Lars Vandenbergh's CubeZoneSpeedcubing taken one step further

One of the unique and interesting properties of the Square1 is that it changes shape when you scramble it. Trying to solve the scrambled puzzle without making it into a cube first can prove to be a tough task since a lot of shapes have very limited options to move pieces around. The state where both layers are square is much more manoeuvrable and it allows us to recognise more easily where each piece belongs.
With this method, the long term goal is to be able to make both layers square in the optimal number of twists of the middle layer. But first we're going to investigate the various shapes that one layer can have, and work out which combinations of bottom layer shapes and top layer shapes are possible.
A layer can have various combinations of corners (large pieces) and edges (small pieces). There are a few restrictions however. Let's say C is the number of corners and E is the number of edges. Since the inner angle of all pieces must add up to 360°, we know that 60C + 30E = 360 or simpler:
2C + E = 12 (constraint 1)
There are also only 8 small pieces in total and only 8 large pieces in total:
0 <= C <= 8 (constraint 2)
0 <= E <= 8 (constraint 3)
If we consider all possible values for C and calculate the value for E from it using constraint 1, we get the following results:
If we discard all possibilities with an invalid value of E using constraint 3, only the following 5 options remain that statisfy all constraints:
One can now work out all possible arrangements of corners and edges for each subcase, which leads to the following 29 shapes:
2 corners and 8 edges (5 shapes) 

44  53  62  71  8 
3 corners and 6 edges (10 shapes) 

222  33  321  312  Left 42  Right 42  411  Left 51  Right 51  6 
4 corners and 4 edges (10 shapes) 

Square  Kite  Barrel  Shield  Left fist  Right fist  Left pawn  Right pawn  Mushroom  Scallop 
5 corners and 2 edges (3 shapes) 
6 corners and 0 edges (1 shape) 

Paired edges  Perpendicular edges  Parallel edges  Star 
Now that we've worked out all possible shapes one layer can have, we can investigate all shapes the whole puzzle can have by considering all possible combinations of 2 shapes (one shape of the top layer and one shape of the bottom layer). Of course, we don't take the state of the middle layer into account. Since the total amount of corners and edges must at all times be exactly 8, we know that:
The last two cases in the list contain the same shapes as the first two cases if you put the cube upside down. So from now on we will only talk about 3 possible configurations:
This page is maintained by Lars Vandenbergh  Last update on 28th May 2007 