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4<title>Rubik's Cube Solutions plus Puzzles and 8-Balls</title>
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10Rubik's Cube Solutions plus Puzzles and 8-Balls<br>
11</font></h2>
12Contents:<br>
13 Introduction <a href="#intro">#intro</a><br>
14 Cube Links <a href="#links">#links</a><br>
15 Basic Cube Group Theory <a href="#basiccubetheory">#basiccubetheory</a><br>
16 Rubik's Cube Solution <a href="#solution">#solution</a><br>
17 Cube Solution Supplement for Patterned Faces <a href="#patternsupplement">#patternsupplement</a><br>
18 Rubik's Revenge Solution <a href="#revengesolution">#revengesolution</a><br>
19 Advanced Cube Group Theory <a href="#grouptheory">#grouptheory</a><br>
20 My Cube and Puzzle Collection <a href="#cubefamily">#cubefamily</a>
21<p><img src="animatedcube.gif" width="53" height="60" align="left"></p>
22<br clear="all"><hr>
23<p><a name="intro"><img src="684racingcube.jpg" width="230" height="256" align="left"></a></p>
24<p>My old racing Rubik's Cube <a href="http://www.rubiks.com">http://www.rubiks.com</a> is well-worn and barely holds together.
25I gave it a lot of wear in June 1982 during the week between high school finals and
26graduation <a href="../../scscards/liberty.html">../../scscards/liberty.html</a>.
27A cube has 43,252,003,274,489,856,000 (over 4 X 10^19) different
28positions <a href="http://www-gap.dcs.st-and.ac.uk/~gap/Intro/rubik.html">http://www-gap.dcs.st-and.ac.uk/~gap/Intro/rubik.html</a>.
29This number is the product of 8! corner position permutations X
3012!/2 edge position permutations X
313^8/3 corner twirls X 2^12/2 edge flips and
32factors as 2^27 X 3^14 X 5^3 X 7^2 X 11.
33If you take a cube apart, there are 12 different ways that you can put it together such
34no set of moves will take the cube from one of the 12 assemblies to the other.
35When I was in practice, I could solve a scrambled cube in about a minute using about 65 moves
36(counting slices as a single move).  I usually do the top corners, three of the top edges,
37the bottom corners, the bottom edges, the last top edge, and then the center edges.<br clear="all"></p>
38<p><a href="http://www.rubiks.com"><img src="685rubiksrevenge.jpg" width="229" height="256" align="left"></a></p>
39<p>A Rubik's Revenge Cube <a href="http://www.rubiks.com">http://www.rubiks.com</a> from 1982.
40A Rubik's Revenge Cube has
417,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 (over 7 X 10^45)
42different positions <a href="http://www.geocities.com/jaapsch/puzzles/cube4.htm">http://www.geocities.com/jaapsch/puzzles/cube4.htm</a>.
43When I was in practice, it took about twice as long and about twice as many moves
44to solve as the original cube.<br clear="all"></p>
45<p><a href="http://vathena.arc.nasa.gov/curric/space/planets/planorbi.html"><img src="solarsystem.jpg" width="95" height="90" align="left"></a></p>
46<p>For an idea of the size of the number of possible positions on a Rubik's Revenge cube,
47note that a Rubik's Revenge cube is 2.6 inches (6.5 cm) per side and the earth's orbit around
48the sun has an average radius of 93 million miles (150 million Km) or about 2.3 X 10^12 cubes.
49If the sun expanded to the size of the earth's orbit, using V = 4/3 Pi r^3,
50it could hold about 50 X 10^36 cubes.
51A giant pile with one cube for every different position would have a radius about
52500 times the radius of the earth's orbit (((7.4 X 10^45)/(50 X 10^36))^(1/3)).
53For comparison, the radius of Neptune's orbit is about 30 times radius of the earth's orbit,
54and the radius of Pluto's orbit is about 40 times the radius of the earth's orbit.<br clear="all"></p>
55<p><img src="001revengecubeinside.jpg" width="200" height="198" align="left"></p>
56<p>A Rubik's Revenge Cube has a round center.<br clear="all"></p>
57<p><img src="002revengecubeparts.jpg" width="200" height="136" align="left"></p>
58<p>Rubik's Revenge parts are fragile, especially the center pieces (on the left).
59The two pieces in the middle are edges, and the two pieces on the right are corners.<br clear="all"></p>
60<hr>
61<p> <a name="links"><b>Some cube links:</b></a><br>
62 Version of this page in Belorussian <a href="http://onlinecasinospotlight.com/web/rubiks-cube-solutions-be">http://onlinecasinospotlight.com/web/rubiks-cube-solutions-be</a><br>
63 Jessica Fridrich's 17 second 56 move solution <a href="http://www.ws.binghamton.edu/fridrich/">http://www.ws.binghamton.edu/fridrich/</a><br>
64 Dan Knight's 17.3 second solution <a href="http://www.knightslab.org/#!speedcubing/cy62">http://www.knightslab.org/#!speedcubing/cy62</a><br>
65 Lars Petrus's 24 second 60 move solution <a href="http://lar5.com/cube/">http://lar5.com/cube/</a><br>
66 Lego Robot Solves Rubik's Cube Puzzle In 3.253 Seconds <a href="http://tech.slashdot.org/story/14/03/19/0333218/lego-robot-solves-rubiks-cube-puzzle-in-3253-seconds">http://tech.slashdot.org/story/14/03/19/0333218/lego-robot-solves-rubiks-cube-puzzle-in-3253-seconds</a><br>
67 How to build a LEGO Speedcuber in under 5 minutes using ARM Powered devices <a href="https://www.youtube.com/watch?v=u7h5KSOFXSc">https://www.youtube.com/watch?v=u7h5KSOFXSc</a><br>
68 Robot solves cube in 12 seconds <a href="http://singularityhub.com/2010/02/17/lego-robot-solves-any-rubiks-cube-in-less-than-12-seconds-video/">http://singularityhub.com/2010/02/17/lego-robot-solves-any-rubiks-cube-in-less-than-12-seconds-video/</a><br>
69 Robot solves larger cubes <a href="http://singularityhub.com/2010/04/28/lego-robot-solves-bigger-and-harder-rubiks-cubes-video/">http://singularityhub.com/2010/04/28/lego-robot-solves-bigger-and-harder-rubiks-cubes-video/</a><br>
70 Android phone solves Rubik's cube in 12.5 seconds <a href="http://deviceguru.com/android-phone-controlled-robot-solves-rubiks-cube/">http://deviceguru.com/android-phone-controlled-robot-solves-rubiks-cube/</a><br>
71 This robot can solve a Rubik's Cube in 1 second <a href="http://thenextweb.com/shareables/2016/01/25/this-robot-can-solve-a-rubiks-cube-in-1-second/">http://thenextweb.com/shareables/2016/01/25/this-robot-can-solve-a-rubiks-cube-in-1-second/</a><br>
72 Robot sets new Rubik's Cube record of 0.637 seconds <a href="http://www.bbc.com/news/technology-37925028">http://www.bbc.com/news/technology-37925028</a><br>
73 Mitsubishi Electric Corp fastest robot to solve a puzzle cube at 0.305 sec (May 2024) <a href="https://www.designfax.net/cms/dfx/opens/article-view-dfx.php?nid=4&amp;bid=1443&amp;aid=14136&amp;et=featurearticle&amp;pn=01">https://www.designfax.net/cms/dfx/opens/article-view-dfx.php?nid=4&amp;bid=1443&amp;aid=14136&amp;et=featurearticle&amp;pn=01</a><br>
74 Purdue ECE students shatter Guinness World Record for fastest puzzle cube-solving robot at 0.103 seconds (May 2025) <a href="https://www.designfax.net/cms/dfx/opens/article-view-dfx.php?nid=4&amp;bid=1534&amp;et=featurearticle&amp;pn=02">https://www.designfax.net/cms/dfx/opens/article-view-dfx.php?nid=4&amp;bid=1534&amp;et=featurearticle&amp;pn=02</a><br>
75 Ben Katz and Jared Di Carlo 0.38 Second Rubik's Cube Solve <a href="http://build-its-inprogress.blogspot.co.uk/2018/03/the-rubiks-contraption.html">http://build-its-inprogress.blogspot.co.uk/2018/03/the-rubiks-contraption.html</a><br>
76 OpenAI's AI-powered robot learned how to solve a Rubik's cube one-handed <a href="https://www.theverge.com/2019/10/15/20914575/openai-dactyl-robotic-hand-rubiks-cube-one-handed-solve-dexterity-ai">https://www.theverge.com/2019/10/15/20914575/openai-dactyl-robotic-hand-rubiks-cube-one-handed-solve-dexterity-ai</a><br>
77 Teenager Lucas Etter sets new fastest time to solve a Rubik's Cube world record of 4.904 seconds <a href="http://www.guinnessworldrecords.com/news/2015/11/confirmed-teenager-lucas-etter-sets-new-fastest-time-to-solve-a-rubiks-cube-wor">http://www.guinnessworldrecords.com/news/2015/11/confirmed-teenager-lucas-etter-sets-new-fastest-time-to-solve-a-rubiks-cube-wor</a><br>
78 Philip Marshall's 65 move solution <a href="http://helm.lu/cube/MarshallPhilipp/">http://helm.lu/cube/MarshallPhilipp/</a><br>
79 Christophe Goudey's CubeLand with solutions, games, contests and links <a href="http://cubeland.free.fr/">http://cubeland.free.fr/</a><br>
80 Michael Reid's links, patterns, and optimal solver <a href="http://www.cflmath.com/Rubik/index.html">http://www.cflmath.com/Rubik/index.html</a><br>
81 Morley Davidson, John Dethridge, Herbert Kociemba, and Tomas Rokicki prove that the best solution is 20 moves <a href="http://www.cube20.org/">http://www.cube20.org/</a><br>
82 Herbert Kociemba's CubeExplorer finds solutions of 19 moves on average <a href="http://kociemba.org/cube.htm">http://kociemba.org/cube.htm</a><br>
83 DeepCubeA finds the best solution 60% of the time <a href="http://deepcube.igb.uci.edu/">http://deepcube.igb.uci.edu/</a> <br>
84 Rubik's Cube solver <a href="https://rubiks-cube-solver.com/">https://rubiks-cube-solver.com/</a><br>
