# 3x3x1 rubik square is isomorphic to ( U2, D2, F2, B2 ) cube subgroup

I hacked up a quick simulator for the 3x3x1 Rubik's Square today and I noticed that the number of positions at each level was the same for the ( U2, D2, F2, B2 ) subgroup of the normal 3x3x3 cube.
```Analysis of ( U2, D2, F2, B2 )
------------------------------

Level   Number of Positions

0             1
1             4
2            10
3            24
4            53
5            64
6            31
7             4
8             1
---
192
```
Everyone agree? I'm not sure if this has been pointed out before. Mark

## Comment viewing options

### The 3x3x1 cuboid is generally

The 3x3x1 cuboid is generally called the "Floppy Cube," although, of course, it's not really a cube.

The isomorphism with <F2, B2, R2, L2> (or equivalently <U2, D2, F2, B2> if you prefer) is mentioned on Jaap's web site in the "Hamilton Paths and Cycles on puzzles" page (http://www.jaapsch.net/puzzles/hamilton.htm).

There are essentially 1543 distinct groups of order 192. In GAP, the Floppy Cube group is isomorphic to SmallGroup(192, 1537). Interestingly, it is so close to to the highest possible index. (As far as I know, the ordering of the groups of a given size in GAP is rather arbitrary, so I don't think this really "means" anything about the group.)

### If you look at what happens w

If you look at what happens when you do only F2,R2,B2,L2 moves on the cube, you will find that none of those moves separate any top or bottom layer piece from its adjacent middle layer piece. The location of the top/bottom layer pieces are therefore completely determined by the middle layer pieces. That middle layer is equivalent floppy cube.

You could even bandage the pieces, gluing the top/bottom pieces to the middle one, so that you get a puzzle with just nine 1x1x3 blocks. This is similar to how larger cubes such as 4x4x4 and 5x5x5 can be bandaged together to make puzzles that are equivalent to various other cuboids such as 3x3x4, 5x5x3, etcetera.

Jaap
Jaap's Puzzle Page: http://www.jaapsch.net/puzzles/