# Message #1831

From: schuma <mananself@gmail.com>

Subject: Re: [MC4D] State graph of MC2D

Date: Sun, 31 Jul 2011 06:42:59 -0000

Melinda, thank you for providing the context about our meeting, and the context about why the MC2D could be interesting.

Out of curiousity, I analyzed another simple puzzle and drew its state graph in a geometrically symmetric manner. The simple puzzle is:

*** 2x2x2 with only 180-degree turns ***

I assume the LBD corner is always solved. Only U2, R2, F2 are the valid moves. Just like MC2D, there are also 24 states. When I asked Mathematica to draw the graph, it didn’t provide a symmetric arrangement for the 24 nodes. Then I manually drew the graph and found a symmetric arrangement:

<http://f1.grp.yahoofs.com/v1/YO80Trg5bQ9DOvCmghWYKmmcstvrJMrSp4ViYt7ay-DHQMPjsF2J30mKITXTSCAeqn267QfnxVO5b2mHWEytX5C56K4Kidcr/Nan%20Ma/MC2D/Torus_2x2x2_180.PNG>

Here is how to look at the above graph. Inside of the large hexagon, there are 24 nodes where thin black lines meet. These 24 nodes represent the 24 states of the 180-degree-turning 2x2x2 puzzle. Every node has three edges, representing U2, R2, F2. The opposite sides of the large hexagon are identified. So topologically it’s a torus. It’s good to think of it as a torus because it’s easy to see why all the nodes are equivalent.

One can also make copies of the large hexagon and tile them on a plane like a honeycomb:

<http://f1.grp.yahoofs.com/v1/YO80TqpgvnVDOvCm6xMkg0_FIRE7YA-QSSfogL5QAb11Ybc9DxG7wtQLOaLhuqPC4EKKpGaDLV76pJcmQT2RdWo80bl9La2G/Nan%20Ma/MC2D/Periodic_2x2x2_180.PNG>

In that way every state (out of 24 states) can find a representitive in each hexagon cell of the honeycomb.

Why are there so many small hexagons? It’s because a 6-move algorithm [U2,R2]x3 always bring you back to the original state. So do [U2,F2]x3 and [F2,R2]x3. Therefore from each starting point there are three hexagonal loops. Note that [R2,U2]x3 is another such algorithm. But it’s simply the inverse of [U2,R2]x3. So these two algorithms are nothing but going along the same loop in different directions. Using this property, one can easily label the thin lines by U2, R2, and F2.

I’m not good at identifying which states belongs to a type (an equivalency class with respect to recoloring and reorientation). It would be interesting to find out whether there’s a bottleneck in the graph of types.

Nan

— In 4D_Cubing@yahoogroups.com, Melinda Green <melinda@…> wrote:

>

> Thank you Nan for posting this to the group. For context, Nan, Brandon

> and I all met for a marathon discussion and brainstorming session a

> little while ago and had a fabulous time. The topic of MC2D came up in