85 How hard is it to scramble Rubik's Cube? <a href="https://phys.org/news/2020-01-hard-scramble-rubik-cube.html">https://phys.org/news/2020-01-hard-scramble-rubik-cube.html</a><br>
86 Cooperman and Kunkle's 26 move solution (May 2007) <a href="http://www.ddj.com/blog/portal/archives/2007/05/rubiks_cube_bus.html">http://www.ddj.com/blog/portal/archives/2007/05/rubiks_cube_bus.html</a>
87and <a href="http://www.sciencedaily.com/releases/2007/05/070531131326.htm">http://www.sciencedaily.com/releases/2007/05/070531131326.htm</a>
88and <a href="http://www.yubanet.com/artman/publish/article_58110.shtml">http://www.yubanet.com/artman/publish/article_58110.shtml</a><br>
89 Jaap's Puzzle Page <a href="http://www.jaapsch.net/puzzles/compcube.htm">http://www.jaapsch.net/puzzles/compcube.htm</a><br>
90 Tomas Rokicki's 25 move solution (March 2008) <a href="http://arxiv.org/abs/0803.3435">http://arxiv.org/abs/0803.3435</a> <a href="http://arxivblog.com/?p=332">http://arxivblog.com/?p=332</a><br>
91 Demaine, Demaine, Eisenstat, Lubiw, and Winslow's Algorithms for Solving Rubik's Cubes (June 2011) <a href="http://arxiv.org/abs/1106.5736">http://arxiv.org/abs/1106.5736</a><br>
92 A deep learning technique to solve Rubik's cube (Feb 2021) <a href="https://techxplore.com/news/2021-02-deep-technique-rubik-cube-problems.html">https://techxplore.com/news/2021-02-deep-technique-rubik-cube-problems.html</a><br>
93 Matthew Monroe's page <a href="http://jjorg.chem.unc.edu/personal/monroe/cube/rubik.html">http://jjorg.chem.unc.edu/personal/monroe/cube/rubik.html</a><br>
94 Josef &quot;Gloom&quot; Jelinek's and Dr. Hana M. Bizek's solution page <a href="http://cube.misto.cz/">http://cube.misto.cz/</a><br>
95 Mark Jeays' solution page <a href="http://jeays.net/rubiks.htm">http://jeays.net/rubiks.htm</a><br>
96 Diego de Pereda's solution page <a href="http://thecube.guru/rubiks-cube-solution/">http://thecube.guru/rubiks-cube-solution/</a><br>
97 R. A. Hoffman's graphic solution <a href="http://www.mts.net/~rhoffman/index.htm">http://www.mts.net/~rhoffman/index.htm</a><br>
98 Peter's Cube Page <a href="http://www.speedcubing.com/peter/">http://www.speedcubing.com/peter/</a><br>
99 Chris Hardwick's Cube Page <a href="http://www.speedcubing.com/chris/">http://www.speedcubing.com/chris/</a><br>
100 Cubeman's page <a href="http://cubeman.vg-network.com/">http://cubeman.vg-network.com/</a> <a href="http://cubeman.org/">http://cubeman.org/</a><br>
101 Ryan Heise's Cube Page <a href="http://www.ryanheise.com/cube/">http://www.ryanheise.com/cube/</a><br>
102 PuzzleSolver.com <a href="http://www.puzzlesolver.com/">http://www.puzzlesolver.com/</a><br>
103 Solution Guide for Beginners <a href="http://www.rubiksplace.com/">http://www.rubiksplace.com/</a><br>
104 Solving the Rubik's Cube Without Human Knowledge <a href="https://arxiv.org/abs/1805.07470">https://arxiv.org/abs/1805.07470</a><br>
105 Jim Cranwell's NxNxN cube solution page <a href="http://www.geocities.com/xulfrepus/rubik/cube.html">http://www.geocities.com/xulfrepus/rubik/cube.html</a><br>
106 4D cube <a href="http://www.superliminal.com/cube/cube.htm">http://www.superliminal.com/cube/cube.htm</a><br>
107 Werner Randelshofer's disassembled cube and pattern collection <a href="http://www.randelshofer.ch/">http://www.randelshofer.ch/</a><br>
108 Joshua's Virtual Puzzle Museum <a href="http://www.twistypuzzles.com/">http://www.twistypuzzles.com/</a><br>
109 W. D. Joyner's cube theory lecture notes <a href="http://web.usna.navy.mil/~wdj/rubik_nts.htm">http://web.usna.navy.mil/~wdj/rubik_nts.htm</a><br>
110 Jim Mahoney's Group Theory &amp; Rubik's Cube lecture notes <a href="http://www.marlboro.edu/~mahoney/courses/Spr00/rubik.html">http://www.marlboro.edu/~mahoney/courses/Spr00/rubik.html</a><br>
111 Speed Cubing page <a href="http://www.speedcubing.com/">http://www.speedcubing.com/</a><br>
112 Gilles and his cubes <a href="http://grrroux.free.fr/">http://grrroux.free.fr/</a><br>
113 Jasmine's Speed Cuber Blog <a href="https://speedcuber.blogspot.com/">http://speedcuber.blogspot.com/</a><br>
114 Georges Helm's page with a 600-item bibliography <a href="http://webplaza.pt.lu/geohelm/myweb/cubeold.htm">http://webplaza.pt.lu/geohelm/myweb/cubeold.htm</a><br>
115 Official Rubik's Cube web site <a href="http://www.rubikscube.com/">http://www.rubikscube.com/</a><br>
116 Meffert's Puzzles sells various type of cubes and puzzles <a href="http://www.mefferts.com/">http://www.mefferts.com/</a><br>
117 Magic Cube 4D <a href="http://www.superliminal.com/cube/cube.htm">http://www.superliminal.com/cube/cube.htm</a><br>
118 Wikipedia has a cube overview <a href="https://en.wikipedia.org/wiki/Rubik%27s_Cube">http://en.wikipedia.org/wiki/Rubik%27s_Cube</a>
119and optimal solutions <a href="https://en.wikipedia.org/wiki/Optimal_solutions_for_Rubik%27s_Cube">http://en.wikipedia.org/wiki/Optimal_solutions_for_Rubik%27s_Cube</a><br>
120 Yahoo Speed Solving Group <a href="http://games.groups.yahoo.com/group/speedsolvingrubikscube/">http://games.groups.yahoo.com/group/speedsolvingrubikscube/</a><br>
121 Discussions on the mathematics of the cube <a href="http://cubezzz.homelinux.org/drupal/">http://cubezzz.homelinux.org/drupal/</a><br>
122 Solving the Rubik's Cube Optimally is NP-complete <a href="http://xxx.lanl.gov/abs/1
122706.06708">http://xxx.lanl.gov/abs/1706.06708</a><br>
123 The most complicated yet elegant artifact <a href="http://shl.stanford.edu:3455/TenThings/417">http://shl.stanford.edu:3455/TenThings/417</a><br>
124 The World's Largest Free Standing Rubik's Cube <a href="https://www.youtube.com/watch?v=1EUDKvShiT8">https://www.youtube.com/watch?v=1EUDKvShiT8</a><br>
125 Children of the Cube (Aug 2018) <a href="https://www.nytimes.com/2018/08/15/sports/cubing-usa-nationals-max-park.html">https://www.nytimes.com/2018/08/15/sports/cubing-usa-nationals-max-park.html</a><br>
126 The Amazing Math Inside the Rubik's Cube (Dec 2019) <a href="https://www.popularmechanics.com/science/math/a30244043/solve-rubiks-cube/">https://www.popularmechanics.com/science/math/a30244043/solve-rubiks-cube/</a><br>
127 A Rubik's microfluidic cube (Jun 2020) <a href="https://phys.org/news/2020-06-rubik-microfluidic-cube.html">https://phys.org/news/2020-06-rubik-microfluidic-cube.html</a><br>
128 J Perm Attempts to Solve the Biggest Rubik's Cube in the World 21x21x21 (Oct 2021) <a href="https://www.youtube.com/watch?v=CoAeCy1R6Qo">https://www.youtube.com/watch?v=CoAeCy1R6Qo</a><br>
129 The 49x49x49 Rubik's Cube (Oct 2024) <a href="https://www.youtube.com/watch?v=4ZeylpCG3IE">https://www.youtube.com/watch?v=4ZeylpCG3IE</a> <a href="https://www.twistypuzzles.com/forum/viewtopic.php?t=39559">https://www.twistypuzzles.com/forum/viewtopic.php?t=39559</a><br>
130 A.I. Solves a 55 x 55 x 55 Rubik's Cube (Dec 2018) <a href="https://www.youtube.com/watch?v=f9smvQ5fc7Q">https://www.youtube.com/watch?v=f9smvQ5fc7Q</a><br>
131 Speedcuber solves Rubik's Cube in 3 seconds, sets new world record (Jun 2023) <a href="https://www.yahoo.com/lifestyle/watch-speedcuber-solves-rubik-cube-220842202.html">https://www.yahoo.com/lifestyle/watch-speedcuber-solves-rubik-cube-220842202.html</a></p>
132<hr>
133<p> <a name="basiccubetheory"><b>Some basic cube theory:</b></a><br>
134 A 3x3x3 cube has 27 pieces: 8 corners with three external sides,
13512 edges with two external sides, 6 centers with one external side,
136and the middle with no external sides.<br>
137 Cube moves work differently than arithmetic.
138On a calculator, if you add 1, then add 2, then subtract 1, and then subtract 2,
139you go back to 0 where you started.
140On a solved cube, if you make a move, then make a second move, then do the first move in reverse,
141and then do the second move in reverse, the cube will usually end up messed up.
142Order matters on a cube so that 1 plus 2 is often not the same as 2 plus 1.
143In group theory terms, two elements commute if they produce the same result when applied
144in either order.  A group is Abelian if every pair of elements in the group commutes.
145The group formed by addition on the set of integers is Abelian,
146while the group formed by moves on a cube is non-Abelian.<br>
147 What helps in finding solutions is that 1 plus 2 is often close enough to 2 plus 1
148that the result of 1 plus 2 minus 1 minus 2 leaves only a few pieces out of place.
149Sequences of moves that leave only a few pieces out of places are building blocks for
150solving the cube.  In group theory, moves of the format X Y X-in-reverse Y-in-reverse
151are called commutators.  I will use the non-standard notation X * Y for X Y X' Y'.
152The usual group theory notation is (X,Y) = X' Y' X Y.<br>
153 If you repeat any sequence of moves, you will eventually return to your starting position.
154Any move affects only a fixed number of pieces, and those pieces have a fixed number of
155different permutations.  Once you repeat the move more times than the number of permutations,
156you must return to a permutation that you saw before.
