN 0-89490-043-9. The three lines of
small white text at the bottom say "The definitive treatise." --
Scientific American Includes a step-by-step solution to the puzzle.
---------------------------------------------------------------------
[686keychaincube]
A cube on a keychain. When you do this cube, it helps to do the face
with keychain first, and then try to keep the corner with the
keychain at LBD.
[003minicube]
A Rubik's Mini-Cube, billed as "The Not-So-Little Challenge". You can
solve it using the same moves as the moves for placing corners on a
Rubik's Cube.
[02cubefamily]
A little cube family. Junior has a piercing.
The 4x4 Rubik's Revenge is about 2.6 inches (6.5 cm) wide.
The 3x3 Rubik's Cube is about 2.2 inches (5.6 cm) wide.
The 2x2 Rubik's Mini-Cube is about 1.5 inches (3.9 cm) wide.
The 3x3 keychain cube is about 1.2 inches (2.9 cm) wide.
[004fingersoncube]
To move pieces around quickly, it helps to have long fingers. In my
younger days, I used to do 60 chins every day from a bar on a
climber. I think that the daily stretch on my wrists helped to keep
me from having repetitive stress carpal tunnel problems from cubing
and from working as a computer programmer. One of the fastest cubists
is a rock climber http://csel.cs.colorado.edu/~knightsd/
[687snoopycube]
I bought this cartoon cube in Buenos Aires from a street vendor in
the park in Recoleta. I suspect that the maker of the cube did not
bother to license the cartoon characters. This cube was not glued
together well. If you look closely, you can see that Mickey Mouse's
head popped off on one of the centers. Cubes with pictures are harder
to solve than regular cubes because you must also rotate the centers
correctly.
[688dragonballzcube]
I bought this Dragonball Z http://www.dragonballz.com/ cube from a
street vendor in Recoleta for $1 during a later trip to Buenos Aires
in February, 2001 ../../200102buenosaires.html. This cube is tricky
to solve because the faces look almost alike (except for the top
face).
[115markets]
I bought this PokeMon http://www.pokemon.com/ cube in Krakow, Poland,
at a stand on the Rynek (market square) for 4 zloty (about $1.03) on
November 27, 2003 ../../poland/poland.html#cube. The front of the
package says "OLD BRAND MAGIC SQUARE" up the left side, "New" inside
a star at the upper right, and "NO. 8999 WARNING: CHOKING HAZARD -
Small parts Not for children under 3 years MADE IN CHINA" across the
bottom.
[pokemoncubeback]
The back has a six-step top-middle-bottom solution with small
diagrams and text in Chinese and English, but the writing is hard to
read because the letters are small and partially rubbed off and
because a Polish "male elementy" small parts safety warning and a bar
code sticker cover the first three steps.
[pokemoncubebackbottom]
The lower right of the back has some interesting text... The upper
left of this picture has a diagram of a cube with the text "the
conbinelion of gix gidea". The upper right has the text "It's OK"
after the sixth step of the solution. The large cube below it says
"big cube". The bold text below that says "The Short Cot Of Playing
Six Sides" (with "Cot" for "Cut"). The diagonal text at the lower
left says "IQ magic cube".
[593phenomenexcube]
I filled in a pharmaceutical crossword puzzle in a Phenomenex http://
www.phenomenex.com/ catalog on February 18, 2004, to win this cube on
April 10, 2004. The sides list the names of their products: white =
phenomenex ...breaking with tradition, black = Zebron, red = explore
LUNA, green = Jupiter Proteo, EZ:faast, Jupiter 300, and BioSep,
yellow = strata, blue = synergi. This cube is an official Rubik's
Cube (made in China) and moves very cleanly. It would be a good
racing cube if it had normal colors. It came addressed to "Rusty
Terrier at Scientific Programmers". Rusty was a dog ../../
wondertwins.html#dogcicles. I use her name when I don't want to get
on a mailing list. Rusty gets lots of mail even though she has been
gone since 1985.
[7058venicecube]
Cube from Venice (January 2008).
See Joshua's Virtual Puzzle Museum http://www.twistypuzzles.com/ for
a large selection of specialty cubes.
