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Square-1 (puzzle) facts for kids

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The Square-1, also known as Back to Square One and Cube 21, is a puzzle similar to the Rubik's Cube. 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 Super Square One and Square Two puzzles have also been introduced. The Super Square One has two additional layers that can be scrambled and solved independently of the rest of the puzzle, and the Square Two has extra cuts made to the top and bottom layer, making the edge and corner wedges the same size.

Square-1 scrambled
The Square-1 puzzle scrambled
Square-1 solved
The same puzzle in its original (solved) state

World records

The world record fastest Square-1 solve is 4.59 seconds, set by Martin Vædele Egdal of Denmark on 5 September 2020 at the Danish Championship 2020 in Roskilde, Denmark.

The world record average of 5 solves (excluding fastest and slowest) is 6.54 seconds, set by Vicenzo Guerino Cecchini of Brazil set at Bernô Feet Friendship 2019 in São Paulo Brazil, with the times of: 6.15 / 7.37 / (6.04) / 6.11 / (DNF).

Top 5 solvers by single solve

Name Fastest solve Competition
Martin Vædele Egdal 4.59s Danish Championship 2020
Jackey Zheng 4.95s Brooklyn 2019
Tijmen van der Ree 4.98s Mental Breakdown Capelle 2019
Vicenzo Guerino Cecchini 5.00s Schoolmark Open 2018
Benjamin Gottschalk 5.11s Washington Championship 2020

Top 5 solvers by average of 5 solves

Name Fastest average Competition
Vicenzo Guerino Cecchini 6.54s Bernô Feet Friendship 2019
Rasmus Stub Detlefsen 6.67s Greve Gymnasium 2020
Michał Krasowski 6.74s Dragon Cubing 2020
Makoto Takaoka (高岡誠) 7.09s Kyoto Open 2019
Aiden Bartlett 7.10s Washington Championship 2020

Super Square One

Super Square1 scrambled-01
The Super Square One, scrambled
Super Square-1 solved
The Super Square One, solved
Super Square-1 scrambled-02
The Super Square One, in half of the turn

The Super Square One is a 4-layer version of the Square-1. Just like the Square-1, it can adopt non-cubic shapes as it is twisted. As of 2009, it is sold by Uwe Mèffert in his puzzle shop, Meffert's.

It consists of 4 layers of 8 pieces, each surrounding a circular column which can be rotated along a perpendicular axis. This allows the pieces from the top and bottom layers and the middle two layers to be interchanged. Each layer consists of 8 movable pieces: 4 wider wedges and 4 narrower wedges. In the top and bottom layers, the wider pieces are the "corner" pieces, and the narrower pieces are the "edge pieces". In the middle two layers, the wider pieces are the "edge" pieces, and the narrower pieces are the "face centers". The wider pieces are exactly twice the angular width of the narrower pieces, so that two narrower pieces can fit in the place of one wider piece. Thus, they can be freely intermixed. This leads to the puzzle adopting a large variety of non-cubic shapes.

Solution

Despite its appearance, the Super Square One is not significantly more difficult to solve than the original Square-1. The middle layers are nearly identical to the top and bottom layers of the Square-1, and may be solved independently using the same methods as the Square-1. The edges of the middle layers are distinguishable because the edges with the same two colors are mirror images of each other, but the centers of each face are interchangeable since they show only one color each.

Square Two

DIY - Square-Two - Finished Again (4273427839)
Square Two made by modification of the Square-1

The "Square Two" is yet another variation of the popular Square-1 puzzle, with extra cuts on the top and bottom layers. It is also currently marketed by Meffert's online store.

The Square Two is mechanically the same as a Square-1, but the large corner wedges of the top and bottom layers are cut in half, effectively making the corner wedges as versatile as the edge wedges. This removes the locking issue present on the Square-1, which in many ways makes the Square Two easier to solve (and scramble) than its predecessor.

Solution

The Square Two, like the Super Square One, isn't much more difficult than the Square-1. In many ways, it's actually easier considering one can always make a slice turn regardless of the positions of the top and bottom layers. Mostly, it's solved just like the Original, merely requiring the extra step of combining the corner wedges. After that, it is solved exactly like the Square-1.

Number of positions

There are a total of 24 wedge pieces on the puzzle.

Any permutation of the wedge pieces is possible, including even and odd permutations. This implies there are 24!=620,448,401,733,239,439,360,000 possible permutations of these pieces.

However, the middle layer has two possible orientations for each position, increasing the number of positions by a factor of 2.

This would theoretically yield a grand total of (24!)*2=1,240,896,803,466,478,878,720,000 possible positions for the puzzle, but since the layers have 12 different orientations for each position, some positions have been counted too many times this way. This reduces the number of positions by 12^2.

The final count is (24!)/72=8,617,338,912,961,658,880,000 total possible positions.

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