157Repeating a sequence of moves often produces cube patterns with only a few cubes out of place,
158especially at repeat counts that evenly divide the number of times that you can repeat the move
159before returning to the initial position.
160These sequences are not good for fast solutions because they can be long and because
161there are often more direct ways to reach the positions, but they are good for exploring
162when you are starting out.
163In group theory, the sequence of moves that you repeated is a generator for the subgroup
164of the moves reachable by repeating that sequence of moves.
165The subgroup is also called the orbit of the element that you used as the generator.
166The number of times that you can repeat the move until you reach the starting position is
167called the order of the subgroup.<br>
168 Once you find a move sequence that shifts a piece from one position to another position,
169you can use it to shift other pieces.  You first do a setup move to place the piece that you
170want to move into the position that your move touches, then you do your move, and then
171you undo the setup move.  It does not matter how much your setup move or your shifting
172move scrambles the cube, except that your setup move should not leave any solved pieces
173in positions that your shifting move will scramble.  Moves of the format
174X Y X-in-reverse are called conjugate moves.
175I will use the non-standard notation X + Y for X Y X'.<br>
176 To continue to more advanced cube group theory, see <a href="#grouptheory">#grouptheory</a>.<br>
177 For an introduction to cube group theory, see <a href="http://members.tripod.com/~dogschool/">
177http://members.tripod.com/~dogschool/</a>.<br>
178 For some basic hints, see <a href="http://remus.rutgers.edu/~rhoads/Fun/Rubik.hints">http://remus.rutgers.edu/~rhoads/Fun/Rubik.hints</a>.</p>
179<p> When you are starting, if you are lost, it is OK to start over by taking the cube apart
180as long as you are careful, but keep in mind that each time you take the cube apart,
181it will get a little looser until it no longer stays together.
182If you feel the need to play a trick on someone, you can put a cube together in the wrong way
183if you must, but never peel off a sticker -- they are glued on and not sticky underneath.<br>
184 No one has yet found the shortest solution, but it is probably in the low 20's of moves.
185This means in reverse, that it takes at least 20 moves to mess up a cube completely.
186When someone who hasn't played with cubes much messes up a cube for me to solve,
187they often do not make enough moves, or enough of the right moves,
188to scramble the cube completely.  If you are careful when solving the first face,
189some of other faces will sometimes solve themselves.<br>
190 The original cube had the pairs of opposite faces as white and blue,
191green and yellow, and red and orange.  You should note the layout of your cube
192before trying to solve it.<br>
193 There is no magic secret to solving the cube quickly.
194Whatever system you use, the more moves that you memorize, the faster you can work.
195The fastest systems require memorizing several hundred moves, plus their inverses
196and reflections.<br>
197 You also need fast fingers.  A friend in high school who could solve cubes in about
19824 seconds had a move where he could flick the right and upper faces in a single motion,
199although he would regularly explode his cube.<br>
200 Avoid cubes with hollowed-out corner pieces.  The edges around the holes can catch
201on other pieces as you make moves.  Better cubes have smooth corners without large
202mold marks.  You can sometimes file down excessive mold marks.<br>
203 I have never found a good cube lubricant.
204Avoid petroleum products -- they dissolve and soften plastic.
205Graphite powder works, but it leaves messy black dust, does not last long,
206and makes a gummy goo if it mixes with any liquid lubricant.
207Some people say that silicone-based lubricants work well.
208The cube itself makes a fine dust of plastic powder as the pieces rub
209against each other when you use the cube.
210You can see some cube dust on the picture of the broken Rubik's Revenge Cube above.
211This cube dust acts as a lubricant when it is dry,
212but it can get gummy if it mixes with a liquid lubricant.</p>
213<p> <a name="solution"><b>My Rubik's Cube moves:</b></a><br>
214 My notation is the standard U/D/R/L/F/B for a clockwise quarter turn of the outer slice
215of the face, U', etc., for a counter-clockwise quarter turn, and
216Us, etc., for a clockwise quarter turn of the middle slice with respect to the face.<br>
217 An anti-slice is combination move of a face and the face opposite to it both
218clockwise or both counter-clockwise.  The anti-slice Ua is UD, Ra is RL, Fa is FB, etc.<br>
219 Read the moves from left to right, so that X Y means to do X and then Y.
220I sometimes add parentheses or spaces for readability or to help with memorization.
221The parentheses and spaces do not correspond to moves.<br>
222 I also use some additional notation:<br>
223 X * Y = X Y X' Y'<br>
224 X + Y = X Y X'<br>
225 X ^ 2 = X2 = X X<br>
226 X ^ 3 = X3 = X X X<br>
227 X ^ 4 = X4 = X X X X<br>
228 X ^ n = repeat X an arbitrary number of times<br>
229 This more compact notation shows the beauty and simplicity of the moves.
230It also helps to show how the moves work.<br>
231 Repeat counts and inverses (') bind tighter than * and + so (X+Y') = (X+(Y')) = X Y' X'.<br>
232 I sometimes temporarily shift the position of the cube in my hands as I make moves,
233especially for moves containing Fs.  Some people have a notation for this
234(for example, x, y, and z for clockwise rotation from R, U, and F),
235but adding the shifting notation would make the move sequences longer
236without helping people who hold the cube differently than I hold it.<br>
237 If you actually read these moves and find a mistake, please let me know.
238My email address is at the bottom of my home page <a href="../../index.html">../../index.html</a>.<br>
239 <b>Getting started:</b><br>
240 I do the top corners and three of the four top edges first.
241I do not spend too much time deciding the first face to solve unless I am counting moves.
242To avoid wasting time, I usually pick the white or blue face,
243whichever is closest to the top when I first pick up the cube,
244unless I happen to notice that one face happens to have two or more correct pieces
245than the face that I would have picked.<br>
246 Most of the web sites listed above have detailed instructions for solving the top layer.<br>
247 <b>Top corners:</b><br>
248 Hold the cube so that the upper face has an unsolved corner in the UFR position.
249Rotate the bottom face so that the corner that belongs at UFR is at DFR.
250Find the position of the color of the upper face on the corner at DFR.<br>
251 If it is at D, use R'D2R DR'D'R = (R'+D2)(D*R').<br>
252 If it is at F, use FDF' = F+D.<br>
253 If it is at R, use R'D'R = R'+D'.<br>
254 If all of the remaining corners for the upper face are on the upper face,
255but in the wrong position, use one of these moves to shift one of the incorrect
256corners from the upper face to the bottom face.  When you shift the corner
257to the bottom, try to avoid leaving the side with the color of the upper face
258on the bottom face.<br>
259 You can sometimes place the first few corners in a single move with
260R or R2, etc., if the adjacent edges and corners are not yet positioned.<br>
261 <b>Top edges:</b><br>
262 Hold the cube so that the upper face has the unsolved edge in the UF position.<br>
263 If the edge that belongs at UF is on the middle slice (between the upper and bottom faces),
264rotate the middle slice to place the edge in the FR position and
265find the position of the color of the upper face on the FR edge.<br>
266 If it is at F, use UL'Us2L = U(L'+Us2).<br>
267 If it is at R, use ULUsL' = U(L+Us).<br>
268 If the edge that belongs at UF is on the bottom face, rotate the bottom face to place
269the edge in the F position and find the position of the color of the
270upper face on the FD edge.<br>
271 If it is at F, use D'Rs'DRs = D'*Rs'.<br>
272 If it is at D, use Rs' D2 Rs = Rs'+D2.<br>
273 If the edge that belongs at UF is on the upper face in the UF position
274with the color of the upper face on F, use F'UsF2Us2F' = (F'+Us)(F+Us2).<br>
275 If the edge that belongs at UF is on the upper face in the UB position
276with the color of the upper face on U, use Rs2 D2 Rs2 = Rs2+D2.<br>
277 If all of the remaining edges for the upper face are on the upper face,
278but in the wrong position,
279move the edge that belongs in UF to RB or LB with R if it is at UR,
280B or B' if it is at UB,
281or L' if it is at UL.
282Then use F or F', depending on the alignment of the edge,
283and then Us, Us2, or Us', depending on the position of the edge,
284to slide it into the slot on the F face, and then restore F and the other face.<br>
285 Remember that you need to place only three of the top four edges.
286The fourth edge will be done in a later step.<br>
287 <b>Positioning the corners:</b><br>
288 Next I turn the cube over, so that the almost-solved face is at the bottom and the
289unsolved face is at the top.  Check the positions of the corners.
290You should be able to turn the top face so that either all of the corners are in place,
291only the two corners at the back are exchanged, or both the two corners at the front and
292the two corners at the back are exchanged.<br>
293 If the two corners in the back are exchanged (4 in 6 chance), use FU'B'UF'U'B U2.<br>
294 If both pairs of corners are exchanged (1 in 6 chance), use F URU'R' F' = F+(U*R).<br>
295 <b>Orienting the corners:</b><br>
296 Next check the orientation of the corners.  Either all of the corners will be correct,
297or two corners will be twirled with one corner clockwise and the other corner counter-clockwise,
298or three corners will be twirled either all clockwise or all counter-clockwise,
299or all four corners will be twirled with two corners clockwise and the other two corners counter-clockwise.<br>
300 If only two corners are not oriented correctly (12 in 27 chance),
301rotate the upper face so that a
302bad corner is in the FR position and the color of its F face should be on the top.
303Use the move R'DR FDF' then rotate the upper face so that the second corner is
304in the FR position and finish with the move FD'F' R'D'R.
305The second half is the inverse of the first half.
306The entire move is ((R'+D)(F+D))*(U^n).<br>
307 If three corners are not oriented correctly (8 in 27 chance),
308place the correctly oriented corner
309in the FL position and look at the corner in the FR position.<br>
310 If the R face of the FR corner has the color that should be on top,
311use the move U2 F'U'FU'F' U2 F.<br>
312 If the F face of the FR corner has the color that should be on top,
313use the move F' U2 FUF'UF U2.  This is just the previous move in reverse.<br>
314 If all four corners are not oriented correctly, rotate the top face so that
315the F face of the FL corner and the B face of the BL corner have the color
316that should go on top.  Then look at the FR corner.<br>
317 If the R face of the FR corner has the color that should be on top (4 in 27 chance),
318use the move F URU'R' URU'R' F' = F+((U*R)^2).<br>
319 If the F face of the FR corner has the color that should be on top (2 in 27 chance),
320use the move R2U2 R U2R2 U = (R2U2+R)U.<br>
321 If any other combinations appear, someone took your cube apart and did not put it
322back together correctly.  You will need to pop it open to orient the corners.<br>
323 <b>Positioning the top and bottom edges:</b><br>
324 Orient the cube so that the first face with all but one solved edge is at the left
325and the second face with only the four solved corners is at the right.
