Union Principle

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The Union Principle makes use of the overlapping and non-overlapping orientation and permutation properties of two or more independent sequences of moves and applies them to a set of states to produce a desired end state. In other words, two or more algorithms can be joined in a union to where, depending on the current case, one can be chosen to be applied to change the current case into a goal state. Informally the Union Principle is referred to as an "algorithm union" or simply a "union".

History

In July of 2020 James Straughan had been looking into ways of reducing the number of cases in the CLL+1 last layer method. He had the idea of making use of the way that the edges are cycled in each algorithm. Straughan thought that there should be some way that the various edge cycles for each of the 12 ZBLL cases per COLL case overlap. After testing, it was proven possible to overlap, or union, two algorithms for each set of 12 cases to ensure a solved edge in every case. This system was applied across both CLL+1 and COLL+1 to greatly reduce the number of algorithms that are required to be memorized. It was then realized that this was a new general concept which can be applied to various situations.

Application

Most notably, unions can be used to solve the problem of how to solve any non-specific piece or group of pieces in a set. Typically when encountering this problem, the approach has been to pick a single piece or piece location to solve. However, by making use of unions, the user can solve any piece and in the most optimal way. Taking COLL+1 for example, someone may decide to always solve the edge that goes to the UF location. This edge can be in any of four locations on the last layer, making the number of COLL+1 cases 4x42 (or around 160). But with the use of unions, any of the four edges can be solved. A union of two algorithms per COLL case can be used and reduces the number of cases to 2x42 (or a total of 83). For CLL+1, unions create a reduction from 8x42 (~330 total) to 4x42 (166 total).

COLL+1 Union.png
  • This image shows that the two algorithms that form a union each solve an edge on their own in certain cases while they both can be used to solve an edge in others (the cases falling in the overlap). Both algorithms solve the same COLL case, but each cycles the edges a different way.

The application of the Union Principle isn't limited to a union of two algorithms. Any number of algorithms can be, or may be necessary to be, placed into a union to achieve the desired state. This can mean a kind of Venn diagram with multiple overlapping sets. Depending on the step to which the Union Principle is applied, the number of algorithms in the unions can be expanded or reduced to achieve desired effects. For the fewest number of algorithms, the unions can be reduced. For easier recognition and potentially more ergonomic algorithms, the unions can be expanded.

Related Systems

In 2006, Lars Petrus developed a 270 algorithm system for solving ZBLL using two short algorithms per case. In 2012, Thom Barlow expanded this system to 1LLL under the name Duplex. Both of these systems are "algorithm combination" systems whereby the user pulls two algorithms from a single pool of algorithms to solve the last layer. The primary difference between these combination systems and the Union Principle is in the depth and execution. Petrus 270 and Duplex make use of a combination of two algorithms from the same pool in both "steps" of the last layer. The Union Principle is a general union concept that makes use of any number of algorithms in a union and any number of unions and includes the element of solving the "any piece" problem.

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