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In 2020 I made another attempt to solve the CLL+1 problem of solving not just a specific edge but any single U layer edge. I eventually discovered a concept where algorithms can be unioned to where one algorithm with a certain kind of piece movement can solve one group of cases, another algorithm with other piece movement properties can solve another group of cases, and both algorithms are able to solve a final group of cases. It works just like a Venn diagram. Eventually I realized that this is like Lars Petrus' 270 alg system for the all edges oriented last layer and Thom Barlow's extension of that to full 1LLL called Duplex. I then eventually realized that it all is part of the larger concept of multi state solving - being able to solve pieces to something other than the solved state. This system allows for solving case sets using fewer and better algorithms. The discovery in my case is the realization of:
Algorithm unions and how it can be taken much further than what was presented in Petrus 270 and Duplex.
How the concept can be applied generally to many other method ideas.
The solving of the "direct solving problem" of "How do you solve any random piece or group of pieces out of a larger group of pieces?"
My creation of the Cayley Table like system of crossing cases and that a lot of information can be derived from the tables. The tables also help the user create almost any kind of algorithm set that they want, whether it is one based on containing their favorite algorithms, cases with the best recognition, and so on.
The big idea I have is to apply the unions and multi state solving concept to large groups of pieces in a solve. For example, when reaching the F2L-1 or EOF2L-1 state, the set of pieces can be recognized then an alg applied that puts the pieces into one of several desired end states. The end states could range from easy solutions such as R U R' or R U' R' through states with the best algorithms such as Sune. Specific styles of end states, such as always ending in L3C is a possibility, but may not have a better overall alg quality compared to ending in the best states.
In April 2022 I developed a program called Union Creator. Using this program you can input a list of algorithms and a list of desired end states and the program will find the unions that allow for solving to those end states using the fewest number of algorithms. Click the link at the top of the page to start using Union Creator.
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