326From here on, the middle slice refers to the slice between the left and right faces.<br>
327 Find an edge in the middle slice that belongs on the right face.
328Turn the middle slice so that the edge is on the bottom face,
329turn the left face so that the single unsolved edge is at the top, and
330turn the right face so that the position where the edge on the middle slice belongs
331is at the top.
332Turn the upper face a quarter-turn clockwise or counter-clockwise,
333depending on the orientation of the edge,
334rotate the middle slice to place the edge in position,
335and then turn the upper face back.<br>
336 If the remaining edges for the right face are all in incorrect positions on the right face,
337rotate the right face to place one of the incorrect edges at the top and then
338use U Rs' U' to push it into the middle slice (or U Rs U' to push it into the left face
339if it has the correct orientation).<br>
340 Very rarely, if all of the R face edges are on the R face but in incorrect positions,
341you can temporarily turn the cube to make the R face into the U face and use moves
342as if you were solving the cube top-middle-bottom.
343You 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),
344a three-face cycle with a flip with Rs' D (R2 Fs U2 Fs') D' Rs (same cycle but flips UR and UF),
345an adjacent edge-pair exchange with Fs2 D' (F2 Rs2 F2 Rs2) D Fs2 (exchanges UF with UR, UL with UB),
346an opposite edge-pair exchange with Fs2 D Rs2 D2 Rs2 D Fs2 (exchanges UF with UB, UL with UR), etc.<br>
347 To place the last edge in the left face, instead of using a middle slice,
348use a three-edge cycle like F2 Rs F2 Rs' to avoid disturbing the right face.
349For example, to place DF in UL, use U' F2 Rs F2 Rs' U = U'+(F2*Rs).<br>
350 If the last edge of the left face ends up in the last position of
351the right edge, you can place both edges at once if you line up the middle slice
352so that you need only a quarter turn.  It is probably faster to do this even when
353one of the edges ends up with the wrong orientation.  You can fix it later.<br>
354 If the last left edge ends up in the last right position, and the last right
355edge ends up in the last left position, you can exchange them with U Rs2 U2 Rs2 U = U+(Rs2*U2).<br>
356 <b>Positioning the middle slice edges:</b><br>
357 Now, only the middle slice is left.
358Orient the cube so that the two solved faces are at the left and right.<br>
359 Place the edges in the correct position.<br>
360 If three edges are out of position (8 in 12 chance),
361hold the cube so that the FD edge belongs in UB,
362the UB edge belongs in UF, and the UF edge belongs in FD, and then use F2 Rs' F2 Rs = F2*Rs'.<br>
363 If four edges are out of position (3 in 12 chance),
364hold the cube so that UF and UB need to be exchanged,
365and DF and DB need to be exchanged, and then use D2 Rs2 D2 Rs2 = D2*Rs2.<br>
366 If two edges are out of position, someone took your cube apart and put it together incorrectly.
367You might as well pop the cube apart now.<br>
368 If one edge is out of position, someone exchanged a piece with another cube.
369You will need to find another cube and swap pieces.<br>
370 <b>Orienting the middle slice edges:</b><br>
371 Now, either the cube will solved, or two edges will be flipped, or all four slice
372edges will be flipped.<br>
373 If only two edges are flipped (6 in 8 chance),
374temporarily reorient the cube so that the
375solved faces are at the front and back and one of the flipped edges is
376at UR and the other is at UL, then use the move R'UsR2Us2R' U2 RUs2R2Us'R U2 = ((R'+Us)(R+Us2))*(U2).
377If the second flipped edge is at DL instead of UL, use L2 before and after the move.<br>
378 If all four edges are flipped (1 in 8 chance),
379use the move F'L'F (RsU RsU RsU RsU) F'LF = (F'+L)+((RsU)^4).<br>
380 If only one edge is flipped, someone has messed with your cube,
381and you will need to take your cube apart to fix it.<br>
382 When I am solving the final edge of the top face, if it happens to be in position but flipped,
383I leave it and then orient it in this step along with the middle slice.  If I end up with three flipped
384edges in the middle slice, I place the flipped edge from the previous step at UL, and I rotate the
385middle slice so that the single correctly flipped edge is at UF, and then I use the move
386RsU RsU RsU RsU.<br>
387 <b>Faster solutions:</b><br>
388 One of the faster methods is to solve the top edges, then the top corners and middle edges
389at the same time, and then the bottom.  The moves to position a top corner in combination
390with the middle edge below it are not much more complicated than just positioning the top corner.
391Finishing the bottom requires four steps: positioning and orienting the edges and positioning
392and orienting the corners.  Faster solutions pair these into two steps that either position
393and then orient everything at once or that solve the edges and then solve the corners at once.
394In the easier, but less efficient, procedure that I use, my moves to position the bottom corners
395also twist them, and most of my moves for positioning and orienting corners also move the edges.
396You can take advantage of this by tracking the other pieces that the moves affect and
397coming up with similar moves for all of the cases.</p>
398<p> <a name="patternsupplement"><b>Centers on Rubik's Cubes with patterned faces:</b></a><br>
399 To solve a cube with pictures on the faces, first solve the cube without worrying about the pictures.
400After you follow the solution for a plain cube, everything should be in place,
401except that some centers might be rotated.<br>
402 Find two centers that need opposite rotations.
403If all of the centers are rotated in the same direction, pick any two centers.
404Make the pattern called &quot;dots&quot; or &quot;boxes&quot; to place one center inside the face of the
405other center, then rotate that face to solve the center (if the center was rotated clockwise
406initially, rotate the face counter-clockwise), then undo the &quot;dots&quot; and unrotate the face.
407I usually start by placing the face with the center under consideration at the front of the cube
408and make &quot;dots&quot; by slicing that center through the top position.<br>
409 Use 6 dots for adjacent centers on F and U,
410(Rs Bs Ls Fs) U^n (Fs' Ls' Bs' Rs') U^n' = (Rs Bs Ls Fs)*(U^n).<br>
411 Use 4 dots for opposite centers on F and B,
412(Rs Bs2 Ls Fs2) B^n (Fs2 Ls' Bs2 Rs') B^n' = (Rs Bs2 Ls Fs2)*(B^n).<br>
413 After repeating this procedure, you will either solve the cube or end up with only one
414rotated center.  The rotated center will always be upside down.  To solve it,
415hold the cube so that the face with the rotated center is on the top, and then
416make the move (RL U2 R'L') U (RL U2 R'L') U = ((Ra+U2)U)^2
417(where U/R/L means rotating the upper/right/left face
418one-quarter turn clockwise, U2 means rotating the upper face one-half turn,
419R' means rotating the right face counter-clockwise,
420and Ra means the right anti-slice RL).</p>
421<hr>
422<p> <a name="revengesolution"><b>My Rubik's Revenge moves:</b></a><br>
423 My notation is the standard U/D/R/L/F/B for a clockwise quarter turn of the outer slice
424of the face, Us, etc., for a clockwise quarter turn of the inner slice of the face,
425and U', etc., for a counter-clockwise quarter turn.<br>
426 Note that slices on opposite faces like Us and Ds' refer to the same slice
427on a Rubik's Cube but to different slices on a Rubik's Revenge.<br>
428 Some people use u for Us, etc., or U3 or U-1 or U- for U', etc.<br>
429 I sometimes add parentheses or spaces for readability or to help with memorization.
430The parentheses and spaces do not correspond to moves.<br>
431 Be careful moving the cube.  When I try to go fast, I often end up jamming the
432cube or knocking out a piece.  The movements are in my fingers, not in my head,
433and require no conscious thought.
434When the cube jams, it is hard to restart a sequence of moves from the middle.<br>
435 After each move, I square-off the cube to keep the faces in alignment.
436This helps to keep the cube from jamming.<br>
437 To reassemble an exploded cube, first do the centers,
438then build out from a corner, and do the final edges last.  The centers have a thin
439leg that is easy to break.  To pop in the last edge, it helps to turn the face 45 degrees.
440I have gone through a few different Rubik's Revenge cubes.
441The centers often break on the cubes with stiffer plastic,
442while the cubes with softer plastic often become too loose and fall apart.<br>
443 I do the same top, bottom, middle routine as on a 3x3 cube, except that I also
444do the top and bottom centers along with the top and bottom corners and edges.<br>
445 Unlike a 3x3 cube, you can not flip edges.  The position of an edge determines its orientation.
446If you take a cube apart, you can see that the edges are rounded so that it is physically not
447possible to pop out a single edge, flip it upside down, and reinsert it.
448If an edge appears to be flipped, what really happened is that you are looking at the edge's pair
449that has the same colors but the opposite orientation.  As a side note, unlike a 3x3 cube,
450it is possible to exchange any two edges.  (In fact, I have the move sequence below as part
451of the solution.)  If you exchange two adjacent edges, you can make a position that
452resembles the impossible single flipped edge on a 3x3 cube.<br>
453 <b>
453Top and bottom centers:</b><br>
454 I first do the top and bottom centers.  You can push the first few centers into position
455easily if you do two adjacent top centers, then do the two adjacent bottom centers across
456from the top centers (for example, if the top centers are at LF and LB, do the bottom centers
457at RF and RB), then turn the top or bottom face a half-turn, and do two more centers.
458Usually some centers are on diagonals instead of adjacent, and you won't get lucky enough
459to place all of the centers easily.<br>
460 For the last few centers, find a center on the middle slice (i.e. Us or Ds) and a hole on
461the U face where the center belongs.  Rotate Us or Ds to bring the center onto the F face,
462and then rotate the F face to bring the center to LD.  Rotate the U face to bring the hole
463to RB.  Then position the center with Rs' F' Rs.<br>
464 You can reduce the number of setup moves by using reflections and inverses of this move:<br>
465 (The center is always on the F face, and the hole is always on the U face.)<br>
466 If the center is at LD, place the hole at RB, and use Rs' F' Rs.<br>
467 If the center is at LU, place the hole at RF, and use Rs' F Rs.<br>
468 If the center is at RU, place the hole at LF, and use Ls F' Ls'.<br>
469 If the center is at RD, place the hole at LB, and use Ls F Ls'.<br>
470 If two centers are at LU and LD, and you can place two holes at RF and RB, use Rs' F2 Rs.<br>
471 If two centers are at RU and RD, and you can place two holes at LF and LB, use Ls F2 Ls'.<br>
472 After you finish the top center, or if you ran out of top center pieces because some
473centers are on the bottom, flip the cube to bring the bottom to the top and use the
474same procedure for positioning centers.<br>
475 You should be one with your cube and know all of the pairs of opposite colors so that
476you can place the correct color on the bottom.  If you are not sure, check the colors
477on the corners.  If you do the corners first, the centers take an extra move to position
478because you must follow Rs' F' Rs with F, and the same for the other moves.<br>
479 <b>Corners:</b><br>
480 Do the top and bottom corners the same as for a normal cube.<br>
481 <b>Top and bottom edges:</b><br>
482 Do the top and bottom edges the same as for a normal cube.
483There are twice as many edges, but the moves are the same.
