Half Turn Reduction

Half Turn Reduction or HTR is a technique invented by Morwen Thistlethwaite in 1981 (yet similar technique called the 3Color Method was developed and used by Michael Feather in 1980). It is employed by computer algorithms, speedsolvers and fewest move solvers to bring the 3x3x3 cube into the Square group so only half turns are required to solve it. This is accomplished by separating the pieces so that each face contains only the same or opposite colors while also avoiding diagonal corner permutation. The most practical way to do this is to perform Domino Reduction on two axes. After the reduction, the cube is always at most 15 moves away from solved.
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Computer algorithms
Half Turn Reduction was initially invented to allow computer algorithms to solve the cube efficiently.
It was first used in 1981 in Thistlethwaite's algorithm, where HTR was the third step. The algorithm was able to perform HTR in 35 moves HTM, resulting in a guaranteed maximum of 52 moves for solving any given legal state.
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 diagonal CP needs to be prevented and since the cube has to be reduced to a Domino state before HTR.
Fewest move solving
Half Turn Reduction is sometimes employed in FMC to finish the solve after Domino Reduction since it often leads to very straightforward finishes. It was used by Harry Savage to set his 17 move WR single and is also described in Alexandros Fokianos' and Tommaso Raposio's "A Domino Reduction Guide". The following should only serve as an overview while the linked guide should be consulted for more indepth information.
Performing Half Turn Reduction
After Domino Reduction, FMC solvers usually tend to perform Half Turn Reduction when the corners are "good", i.e. easy to solve. This is due to the fact that in such situations, diagonal CP after HTR tends to be very rare.
To reach a half turn only state, one must perform Domino Reduction twice on two different axes. The second DR works exactly like the first one, except that edges are already oriented and the solver is only allowed to perform quarter moves on two instead of four faces to preserve the first DR. This means that it is not possible to set up to DR triggers like R' F R F', R U' R', R U R' etc., though R U2 R'/L F2 L' and R can still be used. This reduction is usually done while ignoring CP. When diagonal CP occurs, that continuation is usually discarded. Although it is possible to use 2e2c Insertions in those cases, this should usually be avoided since normal HTR finishes usually tend to be more efficient.
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
 3axis EO
 Thistlethwaite's algorithm
 FMC
 3Color Method