Rubik Cube

Agnishom · 2013-12-29 19:53:47

This link was actually shared by roniman but I did not know how cool it was until my uncle gifted me a cube yesterday.

Bob · 2013-12-29 20:30:51

hi Agnishom,

Thanks for the link. My preferred way to solve the cube uses moves that work independently on edge and corner pieces. I have four moves that allow any edge swop, any corner swop, any edge re-orientation, and any corner re-orientation, subject to what is possible due to the constraints in-built for the cube.

That allows some 'pretty' patterns as well as a straight restoration.

Bob

anonimnystefy · 2013-12-29 20:33:28

I find the comutators and prmutators method quite interesting.

Agnishom · 2013-12-29 23:21:27

I have four moves that allow any edge swop, any corner swop, any edge re-orientation, and any corner re-orientation, subject to what is possible due to the constraints in-built for the cube.

What does this mean?

comutators and prmutators method

What are they?

anonimnystefy · 2013-12-29 23:46:24

You can find about them on the net and there is even a post here on MIFF about them.

Agnishom · 2013-12-29 23:56:56

Where is that post?

anonimnystefy · 2013-12-30 00:01:55

Sorry, I wrote permutators instead of conjugates.

It's here.

Bob · 2014-01-01 05:35:53

Agnishom wrote:

What does this mean?

I have a move that just swops edge A with edge B; and edge C with edge D. Another will do A - > B -> C

I have another that will swop two pairs of corners.

After these moves all else is back where it was.

If, say the red/white edge is in place but with the colours reversed, I have a move that will put this right without moving anything.

And finally a more complicated move that twists corner pieces so the colours point the right way.

Bob

Agnishom · 2014-01-01 06:01:31

Did you discover them yourself or did you learn from someone?

By the way, that guide also has a few moves for twisting the corners

Bob · 2014-01-01 11:08:13

No I didn't discover them although I'd like to have the time again to do that. They came from the Singmaster book.

If you want to discover your own I can give you a hint that will show you how you can start. (Only a small step on the way so there's plenty of challenge remaining. )

I have a 5 x 5 x 5 cube and was able to find an extra move that helps to solve it using this hint.

Bob

Agnishom · 2014-01-01 16:25:46

Could you please provide that hint?

Bob · 2014-01-01 20:52:36

hi Agnishom,

I'll have a go:

If you set up a sequence of turns (I'll call it X) then some pieces obviously move to new positions whereas some pieces may not move at all.

eg, R followed by F move the pieces on those faces but pieces on the left or back are not affected.

So what happens to a particular piece? Let's say piece A moves to position B

Where does B go? Clearly B must move because it's space is occupied by A. It may move into A's position.

If so, this is called a 2-cycle. A -> B and B -> A.

But what if B doesn't move there? Let's say it moves to position C.

Where does C go? If it moves to position A this is called a 3-cycle. A -> B, B -> C, C-> A

If you repeat X then all three pieces move on to the next place in the 3-cycle and after the third repeat, everything is back where it started!

But if C doesn't move to A, let's say it moves to D.

Where does D go? If D -> A then this is called a 4 cycle.

And so on ....

Every move like X that you make to the cube must consist of cycles.

And if you repeat an n-cycle n times the pieces in the cycle move back to their start positions.

OK. So let's say you find a move, X, that has some pieces on 3-cycles and some on 2-cycles. If you do X three times, all the 3-cycles return their pieces to their start positions, so only the 2-cycle pieces swop over. Thus you have a very simple move ( XXX ) that swops pieces and leaves everything else alone.

There is such a move that swops pairs of edges.

But what if the edges you want to swop aren't in the right places for the move?

You can still swop the pairs as follows:

(i) Do any turns that move the pieces you want into the right positions. I'll call that sequence of turns, Y

(ii) Then you do X to swop just the pieces you want to swop.

(iii) Then you do inverse Y to take everything back to where it was.

Sorry if that explanation is a bit muddly. I was trying to avoid giving away a particular X to give you the chance to discover it for yourself.

Bob

Last edited by Bob (2014-01-01 20:58:38)

Agnishom · 2014-01-02 02:49:18

That was a very nice explanation but I do not know where to begin.

Bob · 2014-01-02 05:00:02

OK. Bigger hint:

I'm assuming you have a restored cube.

Using the "youcandothecube" notation do this:

F F R R

Identify which pieces have moved like this:

UF (upper front edge piece) DF etc

Start with one of these and work out where it went. Then where that piece went. Then where ..... until you get back to where you started. That makes a cycle.

eg. UF -> DF -> UF 2-cycle.

You should be able find several cycles like this. Post back your answers.

Bob

Math Is Fun Forum

#1 2013-12-29 19:53:47

Rubik Cube

#2 2013-12-29 20:30:51

Re: Rubik Cube

#3 2013-12-29 20:33:28

Re: Rubik Cube

#4 2013-12-29 23:21:27

Re: Rubik Cube

#5 2013-12-29 23:46:24

Re: Rubik Cube

#6 2013-12-29 23:56:56

Re: Rubik Cube

#7 2013-12-30 00:01:55

Re: Rubik Cube

#8 2014-01-01 05:35:53

Re: Rubik Cube

#9 2014-01-01 06:01:31

Re: Rubik Cube

#10 2014-01-01 11:08:13

Re: Rubik Cube

#11 2014-01-01 16:25:46

Re: Rubik Cube

#12 2014-01-01 20:52:36

Re: Rubik Cube

#13 2014-01-02 02:49:18

Re: Rubik Cube

#14 2014-01-02 05:00:02

Re: Rubik Cube

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