484When you finish the bottom edges, you can sometimes slide in both edges at the same time.<br>
485 While you are sliding in edges, if you are only sliding in one edge into the target face,
486but the adjacent position in the target face has an incorrectly positioned edge for the face,
487you can do a double slice to shift the incorrect edge to a center slice with the same move
488that you position the first edge.<br>
489 It sometimes helps to do the last top and bottom edges together.
490Place one of the top edges in the bottom, and place the bottom edge in a middle slice,
491so that when you slice the middle edge into the bottom, the edge in the bottom goes
492into position on the top.<br>
493 <b>Left slice:</b><br>
494 After I finish the top and bottom, I reorient the cube so that the solved faces
495are at the right and left.  I work on the left slice first and pick the orientation
496that gives me the most pieces in correct positions.  To do the first three left
497slice edges, I orient the cube so that the first good edge piece (if present) is at
498the upper front of the Ls and the second good edge (if present) is at the upper back
499of the Ls.  I then rotate the Rs so that the edge piece that belongs at the lower front
500of the Ls is at the upper back position of the Rs with the color of the front face
501showing on its upper face.  The move is then D2 Rs D2.  If the edge that you need
502is on the Rs, depending on how far you are, you can either do this move in reverse
503to kick an edge from the Ls to the Rs, or you can rotate the Ls for a better starting
504position, or you can reorient the left and right sides of the cube in your hands.<br>
505 To place the final Ls edge, orient the cube so that the open position on
506the Ls is at the upper front, and then rotate the Rs so that the piece for
507that position is at the upper back.
508The move is then (D2 Rs D2 Rs') U2 (Rs D2 Rs' D2) U2 = (D2*Rs)*(U2).
509To make the move, hold the cube with your left hand high on the F and B faces,
510and do the moves with your right hand.
511To help remember the move, note that it starts D2 Rs the same as move for the
512other edges and has a simple formula in commutator notation.
513I must admit here that I have forgotten and rederived this move several times
514because in 1984, my boss threatened to glue my cube together if he saw it one more time,
515and as I write this in 2002, I've played with cubes less than once a year since.
516Hopefully now, I will be able to find the move on this page whenever I need it.<br>
517 <b>Right slice:</b><br>
518 To do the Rs, I place the two pieces that I want to exchange on the
519upper face and use the move F2 (D2 Rs D2 Rs') U2 (Rs D2 Rs' D2) U2 F2 Rs.
520This move is just the previous move for the final Ls edge wrapped by F2.<br>
521 If the two pieces are at diagonals, you can either do the move above twice or
522place the two edges at UB and FD and use a setup sequence like (D' Rs' D Rs)
523before the move above and (Rs' D' Rs D) after, or (D'*Rs')+((F2+((D2*Rs)*(U2)))Rs) in total.<br>
524 <b>Centers:</b><br>
525 Now only the centers are out of place.  The setup moves will usually require
526rotating faces to place pieces in position for the moves.<br>
527 If two opposite faces have two adjacent centers that it would
528correctly position at least two centers to exchange,
529place the centers on the Ls of the U and D faces and use Ls2 Fs2 Ls2 Fs2 = Ls2*Fs2.<br>
530 If two adjacent faces have centers that need to be exchanged,
531orient the cube with the faces at the L and B.
532Rotate the faces so that the L center is nearest U and F faces,
533and the B center is nearest the U and R faces, and then make the move
534B2 (Rs' Fs Rs) B2 (Rs' Fs' Rs) = B2*(Rs'+Fs).
535You can do four pieces at a time by exchanging the Fs with FsBs' and Fs' with Fs'Bs.<br>
536 If three faces have centers that need to be cycled, orient the cube so
537that the center in the F face needs to go to the L face, the center on the L face
538needs to go to the B face, and the center on the B face needs to go to the F face.
539Rotate the faces so that the F center is nearest the R and D faces,
540the L center is nearest the F and U faces, and
541the B center is nearest the D and L faces, and then make the setup move Rs2 and
542then the previous move for two faces.
543You can see now why it helps to start the previous move with an extra B2.
544As with the previous move,
545you can do six pieces at a time by exchanging Fs with FsBs'.
546When I make the setup moves, I usually set up the F and B faces and then the L face.
547After doing the exchange, I let the colors of the faces guide me in undoing the setup moves.<br>
548 If you end up with the last two centers on opposite faces, do a three center
549exchange to move the last two centers to adjacent faces, and then do a two
550center exchange.  To reduce the chances of ending up this way, you should work
551on centers that are on the face opposite to where they belong by using the three
552center exchange before working on centers that are on the face adjacent to where
553they belong.  If the centers are positioned so that a three center exchange
554will solve at least two centers without leaving a center opposite to its face,
555the exchange is probably worth doing.<br>
556 <b>Other solutions:</b><br>
557 It is possible to solve a Rubik's Revenge by positioning the centers and
558pairing the edges and then treating it like a Rubik's Cube.
559The tricky part about pairing the edges is that you can end up with the equivalent
560of the impossible single flipped edge on a Rubik's Cube, and then you have to
561start over again.<br>
562 It is also possible to solve a Rubik's Revenge by layers as top-middle-bottom,
563but there are more middle edges than on a Rubik's Cube, and it is hard to move
564edges without scrambling centers.  Also, the final layer has more possible edge
565permutations than a Rubik's Cube.</p>
566<p> Notice that the last few steps of both cubes require only a few move sequences,
567although the move sequences are relatively long.  Since you need to look at the
568cube for just a second to decide what move to apply next, it is possible to
569solve a cube while having a conversation with someone and making eye contact
570with them and not the cube.  This often surprises non-cubists, although I
571guess after writing this, I can't claim to just have been lucky when I find
572a scrambled, dusty cube at a yard sale or on someone's bookshelf and
573accidentally solve it.</p>
574<hr>
575<p> <a name="grouptheory"><b>More group theory:</b></a><br>
576 For some optional additional group theory (which is OK to skip now if you promise to come back later),
577define a group G as a set with an associative binary operator,
578with an identity element, and with all of its elements invertible.
579Groups are usually written in uppercase letters like &quot;G&quot; and &quot;H&quot;,
580while elements of groups are usually written in lowercase letters like &quot;g&quot; and &quot;h&quot;.
581The operator is often written in a product format (and assumed between adjacent elements),
582and the identity element is often written as '1' (but sometimes as '0' for
583groups that are written in additive format,
584and sometimes as 'I' for groups that are transforms).
585Associative means that x*(y*z)=(x*y)*z,
586for example, in multiplication over the set of integers, 2*(3*4)=(2*3)*4=24.
587Lie and Jordan algebras have non-associative multiplication and are sometimes used in topology.
588An element g in G is invertible if there is a g' in G such that g*(g')=1.
589I will usually write g' for the inverse of g,
590but the inverse can also be written as g^-1 or -g.
591A ' on a group instead of on an element as in G' means another group,
592often somehow derived from G, according to context.<br>
593 In the set of integers over addition,
594the identity element is 0, and the inverse of every integer is its negation,
595for example, 3+(-3)=0.<br>
596 A group is Abelian (or commutative) if a*b=b*a for all elements a and b in the group.
597Addition over the set of integers is commutative, for example, 3+5=5+3=8.<br>
598 For a group G,
599define the commutator subgroup G' as the subgroup of G generated
600by all of the commutators (g,h) = g'h'gh where g and h are in G.<br>
601 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.<br>
602 If K is a normal subgroup of G, then K' is also a normal subgroup of G.
603For example, if f is a homomorphism from G to K,
604f(g,h) = f(g'h'gh) = f(g')f(h')f(g)f(h) = (f(g),f(h)),
605so f(G') is contained in K'.
606If f covers all of K, every commutator (f(g),f(h)) is in f(G') and f(G') = K'.
607For 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 =
608(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',
609so K' is normal.<br>
610 G/K is defined as the quotient group of G relative to the normal subgroup K.
611Elements in G/K are equivalence classes of G.  Each element g of G maps to the
612element gK in G/K where gK is the set of all elements of the form gk where g is in G
613and k is in K.  For example, the group of addition of hours on a 12-hour clock
614has the form Z/12Z, where Z is the set of integers or 0, 1, -1, 2, -2, etc.,
615and 12Z is the set of all integral multiples of 12.
616Z/nZ is sometimes written as Zn.  For example, Z12 is short for Z/12Z.<br>
617 <b>|</b> S <b>|</b> is defined as the number of elements in the set S.
618For example, |Z12| is 12.<br>
619 For the elements g1, g2, ..., gn in G, &lt;g1,g2,...,gn&gt; defines the subgroup generated
620by all of the products of powers of g1, g2, ..., gn.
621Each of the gi's are called generators.
622For example, &lt;F,B,U,D,R,L&gt;, the set of moves generated by quarter-turns on the
623six faces of a cube, generates all of the possible moves on a cube.<br>
624 If a group of n items can be generated by powers of a single generator,
625the group is Abelian, and it is isomorphic to the cyclic group of n items.
626For example, Z12 can be generated by 1.  Z12 is Abelian, and it is isomorphic
627to the cyclic group of 12 items.<br>
628 Two numbers that do not share any common factors are called <i>relatively prime</i>
629or <i>coprime</i>.  The probability that two numbers are coprime is related
630to the Riemann zeta function <a href="http://www.cacr.caltech.edu/~roy/upi/coprime.html">http://www.cacr.caltech.edu/~roy/upi/coprime.html</a>.<br>
631 If a finite cyclic group of n items has a generator, it has phi(n) elements that are generators,
632where phi(n) is the Euler phi-function, which is the number of positive integers
633between 1 and n that do not share any factors with n.
634If n is written as a unique product of prime numbers p1^i1 X p2^i2 X ... X pN^iN,
635phi(n) is (p1^i1 - p1^(i1-1)) X (p2^i2 - p2^(i2-1)) X ... X (pN^iN - pN^(iN-1)).
