View source for Domino Reduction

{{Substep Infobox
|name=Domino Reduction
|image=DR.png
|proposers=[[Morwen Thistlethwaite]]
|year=1981
|anames=DR, Domino Phase, H1 (Thistlethwaite), G1 (Kociemba), Phase 1 (Kociemba)
|variants=[[Partial Domino Reduction]], [[Half Turn Reduction]]
|subgroup=
|moves=9.53 (optimal)
|algs=0-2217 million (number of positions)
|purpose=<sup></sup>
* [[:Category:Computer algorithms|Computer algorithms]]
* [[Fewest Moves]]
* [[Speedsolving]]
|previous=[[Scrambled cube state]]
|next=[[Domino cube state]]
}}
'''Domino Reduction''' or '''DR''' is a technique invented by [[Morwen Thistlethwaite]]. It is employed by some [[:Category:Computer algorithms|computer algorithms]], [[Speedsolving|speedsolvers]] and [[FMC|fewest move solvers]] to bring the [[3x3x3 cube]] into a state similar to the [[Rubik's Domino]]. This is accomplished by [[CO|orienting all the corners]] and performing [[2-axis EO]], reducing the cube from <U,D,L,R,F,B> to <U,D,L2,R2,F2,B2>. [[God's number]] for DR is 12.

== Computer algorithms ==
Domino Reduction was initially invented to allow computer algorithms to solve the cube efficiently.

The first computer algorithm to utilize Domino Reduction was [[Thistlethwaite's algorithm]] in 1981. Due to hardware limitations back then, Domino Reduction is performed in two steps. The algorithm firstly orients the edges and then proceeds to orient the corners and separate the E-slice edges in one step. Using this approach, Domino Reduction can be reached in a maximum of 20 moves. Using two more steps after Domino Reduction, this algorithm was able to solve a cube in a maximum of 52 moves [[HTM]].

11 years later, in 1992, [[Herbert Kociemba]] invented [[Kociemba's Algorithm]]. It's main difference from [[Thistlethwaite's algorithm]] is that the first two steps and the last two steps are combined into one step each, solving the cube in two "phases". Because of this, it is also called the Two-Phase-Algorithm. It was possible to store all positions using the more advanced technology and various optimizations like symmetry reduction described on [http://kociemba.org/cube.htm Kociemba's website]. This algorithm was able to solve Domino Reduction optimally in at most 12 moves and the whole cube in 30 moves.

== Speedsolving ==
Although Domino Reduction is very rarely used in speedsolving, various methods have been invented to allow human solvers to reduce to a domino state quickly.

The first one was [[Human Thistlethwaite]], an adapted version of [[Thistlethwaite's algorithm]] for humans, proposed by [[Ryan Heise]] in 2002. Most of the reduction steps are either done intuitively or using a small set of pre-made algorithms. Because of the amount of thinking required and the relatively bad ergonomics of the method, virtually no one uses [[Human Thistlethwaite]] as their main speedsolving method.

An unrelated method, [[SSC|Shadowslice Snow Columns]], was invented in 2015 by [[Joseph Briggs]]. It starts with [[EOEdge]], where [[EO|edges are oriented]] and two E-slice edges are solved. The solver then proceeds by building a pseudo-pair and a pseudo-triplet and performing one of 23 algorithms. This solves the last two E-slice edges and orients the corners, effectively reducing the cube to a domino state with all E-slice edges solved. After this step, there are multiple ways to finish. The finish usually has pretty good ergonomics (R2, U, D and M moves) and makes the solve 40 to 50 moves STM long. SSC is a much more viable method for speedsolving than Human Thistlethwaite, although it's still rarely used because it drastically differs from mainstream speedsolving methods and since a good way to finish the cube after DR still needs to be found.

Although Domino Reduction might be viable for speedsolving, it is still not developed enough and too different from other methods to be adapted by more people in the near future.

== Fewest move solving ==
Domino Reduction in an [[FMC]] context has been mentioned as early as 2007[https://www.speedsolving.com/threads/fewest-moves-tips-and-techniques.1566/#post-15795]. However, it only started to be widely used in 2019. In that year, Domino Reduction became so popular that it was adopted by virtually all top solvers as their main way to do FMC because it usually yields better results than standard blockbuilding, allowing for consistent sub-30 solves in the hands of an expert. The technique also lead to multiple records in that year, such as [[Harry Savage]]'s [https://www.speedsolving.com/threads/harry-savage-17-fmc-single-sebastiano-tronto-24-00-mean.72299/ 17 move WR single] [[Sebastiano Tronto]]'s [https://www.speedsolving.com/threads/wrs-sebastiano-tronto-22-mean-and-16-single-at-fmc-2019.74270/ 16 move WR single]. Inspired by the records and frustrated by the lack of documentation, [[Alexandros Fokianos]] and [[Tommaso Raposio]] documented Domino Reduction for FMC in [https://www.speedsolving.com/threads/a-domino-reduction-guide-for-fmc.74828/ "A Domino Reduction Guide"] on August 7, 2019. Because the guide already explains Domino Reduction in great detail and should be the primary source to learn from, only an overview is given here.