[690rubiklinkfront]
The circles join in the solved position of this flat Rubik's Magic
puzzle. The puzzle has a 1986 copyright by Rubik for the artwork and
has an E. Rubik signature printed on the bottom. Matchbox made the
puzzle in China. The puzzle has a few lengths of clear fishing line
that hold the pieces together. The ends of the line are joined with
metal clips. As you move the puzzle, the line (and the clips along
with it) shifts position. Once a clip shifts between two pieces, the
line becomes stuck, and the puzzle no longer works.
Christian Eggermont's page http://web.inter.nl.net/users/C.Eggermont/
Puzzels/Magic/ shows other shapes but includes the warning: "To make
the cube you have to be very careful since you probably stretch some
of the wires trying to make the cube... which is not smart."
Jaap's puzzle page http://www.geocities.com/jaapsch/puzzles/magic.htm
explains how the looped cords work to hold the puzzle together.
TwistyPuzzles http://twistypuzzles.com/articles/
magic-restringing.shtml has restringing instructions.
[691rubiklink]
The back of the puzzle has closed links.
[693rubiklinkstar]
The puzzle in a star configuration. A few of the clips have jammed.
This is one of the few shapes that the puzzle can still form.
[577tangram]
My mother bought this tangram puzzle in March 2004 to give to one of
her grandchildren (i.e. my cousins). She took it apart to test it and
couldn't get it back together. I played with it for a few hours with
no more success. I was already considering writing a program to solve
it. We returned from visiting my brother Joel on April 6, 2004. I was
exhausted that night and played with the puzzle at 3 am before going
to bed, and somehow by luck, the pieces fell into place and I solved
it in a few minutes. I took this picture.
[insandoutsofpegsolitaire]
My copy of "The Ins & Outs of Peg Solitaire" by John D. Beasley,
1985, ISBN 0-19-853203-2. This book is one of the best references for
peg solitaire.
For some other on-line references, see
http://www.cst.cmich.edu/users/graha1sw/Pub/PegSol/PegSolIntro.html
http://mathworld.wolfram.com/PegSolitaire.html
http://www.gpj.connectfree.co.uk/gpjj.htm
http://www.johnrausch.com/PuzzleWorld/
http://www.ionet.net/~bkeltch/pegs.html
http://homepage.sunrise.ch/homepage/pglaus/Solitaire/solitaire.htm#
InsOuts#InsOuts
http://www.durangobill.com/PegSolitaire.html
http://www.math.temple.edu/~xysun/peg/peg.htm
http://www.mathematische-basteleien.de/solitaire.htm
[692cdw8ballnotabug]
A CDW http://www.cdw.com Magic 8-Ball for programmers showing the
answer "IT'S NOT A BUG - IT'S A FEATURE". The bottom of the ball says
"For your most commonly asked IT requests."
Here are the answers. I think that I have told people all of these,
except the ones that are ads for CDW.
CDW HAS YOUR COMPUTING SOLUTIONS
I CAN'T TEST EVERYTHING
IT WORKED YESTERDAY
IT WORKS LIKE I PROGRAMMED IT
IT WORKS ON MY MACHINE
IT'LL BE FIXED IN THE NEXT RELEASE
IT'S A BETA - WHAT DID YOU EXPECT
IT'S AN UNLIKELY COINCIDENCE
IT'S JUST AN ISOLATED INCIDENT
IT'S NOT A BUG - IT'S A FEATURE
IT'S NOT SUPPOSED TO DO THAT
LOG ON TO WWW.CDW.COM
PLEASE SUBMIT A FORMAL REQUEST
PLUG IT IN
PROGRAM WORKS - MUST BE USER ERROR
REBOOT
SOMEONE CHANGED MY CODE
YOUR CDW ACCOUNT MANAGER CAN HELP
Other people use balls like this http://www.ericmackonline.com/ica/
blogs/emonline.nsf/dx/how-does-he-answer-so-quickly
[drilling8ball]
The Unofficial Magic 8-Ball Home Page http://8ball.ofb.net/
demonstrates the dissection of an 8-ball in full color.
---------------------------------------------------------------------
[circuits]
Return to the museum ../museum.html
[wb64]
Return home ../../index.html
---------------------------------------------------------------------
http://williambader.com - Revised June 16, 2023 06:22:41 PM.
Copyright (c) 2023 William Bader.