636For 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.
637This means that Z12 has exactly four elements that are generators.
638The generators are 1, 5, 7, and 11.
639The group generated by clockwise turns of the right face of a cube, &lt;R&gt;,
640is cyclic with four elements, R, R2, R3(=R'), and R4(=I).
641phi(4) = 2^2-2^1 = 4-2 = 2, so {R,R2,R3,I} has two elements that are generators.
642One is R, and the other is R3.  Someone who is left-handed could express cube
643moves in terms of &lt;F',B',U',D',R',L'&gt; instead of &lt;F,B,U,D,R,L&gt;.
644R2 has order 2 because &lt;R2&gt; has only two elements, R2 and I.
645&lt;R2&gt; is a subgroup of &lt;R&gt;.
646The number of elements in a subgroup always divides evenly into the number of elements of its group.
647The cube group contains some larger cyclic subgroups.
648For example, &lt;RU&gt; has order 105.  Each RU move cycles 7 edges, 5 corners, and twists 1 corner,
649so the period of the entire move is the least common multiple, 7 * 5 * 3 = 105.
650&lt;RU'&gt; has order 42 because each RU' move cycles 7 edges and 6 corners.<br>
651 A function f that maps elements from group G to group H such that for all a and b in G,
652f(ab) = f(a) * f(b) is called a homomorphism.
653If f maps from G into itself, it is called an endomorphism.
654If f maps 1-to-1, it is called an isomorphism.
655If f maps 1-to-1 from G into itself, it is called an automorphism.<br>
656 Consider a function that maps i in Z12 to R^i on a cube.
657This map takes 0,1,2,3,4,5,6,7,...,11 to I,R,R2,R3,I,R,R2,R3,...,R3.
658You can check by hand that it is a homomorphism.
659For example, f(5+6) = f(11) = R^11 = R^3 = R^1 * R^2 = R^5 * R^6 = f(5) * f(6).
660This mapping works because Z12 is isomorphic to Z3 X Z4, so it contains Z4 as a subgroup,
661and Z4 and &lt;R&gt; are both isomorphic to the cyclic group of 4 elements.
662The same map f(i) = R^i in Z10 instead of Z12 is not a homomorphism because
663f(5+6) = f(11) = f(1) = R, but f(5) * f(6) = R^5 * R^6 = R^1 * R^2 = R^3.<br>
664 The kernel of the homomorphism f from G to H is the set of elements in G that
665map to the identity element in H.  In the previous example mapping Z12 to &lt;R&gt;,
666the kernel is the set {0,4,8}.  The kernel is a normal subgroup of G,
667because for g in G and k in the kernel, f(g'kg) = f(g')*f(k)*f(g) = f(g)'*I*f
667(g) = I,
668so g'kg is in the kernel.
669Similarly, from a normal subgroup K of a group G, you can construct a mapping
670f(g) from g in G into gK in G/K,
671where gK is the set of products gk for all g in G and k in K.
672Each gK is an equivalence class of elements in g that map to the same element in G/K.
673For example, in Z12 with K = {0,4,8},
6740, 4, and 8 map to 0+K or {0,4,8},
6751, 5, and 9 map to 1+K or {1,5,9},
6762, 6, and 10 map to 2+K or {2,6,10}, and
6773, 7, and 11 map to 3+K or {3,7,11}.
678Z12/K is isomorphic to Z4 with the map i+K in Z12/K to i in Z4.<br>
679 Consider a group G and its commutator subgroup G'.
680G/G' is Abelian and every normal subgroup K such that G/K is Abelian contains G'.
681G is Abelian if and only if G' = 1 because for any g and h in
682an Abelian group G, (g,h) = g'h'gh = g'gh'h = 1 and
683if G' = 1 then every pair of elements in G must commute.
684For any g and h in G, and K normal in G, (gK,hK) = (gK)'(hK)'gKhK = (g'K)(h'K)gKhK = (g,h)K,
685so that for every g and h, (gK,hK) = K (= 1 in G/K, i.e. G/K is Abelian)
686if and only if (g,h) is in K (i.e. K contains G').
687Therefore, G/K is Abelian if and only if K includes all of the commutators of G, which happens
688if and only if K includes G'.<br>
689 I think that the largest Abelian subgroup on a cube is the set of edge flip moves,
690plus the set of corner twirl moves, plus the set of moves for exchanging opposite centers.
691By the theorem above, none of those moves can be expressed as commutators.<br>
692 The center of a group is the set of elements that commute with all elements in the group.
693The identity move and the move that flips all 12 edges are the only moves that commute
694with all other moves.  They form the center of the cube group.<br>
695 G is defined as solvable if G contains a series of normal subgroups G, G1, G2, ..., Gn,
696such that G1 = G, Gn = 1, and Gi+1 is normal in Gi, and all of the quotient groups
697Gi/Gi+1 are Abelian.
698A normal series constructed from commutator subgroups of G
699with G, G', G'', ..., G'''...' = 1 satisfies this condition.
700Sn, the group of permutations of n items, is solvable only for 4 items or less.
701Consider the subgroup of even permutations of n items, called An or
702the alternating group of degree n.  For n &gt;= 3, An is generated by 3-cycles.
703An is not Abelian for n &gt;= 4 since the permutations (123) and (234) do not commute.
704For n &gt;= 5, if K is a normal subgroup of An and K is not 1, K must contain a 3-cycle,
705but if it contains one 3-cycle, it must contain them all, so K = An.
706Suppose that K contains the 3-cycle (123), then for any i, j, and k,
707construct an even permutation g in An, n &gt;= 5, of the form (1i)(2j)(3k)(4l)(5m)...
708or (lm)(1i)(2j)(3k)(4l)(5m)... (taking whichever one is even),
709and then g(123)g' = (ijk) is in K, so K = An.
710Suppose that K contains an element k that is not a three-cycle.
711Select the k that affects the smallest number of elements and is not the identity.
712Since k is even and is not a 3-cycle, it must have the form (12345...)... or (12)(23)...
713Let g = (345).  kgk'g' leaves 1 unchanged without affecting any more elements than k,
714but the existence of a permutation that affects fewer elements than k affects contradicts
715the assumption, so K contains a 3-cycle.
716People discovered the quadratic formula in ancient times.
717Del Ferro discovered the formula for some cubic equations around 1501,
718Scipione discovered the formula for general cubic equations in 1515,
719and Tartaglia later rediscovered the formula in 1535.
720Cardano published Tartaglia's procedure in 1545,
721and Cardano's student Ferrara discovered the formula for quartic equations in 1545,
722but for the next 300 years no one made any further progress.
723In 1830, Galois reduced the question of solving an equation to solving the permutation
724group of its roots.  Since no permutation group of 5 or more items is solvable,
725there is no general formula for solving equations of fifth degree or higher.<br>
726 Returning to the normal series of finite groups G, G1, ..., Gn,
727where G = G1, Gn = 1, and each Gi+1 is normal in Gi, the series is called
728a composition series if each Gi+1 is maximal normal in Gi such that there
729does not exist a normal subgroup H of Gi that contains Gi+1.
730Each Gi/Gi+1 is then called a composition factor,
731and by the Jordan-H&ouml;lder Theorem,
732every composition series for a group has the same composition factors (to isomorphism)
733except that they might be in a different order.
734The original group G is solvable if and only if the factors are all cyclic groups of prime order.
735This can simplify proofs on some classes of groups to just examining cyclic groups of prime order.<br>
736 If you've read this far and understood everything, either you are a math major
737or maybe I shouldn't have joined the dark side and become an EE...
738I've also noticed how much I've forgotten since graduating from college in 1986.<br>
739 People have tried to apply similar decompositions to the group of cube moves.
740For example, David Singmaster's book from 1981 (I have a picture of the book below)
741lists an algorithm developed by the British mathematician Morwen Thistlewaite
742based on decomposing the cube group G = &lt;L,R,F,B,U,D&gt; into
743G1 = &lt;L,R,F,B,U2,D2&gt;, G2 = &lt;L,R,F2,B2,U2,D2&gt;, G3 = &lt;L2,R2,F2,B2,U2,D2&gt;,
744and G4 = {I}.
745The total procedure takes at most 52 moves, but I think that it requires a
746computer to follow because it is hard to tell the different groups apart by eye.
747I suspect that solutions that are restricted from using certain moves after a given
748point can not be optimal.
749I think that better solutions have fewer phases because the moves in one phase
750often touch pieces positioned by previous phases.
751Some pieces take different numbers of moves to exchange depending on the number
752of other pieces that must remain in position.
753Pieces that can be exchanged only with short moves that affect a lot of other
754pieces or with very long moves should be positioned first,
755while pieces that can be exchanged with short moves that do not affect other
756pieces can be positioned last.<br>
757 The cube group also decomposes into a direct product of two wreath product groups.
758A group H acts on a group K by automorphisms if H acts on K such that for every
759h in H, the map of K taking k to hk is an automorphism.
760This means that elements of H are permutations of elements of K that preserve K's structure.
761The semi-direct product of K and H is a group on the product set KxH
762with the binary composition rule (k1,h1)(k2,h2) = ((h2'k1)k2,h1h2).
763Now consider a set S, a transformation group H that acts on S, and a group G.
764Define the group G^S (the S-direct power of G) as the set of maps from S into G
765with the composition rule (f1f2)(s)=f1(s)f2(s) for fi in G^S and s in S.
766For h in H and f in G^S, define (hf)(s)=f(h's).
767The wreath product G wr H is the semi-direct product of H and G^S.<br>
768 The group for a Rubik's Cube is contained in (Z3 wr S8) X (Z2 wr S12),
769where Z3 is the group of integers mod 3 over addition,
770S8 is the group of permutations of 8 items,
771Z2 is the group of integers mod 2 over addition,
772and S12 is the group of permutations of 12 items.
773This group also covers all of the possible ways that a cube may be taken apart and re-assembled.
774The first factor represents the 8 corners, and the second factor represents the 12 edges.
775The number of elements of the group is (3^8 X 8!) X (2^12 X 12!).
776The wreath product accounts for the way that the positions and orientations of
777edges and corners are related as the cube's faces turn.
778This factoring shows why it is natural, although not maybe the most efficient,
779to solve the cube in phases for corners and edges and in subphases for positioning and orienting pieces.<br>
780 Solomon Golomb of the University of Southern California relates twists
781on corner pieces to elementary particles made from quarks.
