How to Solve the Square-1
Everyone and their mom knows about the Rubik's cube, and a considerable subset of everyone and their mom knows how to solve a Rubik's cube. But what about the Rubik's cube's anemic red-headed cousin, the Square-1?
I would like to start by pointing out that there are other tutorials on the web that are much more comprehensive and helpful than my own (might I suggest Jaap Scherphuis's puzzle site or Lars Vandenbergh's CubeZone?). So why does this tutorial exist? In short, I got a Square-1 one day and discovered that I'm completely useless at learning Square-1 algorithms. After a few attempts with online guides, I gave up and decided to figure out how to solve it on my own. This is the very elementary method I worked out, using algorithms I made up as I bumbled my way to a solution. The algorithms provided will help you solve your Square-1, but not particularly efficiently. It's called the SSS1M, short for Shellie's Sillie Square-1 Method. Yes, it is so silly it involves a misspelling of my own name as well as the word silly.
Initially I never meant to publish this method, but Lucas Garron convinced me it would be funny. As such, I slapped it together at work one day. There are no pictures. Sorry. Use your imagination or find a better tutorial.
(As a side note, despite the self-proclaimed silliness of this method, Dan Dzoan learned it and was faster than me within 24 hours. Stephanie Chow has also since become a remarkably successful student of this tutorial. Take that as you will.)
Part 0: Finding your way around the Square-1
The top and bottom layers of the Square-1 consist of corner (big) and edge (small) pieces and can be rotated freely. The middle layer is divided into two halves, and the puzzle can be rotated vertically if all the pieces on the top and bottom layers line up along the cut. Notice that the radial angle of a corner piece is twice the radial angle of an edge piece.
Square-1 algorithms look like pairs of numbers in parentheses separated by forward slashes. Each number denotes the multiple of 30 degrees to turn the top or bottom layer (first number for top layer, second number for bottom layer; positive number for clockwise, negative number for counterclockwise). An easier way to think of this is to remember that each big piece counts as 2 and each small piece counts as 1. Each / means to turn the right side vertically 180 degrees, swapping the upper and lower layer pieces on the right half of the puzzle. When following the algorithms, hold the puzzle so the middle layer has a smaller section on the left side.
Part 1: Making both layers square
Your first objective is to make the top and bottom layer square. I find it is easier to do this by first grouping all eight edges together on the top layer. For the most part, this can be done intuitively, but if you get stuck, you should check out a more comprehensive guide. I recommend you practice doing this step intuitively before learning to solve the rest of the puzzle; it makes things a lot easier.
Once you have all the edges grouped on the top layer, you should have a "horseshoe" shape on top and a "star" shape on the bottom. Position the top layer so that the line of symmetry of the "horseshoe" lies along the center cut and turn.
Now you should have "paw" shapes on the top and bottom layer. Position them so that the line of symmetry lies along the center cut and both "paws" point in the same direction and turn.
Now you have two "barrel" shapes. Position one vertically and one horizontally and turn.
Now you have two "stingrays". Position them so that the center cut is along the long axis of the stingray (head to tail) and both "stingrays" are pointing in the same direction and turn to complete the square. Dont worry about the shape of the middle layer until the end.
In shorthand, start with the ends of the "horseshoe" pointing away from you and do / (-2,-4) / (-1,-2) / (-3,-3) /
Part 2: Separate pieces into their proper layer
Start by putting the corners in their proper layer. This can be done intuitively. If you do a (1,0) or (0,-1) before you start and only turn the top and bottom layers in multiples of 90 degrees (multiples of 3 in Square-1 algorithm notation), you can preserve the square shape and treat the corners as a 2x2 cube.
Once the corners are in their proper layer, do the same for the edges. Position "wrong" edges at UB and DF and do (0,-1) / (-3,0) / (4,1) / (-4,-1) / (3,0) /
You can also swap two pairs of "wrong" edges at a time by positioning them at UF, UB, DF and DB and doing (1,0) / (-1,-1) /
Part 3: Solve corners
I use only one algorithm for this. It switches ULB/URB and DLB/DRB:
(1,0) / (3,0) / (3,3) / (3,0) /
Faster variant of this algorithm which I learned from Nick Young (it deviates briefly from the square shape, but don't worry):
/ (3,0) / (-3,-3) / (0,3) /
You may have to do the algorithm twice. Another one switches diagonal corners on both layers, but this case is rare:
/ (-3,3) / (3,-3) /
Part 4: Solve edges
I use two algorithms to solve edges. To swap UF/UB and DF/DB:
(1,0) / (-1,-1) / (6,0) / (1,1) /
The following does the equivalent of a Z-permutation on the Rubik's cube. It swaps UL/UF and UB/UR edges. Combine it with a / setup move to swap bottom layer edges if necessary.
(1,0) / (-1,-1) / (-3,0) / (1,1) / (3,0) / (-1,-1) /
Note: Try (0,-1) / (-3,0) / (1,1) / (2,-1) / to swap UL/UF and DF/DR. It's the same as the pseudo-Z-permutation above but with the additional setup move and with resulting extraneous moves truncated.
Part 5: Parity
Sometimes you will end up with parity, where you have exactly one pair of edges that need to be swapped. There are good ways to solve parity. This isn't one of them. Go back to Part 1 and work backward from the square to the "paw" shapes. Swap 3 corners on the top with 3 corners on the bottom with a / turn, go back to your square shape, and start the solve over.
I told you this was an inefficient method.
Ideally, you should be able to spot whether or not you have parity before beginning step 4 (or step 3 if you're good at recognizing PLLs), so you don't have the entire solve wasted.
Part 6: Fixing the middle layer
Once your top and bottom layers are complete, fix the middle one if necessary. This part is intuitive, but I've included algorithms for those who can't figure it out on their own. To make the middle layer square if it isn't already, do / (6,0) / (6,0) /. If the middle layer is upside down (which really means your top and bottom layer need to be swapped, do / (6,6) /.