Half Turn Reduction
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Half Turn Reduction or HTR is a technique invented by Morwen Thistlethwaite in 1981 - however, it has been actively used since 1980 as a part of the 3-Color Cube method by Michael Feather. It is employed by some computer algorithms and fewest move solvers to bring the 3x3x3 cube into the Square group so that only half turns can be used to solve it. This is accomplished by separating the pieces so that each face contains only the same or opposite colors while avoiding a so called fake HTR - states characterized by either 2-swap of corners and 2-swap of edges (such as N permutation) or a 3-cycle of corners (such as (UFL, URB, DFR)).
After HTR, the cube is always at most 15 moves away from solved when using the <U2,D2,R2,L2,F2,B2> move set, or at most 13 moves away from solved in HTM when using the <U,D,R,L,F,B> move set. Fake HTR states can not be solved using half-turns only.
HTR together with fake HTR form the 3-color cube reduction as can be seen on the following diagram. Fake HTR can be further divided into two subsets, details can be found elsewhere.
Computer algorithms
Half Turn Reduction was initially invented to allow simplification of the 3x3x3 cube.
It was used in 1981 in Thistlethwaite's algorithm, where HTR was the third step. The algorithm was able to bring the cube into the HTR state in 35 moves HTM, resulting in a guaranteed maximum of 52 moves for solving it.
Depending on used solver variant, Feather's algorithm can utilize either 117,265 out of 663,552 HTR states, or all 663,552 HTR states along with all 3,317,760 fake HTR states. It can also utilize all 24 HTR states on a 2x2x2 cube efficiently.
Thistlethwaite's algorithm is using the <U2,D2,R2,L2,F2,B2> move set whereas Feather's algorithm is using the <U,D,R,L,F,B> move set to finish a solve, meaning that the solution length found by Thistlethwaite's algorithm in phase 4 ≥ the solution length found by Feather's algorithm in phase 2.
Speedsolving
Although attempts like Human Thistlethwaite have been made to use HTR in speedsolving, no one who mains a method based around HTR for speedsolving is known. This is mainly due to the bad ergonomics in the finish (half turns usually take longer to perform than their quarter equivalents and regrips tend to be common) and the reduction being too complicated because fake HTR needs to be prevented.
Fewest move solving
Half Turn Reduction is often employed in FMC to finish the solve after Domino Reduction since it often leads to very straightforward finishes. It was used, among others, by Harry Savage to set his 17 move former WR single and is also described in Alexandros Fokianos' and Tommaso Raposio's "A Domino Reduction Guide".
Finishing after Half Turn Reduction
Since the cube usually is not many moves away from solved, HTR finishes are often trivial. The general approach is just to blockbuild and try lots of possibilities until the cube is solved.
Another more defined approach is HTORoux, although this method may not always be the most efficient.
See also
- Morwen Thistlethwaite
- Square group
- Domino Reduction
- 3-axis EO
- Thistlethwaite's algorithm
- FMC
- 3-Color Cube method
- Feather's algorithm
