# Message #550

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Something interesting and strange about permutations

Date: Mon, 11 Aug 2008 22:13:02 -0700

David Vanderschel wrote:

> […]

>> For me, the most interesting thing about MC2D as

>> implemented is that one can easily sketch the entire

>> state graph for the puzzle (8 states!) and thereby

>> begin to get an idea of what the topology of other

>> similar puzzles might look like.

>>

>

> Where does this "8 states!" come from? Orientation of

> a corner 2-cubie depends only on its position.

> However, the 4 corner 2-cubies can be permuted in all

> 24 different ways. In what sense can 3 different

> permutations all be regarded as the same state? I had

> pointed out the apparent discrepancy in Melinda’s

> analysis in somewhat greater detail a couple years

> ago:

> http://games.groups.yahoo.com/group/4D_Cubing/message/330

> Since Melinda is now repeating the dubious claim, I

> wonder if she ever saw my old message replying to

> hers. In that old message, I also touched on some of

> the other issues which have arisen again in the

> current discussion as well as some other issues which

> have not rearisen (yet).

David,

I arrived at the conclusion that MC2D contains exactly 8 states by

simply recording every sticker pattern that I was able to produce using

the puzzle, ignoring all duplicates due to color or positional

symmetries. I’m pretty sure there are only 8 of them. I gave each of

them a number and then mapped out all the possible transitions between

the those states to produce the complete graph for the puzzle. Perhaps

there are 24 states if you don’t mind duplicates, but since arriving at

one state gives exactly the same options as arriving at symmetrical

twins, it seems best to collapse all duplicates into the same logical

state and leave only a graph containing all the truly unique states. I

attempted to include an ASCII diagram in the post that you replied to

(http://games.groups.yahoo.com/group/4D_Cubing/message/329) but

unfortunately Yahoo stripped out my spacing characters and left a bit of

a mess. I should probably sketch it up again in Visio or other

diagraming tool for clarity.

-Melinda