782See <a href="http://fusionanomaly.net/rubikscube.html">http://fusionanomaly.net/rubikscube.html</a>
783and <a href="http://www.puzzlesolver.com/rubik/cube/docs/history.html">http://www.puzzlesolver.com/rubik/cube/docs/history.html</a>
784and <a href="http://www.math.niu.edu/~rusin/known-math/95/rubik.physics">http://www.math.niu.edu/~rusin/known-math/95/rubik.physics</a>
785and <a href="http://www.lysator.liu.se/~enok/rubics.quark.html">http://www.lysator.liu.se/~enok/rubics.quark.html</a>
786and <a href="http://www.chemistrydaily.com/chemistry/Rubik's_Cube">http://www.chemistrydaily.com/chemistry/Rubik's_Cube</a>
787and <a href="http://www.jatiyah.freeserve.co.uk/knots.htm">http://www.jatiyah.freeserve.co.uk/knots.htm</a>.<br>
788 Quarks have fractional electric charges of 1/3 and 2/3 the same way that corner pieces
789on a cube can have twists of 1/3 and 2/3.  Positive charges correspond to clockwise twists,
790while negative charges correspond to counter-clockwise twists.
791Quarks always combine into particles with an integral charge;
792a quark and anti-quark pair form a meson,
793while three quarks form a baryon.
794For example, a proton is a baryon with two up quarks and a down quark,
795while a neutron is a baryon with one up quark and two down quarks.<br>
796 In the same fashion, unless you take a cube apart, the twists on the corners
797always add up to an integral value.  You can twist one corner clockwise and another
798counter-clockwise, or you can twist three corners in the same direction,
799but you can not twist a single corner alone, and you can not twist only two corners
800in the same direction.  This is the Rubik's Cube analog to quark confinement.<br>
801 It is possible that someone living in a higher dimensional space could reach
802into our universe, extract a lone quark, and return it in a different state,
803the same way that you can remove a corner from a cube, twist it, and then
804put it back.
805A person (or, in the case of some of my cubes, a <i>dust bunny</i>) living in
806the world of a cube could calculate the patterns that could be made from a cube
807with a twisted corner without ever having the means to construct a cube with a
808twisted corner.<br>
809 For more quark theory, see the slide shows at <a href="http://www.phys.ufl.edu/~acosta/phy3101/quarks/">http://www.phys.ufl.edu/~acosta/phy3101/quarks/</a>
810and <a href="http://umaxp1.physics.lsa.umich.edu/~bloom/smp_L1/">http://umaxp1.physics.lsa.umich.edu/~bloom/smp_L1/</a>.</p>
811<p> <a name="primality">In August 2000,</a> a group of mathematicians in India discovered the first deterministic
812polynomial-time algorithm for determining whether or not a number is prime or
813composite;
813 see <a href="http://www.cse.iitk.ac.in/news/primality.html">http://www.cse.iitk.ac.in/news/primality.html</a>
814and <a href="http://crypto.cs.mcgill.ca/~stiglic/PRIMES_P_FAQ.html">http://crypto.cs.mcgill.ca/~stiglic/PRIMES_P_FAQ.html</a>.
815Mathematicians have been studying prime numbers since ancient times,
816and in modern times, fast tests for prime numbers have important applications in cryptography.
817The cube-related group theory above is complete enough that with just a few definitions below,
818the prime number paper is understandable.<br>
819 A <b>ring</b> R is a non-empty set with operators for addition and multiplication,
820where the set and the addition operator form an abelian group,
821the set and the multiplication operator form a monoid (similar to a group except
822that some elements do not have inverses),
823and multiplication distributes over addition (i.e., for all a, b, and c in the ring,
824a(b+c)=ab+ac and (b+c)a=ba+ca). For example, Z12 forms a ring with the additive operator
825as addition mod 12 and the multiplicative operator as multiplication mod 12.<br>
826 For a and b in a ring R, a divides b (written a <b>|</b> b) if a is a factor of b.
827For example, in Z, the ring of integers, 3|6 but 3 does not divide 5.<br>
828 A <b>field</b> F is a ring where the set of non-zero elements forms a commutative group.
829For example, Z12 is ring but not a field because the elements 2, 3, 4, 6, 8, and 9 do not
830have multiplicative inverses.  Z5 is a ring because multiplication mod 5 is commutative
831and 2*3 = 1 and 4*4 = 1 so every non-zero element has a multiplicative inverse.<br>
832 A <b>polynomial ring</b> R[x] is a ring of polynomials in the variable <i>x</i> with
833coefficients in the ring R.  A typical element has the form a0 + a1*x + a2*x^2 + ... + an*x^n
834where each ai is in R.  For example, in the polynomial ring Z5[x],
835(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,
836and (x + 2)*(2x + 3) = 2x^2 + (4+3)*x + (2*3) = 2x^2 + 2x + 1.<br>
837 The <b>degree</b> of a polynomial in x is the exponent of the highest power of x.
838For example, the degree of 3x^2 + x + 1 is 2.<br>
839 For polynomials f(x) and g(x) in the polynomial field F[x] and with g(x) not zero,
840there exists a unique q(x) and r(x) in F[x]
841such that f(x) = q(x)*g(x) + r(x) and the degree of r(x) is less than the degree of q(x).
842The division ring F[x]/g(x) contains equivalence classes of elements of F[x] based
843on their remainders, r(x), when they are divided by g(x).
844If g(x) is unfactorable in F[x], then F[x]/g(x) is a field.
845For example, in the field of real numbers, <i>R</i>, the polynomial x^2 + 1 has no factors,
846so <i>R</i>[x]/(x^2+1) is a field (and happens to correspond to the field of complex numbers
847where x maps to <i>i</i> (the square root of -1) because x^2+1=0 so x^2=-1).<br>
848 This should provide enough background to read the paper on prime numbers.<br>
849 Cyclic Redundancy Check (CRC) algorithms work in the polynomial ring Z2[x].
850To check a block of bits with a CRC, you append a CRC checksum such that
851if you treat the block plus the checksum as coefficients of a polynomial in Z2[x]
852and then divide that polynomial by the CRC polynomial, the result is 0.
853If the degree of the CRC polynomial is n and you treat the block of bits as coefficients
854of a polynomial in Z2[x], then the checksum is the n bits of the coefficients
855of the remainder when you divide the polynomial by the CRC polynomial.
856CRCs are good for detecting small numbers of random bit errors in short data blocks,
857but they can fail to detect large bursts of errors.
858CRCs are easy to calculate which makes them good for detecting random noise
859in data connections but insecure for cryptography.
860For more on CRCs, see <a href="http://www.rad.com/networks/1994/err_con/crc.htm">http://www.rad.com/networks/1994/err_con/crc.htm</a>
861or <a href="http://www.ciphersbyritter.com/ARTS/CRCMYST.HTM">http://www.ciphersbyritter.com/ARTS/CRCMYST.HTM</a>.
862The DEC VAX, a classic CISC design, has a CRC instruction <a href="http://rtfm.net/vax/vax-crc.html">http://rtfm.net/vax/vax-crc.html</a>.</p>
863<p> For some math quotes, see <a href="../../aksnyderquotes.html">../../aksnyderquotes.html</a>.</p>
864<p><img src="002cubenotes.jpg" width="200" height="152" align="left"></p>
865<p>My old notes from 1981 and 1982 with moves calculated from permutations.
866Check out the old fan-fold green-bar computer paper.<br clear="all"></p>
867<p><a href="https://www.amazon.com/exec/obidos/search-handle-url/index%3Dbooks%26field-author%3DSingmaster%2C%20David"><img src="001singmasterbook.jpg" width="200" height="312" align="left"></a></p>
868<p>My brother Joel gave me this classic book <b>NOTES ON RUBIK'S MAGIC CUBE</b> by David Singmaster,
8691981, ISBN 0-89490-043-9.  The three lines of small white text at the bottom say
870&quot;The definitive treatise.&quot; -- <i>Scientific American</i>
871Includes a step-by-step solution to the puzzle.<br clear="all"></p>
872<hr>
873<p><a name="cubefamily"><img src="686keychaincube.jpg" width="242" height="256" align="left"></a></p>
874<p>A cube on a keychain.
875When you do this cube, it helps to do the face with keychain first,
876and then try to keep the corner with the keychain at LBD.<br clear="all"></p>
877<p><img src="003minicube.jpg" width="131" height="150" align="left"></p>
878<p>A Rubik's Mini-Cube, billed as <i>&quot;The Not-So-Little Challenge&quot;</i>.
879You can solve it using the same moves as the moves for placing corners on
880a Rubik's Cube.<br clear="all"></p>
881<p><img src="02cubefamily.jpg" width="200" height="164" align="left"></p>
882<p>A little cube family.
883Junior has a piercing.<br clear="all"></p>
884<p> The 4x4 Rubik's Revenge is about 2.6 inches (6.5 cm) wide.<br>
885 The 3x3 Rubik's Cube is about 2.2 inches (5.6 cm) wide.<br>
886 The 2x2 Rubik's Mini-Cube is about 1.5 inches (3.9 cm) wide.<br>
887 The 3x3 keychain cube is about 1.2 inches (2.9 cm) wide.</p>
888<p><img src="004fingersoncube.jpg" width="200" height="172" align="left"></p>
889<p>To move pieces around quickly, it helps to have long fingers.
890In my younger days, I used to do 60 chins every day from a bar on a climber.
891I think that the daily stretch on my wrists helped to keep me from
892having repetitive stress carpal tunnel problems from cubing and from
893working as a computer programmer.
894One of the fastest cubists is a rock climber <a href="http://csel.cs.colorado.edu/~knightsd/">http://csel.cs.colorado.edu/~knightsd/</a><br clear="all"></p>
895<p><img src="687snoopycube.jpg" width="229" height="256" align="left"></p>
896<p>I bought this cartoon cube in Buenos Aires from a street vendor in the park in Recoleta.
897I suspect that the maker of the cube did not bother to license the cartoon characters.
898This cube was not glued together well.  If you look closely, you can see that
899Mickey Mouse's head popped off on one of the centers.
900Cubes with pictures are harder to solve than regular cubes because
901you must also rotate the centers correctly.<br clear="all"></p>
902<p><img src="688dragonballzcube.jpg" width="232" height="256" align="left"></p>
903<p>I bought this Dragonball Z <a href="http://www.dragonballz.com/">http://www.dragonballz.com/</a> cube
904from a street vendor in Recoleta for $1 during a later trip
905to Buenos Aires in February, 2001 <a href="../../200102buenosaires.html">../../200102buenosaires.html</a>.
