The Square-1 (full name "
Back to Square One") or alternatively, "
Cube 21", was invented by Karel Hršel and Vojtěch Kopský in 1990. Application for a Czechoslovak patent was filed on 8 November 1990, then filed as a "priority document" on January 1, 1991. The patent was finally approved on 26 October 1992, with patent number CS277266 . On March 16, 1993, the object itself was patented in the US with patent number US5,193,809 [http://patft.uspto.gov/netacgi/nph-Parser?u=/netahtml/srchnum.htm&Sect1=PTO1&Sect2=HITOFF&p=1&r=1&l=50&f=G&d=PALL&s1=5193809.WKU.&OS=PN/5193809&RS=PN/5193809 then its design was also patented, on October 5, 1993, with patent number D340,093.
Description The Square-1 consists of three layers. The upper and lower layers contain
kite and
triangular pieces. They are also called
corner and
edge pieces, respectively. There are altogether 8 kite and 8 triangular pieces. The kite pieces are 60 degrees wide, while the triangular pieces are 30 degrees wide, relative to the center of the layer. The middle layer contains two
trapezoid pieces, which together may form an irregular
hexagon or a
square. Each layer can be rotated freely, and if the boundaries of pieces in all layers line up, the puzzle can be twisted vertically, interchanging half of the top layer with half of the bottom. In this way, the pieces of the puzzle can be scrambled. Note that because the kite pieces are precisely twice the angular width of the triangular pieces, the two can be freely intermixed, with two triangular pieces taking the place of a single kite piece and vice versa. This leads to bizarre shape changes within the puzzle at any point. For the puzzle to be in cube shape, the upper and lower layers must have alternating kite and triangular pieces, with 4 kite and 4 triangular pieces on each layer, and the middle layer must have a square shape. However, since only two shapes are possible for the middle layer, there is a quick sequence of twists that changes the shape of the middle layer from one to the other without touching the rest of the puzzle. Once the puzzle has a cube shape, the upper and lower layers are cut in an
Iron Cross-like fashion, or equivalently cut by two
concentric (standard)
crosses, that make an angle with each other. Like Rubik's Cube, the pieces are colored. For the puzzle to be solved, not only does it need to be in cube shape, but each face of the cube must also have a uniform color. In its solved (or original) state, viewing the cube from the face with the word "Square-1" printed on it, the colors are: white on top, green on the bottom, yellow in front, red on the left, orange on the right, and blue behind. Alternative versions of the Square-1 may have different color schemes, as well as different types of materials.
Solutions A good number of solutions for this puzzle exist on the Internet. Some solutions employ the classical layer-by-layer method, while other approaches include putting the corner pieces in place first, then the edge pieces, or vice versa. Some solutions are a combination of these approaches. Although these solutions use different approaches, most of them try to restore the cube shape of the puzzle first, regardless of the placement of the pieces and the parity of the middle layer, and then proceed to put the pieces in their correct places while preserving the cube shape. The shape is often restored first because it allows for the greatest range of possible moves at any one time – other shapes have fewer moves available. The majority of solutions provide a large set of
algorithms. These are sequences of turns and twists that will rearrange a small number of pieces while leaving the rest of the puzzle untouched. Examples include swapping two pieces, cycling through three pieces, etc. Larger scale algorithms are also possible, such as interchanging the top and bottom layers. Through the systematic use of these algorithms, the puzzle is gradually solved. Like solutions of the Rubik's Cube, the solutions of Square-1 depend on the use of algorithms discovered either by trial and error or by using computer searches. However, while solutions of the Rubik's Cube rely on these algorithms more towards the end, they are heavily used throughout the course of solving the Square-1. This is because the uniform shape of the pieces in the Rubik's Cube allows one to focus on positioning a small subset of pieces while disregarding the rest, at least at the beginning of a solution. However, with the Square-1, the free intermixing of corner and edge pieces can sometimes cause a certain desired operation to be physically blocked; so one must take all pieces into account right from the beginning. Some solutions of the Square-1 rely solely on the use of algorithms.
Number of positions If different rotations of a given permutation are counted only once while reflections are counted individually, there are 170 × 2 × 8! × 8! = 552,738,816,000 positions. If both rotations and reflections are counted only once, the number of positions reduces to 15! ÷ 3 = 435,891,456,000. Also, it always can be solved in a maximum of 13 twists. If instead we wish to count only all those positions where there are no corner pieces in the way of twisting the halves, there are 3678·2·8!·8! = 11,958,666,854,400 twistable positions, and always can be solved in a maximum of 31 face turns. ==Notation==