Square one (puzzle)

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The Square One, also known as Back to Square One, is a puzzle similar to the Rubik's Cube, invented by Karel Hršel and Vojtěch Kopský around 1990. Its distinguishing feature among the numerous Rubik's Cube variants is that it can change shape as it is twisted, due to the way it is cut, thus adding an extra level of challenge and difficulty.

The Square One puzzle in one of its unsolved states.

The same puzzle in its original (solved) state.

1 History
2 Description
3 Solutions
4 Number of positions
5 See also
6 External links

[edit] History

The Square One, with the full name "Back to Square One", and alternative name "Cube 21", was invented by Karel Hršel and Vojtěch Kopský around 1990. Application for Czechoslovak patent was filed on 8 November 1990, the patent was approved on 26 October 1992. On March 16, 1993, it was patented in the USA with patent number US5,193,809.[1] Its design was also patented on October 5, 1993, with patent number D340,093.

[edit] Description

The Square One 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 all together 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 in the puzzle.

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 which 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 coloured. 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 colour. In the solved state, viewing from the face with the word "Square-1" printed on it, the colours are: white on top, green at bottom, yellow in front, red on the left, orange on the right, and blue behind. Alternative versions of the Square One may have different colour schemes.

[edit] 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 majority of solutions provide a large set of special operations. 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 operations are also possible, such as interchanging the top and bottom layers. Through the systematic use of these operations, the puzzle is gradually solved.

Like solutions of the Rubik's Cube, the solutions of Square One depend on the use of special operations, discovered either by trial and error, or by using computer searches. However, while solutions of Rubik's Cube rely on these special operations more towards the end, they are heavily used throughout the course of solving the Square One. 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 One, 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 One rely solely on the use of special operations.

[edit] 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 once only, the number of positions reduces to 15! ÷ 3 = 435,891,456,000.