906This cube is tricky to solve because the faces look almost alike (except for the top face).<br clear="all"></p>
907<p><img src="../../poland/115marketsquarepokemoncube.jpg" align="left"></p>
908<p>I bought this PokeMon <a href="http://www.pokemon.com/">http://www.pokemon.com/</a> cube in Krakow, Poland, at a stand on the Rynek (market square)
909for 4 zloty (about $1.03) on November 27, 2003 <a href="../../poland/poland.html#cube">../../poland/poland.html#cube</a>.  The front of the package
910says &quot;<i>OLD BRAND MAGIC SQUARE</i>&quot; up the left side, &quot;<i>New</i>&quot; inside a star at
911the upper right, and &quot;NO. 8999 WARNING: CHOKING HAZARD - Small parts
912Not for children under 3 years MADE IN CHINA&quot; across the bottom.<br clear="all"></p>
913<p><img src="pokemoncubeback.jpg" width="177" height="300" align="left"></p>
914<p>The back has a six-step top-middle-bottom solution with small diagrams and text in Chinese and English,
915but the writing is hard to read because the letters are small and partially rubbed off
916and because a Polish &quot;male elementy&quot; <i>small parts</i> safety warning and a bar code sticker
917cover the first three steps.<br clear="all"></p>
918<p><img src="pokemoncubebackbottom.jpg" width="300" height="166" al
918ign="left"></p>
919<p>The lower right of the back has some interesting text...
920The upper left of this picture has a diagram of a cube with the text &quot;the conbinelion of gix gidea&quot;.
921The upper right has the text &quot;It's OK&quot; after the sixth step of the solution.
922The large cube below it says &quot;big cube&quot;.
923The bold text below that says &quot;The Short Cot Of Playing Six Sides&quot; (with <i>&quot;Cot&quot;</i> for <i>&quot;Cut&quot;</i>).
924The diagonal text at the lower left says &quot;IQ magic cube&quot;.<br clear="all"></p>
925<p><a href="http://www.phenomenex.com/"><img src="593phenomenexcube.jpg" width="230" height="267" align="left"></a></p>
926<p>I filled in a pharmaceutical crossword puzzle in a Phenomenex <a href="http://www.phenomenex.com/">http://www.phenomenex.com/</a> catalog
927on February 18, 2004, to win this cube on April 10, 2004. The sides list the
928names of their products: white = phenomenex ...breaking with tradition, black = Zebron,
929red = explore LUNA, green = Jupiter Proteo, EZ:faast, Jupiter 300, and BioSep, yellow = strata, blue = synergi.
930This cube is an official Rubik's Cube (made in China) and moves very cleanly.
931It would be a good racing cube if it had normal colors.
932It came addressed to &quot;Rusty Terrier at Scientific Programmers&quot;.
933Rusty was a dog <a href="../../wondertwins.html#dogcicles">../../wondertwins.html#dogcicles</a>.
934I use her name when I don't want to get on a mailing list.
935Rusty gets lots of mail even though she has been gone since 1985.<br clear="all"></p>
936<p><img src="7058venicecube.jpg" width="173" height="251" align="left"></p>
937<p>Cube from Venice (January 2008).<br clear="all"></p>
938<p>See Joshua's Virtual Puzzle Museum <a href="http://www.twistypuzzles.com/">http://www.twistypuzzles.com/</a> for a large
939selection of specialty cubes.</p>
940<p><img src="690rubiklinkfront.jpg" width="256" height="133" align="left"></p>
941<p>The circles join in the solved position of this flat Rubik's Magic puzzle.
942The puzzle has a 1986 copyright by Rubik for the artwork and
943has an <i>E. Rubik</i> signature printed on the bottom.
944Matchbox made the puzzle in China.
945The puzzle has a few lengths of clear fishing line that hold the pieces together.
946The ends of the line are joined with metal clips.
947As you move the puzzle, the line (and the clips along with it) shifts position.
948Once a clip shifts between two pieces, the line becomes stuck,
949and the puzzle no longer works.<br clear="all"></p>
950<p> Christian Eggermont's page <a href="http://web.inter.nl.net/users/C.Eggermont/Puzzels/Magic/">http://web.inter.nl.net/users/C.Eggermont/Puzzels/Magic/</a>
951shows other shapes but includes the warning: &quot;To make the cube you have to be very careful
952since you probably stretch some of the wires trying to make the cube... which is not smart.&quot;<br>
953 Jaap's puzzle page <a href="http://www.geocities.com/jaapsch/puzzles/magic.htm">http://www.geocities.com/jaapsch/puzzles/magic.htm</a> explains how the looped cords work
954to hold the puzzle together.
955TwistyPuzzles <a href="http://twistypuzzles.com/articles/magic-restringing.shtml">http://twistypuzzles.com/articles/magic-restringing.shtml</a>
956has restringing instructions.</p>
957<p><img src="691rubiklink.jpg" width="256" height="134" align="left"></p>
958<p>The back of the puzzle has closed links.<br clear="all"></p>
959<p><img src="693rubiklinkstar.jpg" width="250" height="256" align="left"></p>
960<p>The puzzle in a star configuration.
961A few of the clips have jammed.
962This is one of the few shapes that the puzzle can still form.<br clear="all"></p>
963<p><img src="577tangram.jpg" width="234" height="233" align="left"></p>
964<p>My mother bought this tangram puzzle in March 2004 to give to one of her grandchildren (i.e. my cousins).
965She took it apart to test it and couldn't get it back together.  I played with it
966for a few hours with no more success.  I was already considering writing a program to solve it.
967We returned from visiting my brother Joel on April 6, 2004.  I was exhausted that night and played
968with the puzzle at 3 am before going to bed, and somehow by luck, the pieces fell into place and I solved
969it in a few minutes.  I took this picture.<br clear="all"></p>
970<p><a href="https://www.amazon.com/exec/obidos/tg/detail/-/019286145X/104-8415059-8067954"><img src="insandoutsofpegsolitaire.jpg" width="204" height="313" align="left"></a></p>
971<p>My <a name="peg">copy</a> of &quot;The Ins &amp; Outs of Peg Solitaire&quot; by John D. Beasley, 1985, ISBN 0-19-853203-2.
972This book is one of the best references for peg solitaire.<br clear="all"></p>
973<p>For some other on-line references, see<br>
974<a href="http://www.cst.cmich.edu/users/graha1sw/Pub/PegSol/PegSolIntro.html">http://www.cst.cmich.edu/users/graha1sw/Pub/PegSol/PegSolIntro.html</a><br>
975<a href="http://mathworld.wolfram.com/PegSolitaire.html">http://mathworld.wolfram.com/PegSolitaire.html</a><br>
976<a href="http://www.gpj.connectfree.co.uk/gpjj.htm">http://www.gpj.connectfree.co.uk/gpjj.htm</a><br>
977<a href="http://www.johnrausch.com/PuzzleWorld/">http://www.johnrausch.com/PuzzleWorld/</a><br>
978<a href="http://www.ionet.net/~bkeltch/pegs.html">http://www.ionet.net/~bkeltch/pegs.html</a><br>
979<a href="http://homepage.sunrise.ch/homepage/pglaus/Solitaire/solitaire.htm#InsOuts#InsOuts">http://homepage.sunrise.ch/homepage/pglaus/Solitaire/solitaire.htm#InsOuts#InsOuts</a><br>
980<a href="http://www.durangobill.com/PegSolitaire.html">http://www.durangobill.com/PegSolitaire.html</a><br>
981<a href="http://www.math.temple.edu/~xysun/peg/peg.htm">http://www.math.temple.edu/~xysun/peg/peg.htm</a><br>
982<a href="http://www.mathematische-basteleien.de/solitaire.htm">http://www.mathematische-basteleien.de/solitaire.htm</a></p>
983<p><a href="http://www.cdw.com"><img src="692cdw8ballnotabug.jpg" width="253" height="256" align="left"></a></p>
984<p>A CDW <a href="http://www.cdw.com">http://www.cdw.com</a> Magic 8-Ball for programmers showing the answer &quot;IT'S NOT A BUG - IT'S A FEATURE&quot;.
985The bottom of the ball says &quot;For your most commonly asked IT requests.&quot;<br clear="all"></p>
986<p> Here are the answers.  I think that I have told people all of these,
987except the ones that are ads for CDW.<br>
988 CDW HAS YOUR COMPUTING SOLUTIONS<br>
989 I CAN'T TEST EVERYTHING<br>
990 IT WORKED YESTERDAY<br>
991 IT WORKS LIKE I PROGRAMMED IT<br>
992 IT WORKS ON MY MACHINE<br>
993 IT'LL BE FIXED IN THE NEXT RELEASE<br>
994 IT'S A BETA - WHAT DID YOU EXPECT<br>
995 IT'S AN UNLIKELY COINCIDENCE<br>
996 IT'S JUST AN ISOLATED INCIDENT<br>
997 IT'S NOT A BUG - IT'S A FEATURE<br>
998 IT'S NOT SUPPOSED TO DO THAT<br>
999 LOG ON TO WWW.CDW.COM<br>
1000 PLEASE SUBMIT A FORMAL REQUEST<br>
1001 PLUG IT IN<br>
1002 PROGRAM WORKS - MUST BE USER ERROR<br>
1003 REBOOT<br>
1004 SOMEONE CHANGED MY CODE<br>
1005 YOUR CDW ACCOUNT MANAGER CAN HELP</p>
1006<p>Other people use balls like this <a href="http://www.ericmackonline.com/ica/blogs/emonline.nsf/dx/how-does-he-answer-so-quickly">http://www.ericmackonline.com/ica/blogs/emonline.nsf/dx/how-does-he-answer-so-quickly</a></p>
1007<p><a href="http://8ball.ofb.net/"><img src="drilling8ball.jpg" width="176" height="145" align="left"></a></p>
1008<p>The Unofficial Magic 8-Ball Home Page <a href="http://8ball.ofb.net/">http://8ball.ofb.net/</a>
1009demonstrates the dissection of an 8-ball in full color.<br clear="all"></p>
1010<hr>
1011<p><a href="../museum.html"><img src="../circuits.gif" width="64" height="64" align="left"></a></p>
1012<p>Return to the museum <a href="../museum.html">../museum.html</a><br clear="all"></p>
1013<p><a href="../../index.html"><img src="../../wb64.jpg" width="64" height="39" align="left"></a></p>
1014<p>Return home <a href="../../index.html">../../index.html</a><br clear="all"></p><br clear="all">
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