=== Performing Domino Reduction ===
Domino Reduction in FMC consists of three substeps:
* [[Edge Orientation]]
* Separating the E-slice edges or [[2-axis EO]] (if EO is already done)
* [[Corner Orientation]]
These can be done at different times, e.g. one could first orient the corners, separate the E-slice edges and then orient the remaining edges. The most widely used approach firstly orients the edges, then separates some E-slice edges (not necessarily all) and then sets up to a so called "DR Trigger", which can then be applied to reduce to a Domino state. A DR Trigger is a usually short, pre-memorized algorithm that places a certain amount E-slice edge into the E-slice and orients certain corners. For example, the move "R" is a DR Trigger which brings the edges on UR and DR to the E-slice and orients the FUR and BDR corners clockwise and the FDR and BUR corners anti-clockwise. Other common DR Triggers include R U R', R U' R' and R U2 R'/L F2 L'. The setup is usually done by trying to get the same amount of bad edges and bad corners as a certain DR Trigger (using the example from before, the trigger "R" needs two bad edges and four bad corners to allow for it to be used). The bad pieces are then placed into the formation of a DR Trigger using U, D, F2, L2, B2 and R2 moves to preserve them and the trigger is applied.

By performing Domino Reduction on 2 axes (also known as Double Domino) you will get the 3-color cube reduction with a total of 3,981,312 states: 1/6 of these states are the HTR states, 1/2 of states are [[HTR|fake HTR]] states characterized by a 2-swap of corners and 2-swap of edges, and 1/3 of states are fake HTR states characterized by a 3-cycle of corners. 3-color cube reduction is the intermediate (phase 1) solution in [[Feather's algorithm]].

=== Finishing after Domino Reduction ===
There are four main ways to solve the cube after Domino Reduction:
* Normal [[Skeleton]]s
*# Solve as many pieces as possible using [[Blockbuilding]]
*# Solve the remaining pieces using [[Insertion]]s
* [[Corners First]]
*# Solve the corners (usually only applicable when it's easy to do so)
*# Solve the edges using [[Insertion]]s and [[Free slices]]
* [[Blockbuilding]] and linear endings
*# Solve all pieces using [[Blockbuilding]]
* [[HTR]] solves
*# Perform [[Half Turn Reduction]]
*# Solve as many pieces as possible
*# If some pieces remain, solve them using [[Insertion]]s

It should be noted here that after Domino Reduction, [[Blockbuilding]] becomes a lot easier than usual (comparable to normal solves and ones with [[Edge Orientation|edges already oriented]]). Additionally, Corner [[Insertion]]s become less efficient due to the high amount of half turns while Edge [[Insertion]]s become more efficient due to [[EO|oriented edges]] and the high amount of double and parallel moves.

=== Partial Domino Reduction ===
Partial Domino Reduction is when only exactly two of the three steps of Domino Reduction ([[CO]], E-slice, [[EO]]) have been performed, so the cube is only ''partially'' reduced to a Domino state. This can be useful when very few moves are required to get to such a state.

Possible ways to continue are to
* immediately reduce to normal Domino Reduction
** E-slice remaining: insert a commutator that brings the edges into the E-slice
** CO remaining: use DR triggers like R U' L2 U R'
** EO remaining: orient the edges using [[Roux]]-like EOs (such as M' U M)
* solve the pieces later using [[Insertion]]s
** E-slice remaining: solve up to a state where only a few edges remain (including the ones that belong in the E-slice) and use Edge [[Insertion]]s to solve them
** CO remaining: insert Corner [[Commutator]]s or [[Sune]]s
** EO remaining: solve up to a state where only a few edges remain (including the unoriented ones) and use Edge [[Insertion]]s to solve them

== See also ==
* [[Morwen Thistlethwaite]]
* [[Rubik's Domino]]
* [[Corner Orientation]]
* [[2-axis EO]]
* [[Thistlethwaite's algorithm]]
* [[Kociemba's Algorithm]]
* [[FMC]]
* [[Human Thistlethwaite]]
* [[SSC]]
* [[Half Turn Reduction]]

== External links ==
* [https://www.speedsolving.com/threads/a-domino-reduction-guide-for-fmc.74828/ A Domino Reduction Guide for FMC by Alexandros Fokianos and Tommaso Raposio]
* [https://www.youtube.com/playlist?list=PLo9h66pPA8XuV6af12Mqtv6W3DAapPl6A FMC Domino Reduction video tutorial by Firstian Fushada]
* [https://drive.google.com/drive/folders/1LWfSjdM0upYto5c7wU14mgQiGS8ckshp Jack314's FMC Domino Reduction resources]
* [http://kociemba.org/cube.htm Kociemba's website, including his Two-Phase-Algorithm]
* [http://kociemba.org/math/distribution.htm Kociemba's distribution of moves required for DR]
* [https://www.jaapsch.net/puzzles/thistle.htm Jaap's page about Thistlethwaite's algorithm]
* [https://docs.google.com/document/d/1oZwr2aSllFBL5lhbLTiWKQWplfk4i0LN0wA0uskeLJs Alexandre Campos' document of FMC solves with Partial Domino Reduction]

[[Category:3x3x3 first substeps]]
[[Category:Puzzle theory]]