HTR-CP: New Method for HTR

[meskelto]

HTR-CP is a method for going from Domino Reduction (DR) to Half-Turn Reduction (HTR) in a decent number of moves, which has simple recognition and solves both corners and edges at the same time. I’m not sure how useful it would be for FMC, though it could be useful in a pinch or as a beginner method. It has been fun to play around with and use for solving in the fewest moves without backtracking. I originally mentioned this idea and called it HTORoux before I figured out the CP part. I think this name is confusing, and so I will now call it HTR-CP.

The idea is to solve corner permutation (CP) with respect to HTR, leaving an HTR corner pair in DL and the rest of HTR solvable using only R2, U, U’, and U2 moves (ignoring slice moves to solve edges):

1. HTR-CP: Solve HTR-CP while influencing edges
2. 2-gen finish: Solve HTR for both corners and edges using (mostly) 2-gen R/U

Cube Orientation: Hold the cube with white on top and green in front.


2-gen Finish

F/B Cases

For any DR state, there are some red or orange corner colours (side colours) on F/B. There are 4 cases (from the Hyper-Parity method for HTR) which can be seen by moving 2-gen from solved HTR via quarter U turns:

• Solved: 0 side colours on F/B
• 1 Face: 4 side colours on F/B such that they can all be placed in U via half turns
• Bars: 4 side colours on F/B such that they cannot all be placed in U via half turns
• 1 Bar: 2 or 6 side colours on F/B

It is a good idea to practice solving HTR 2-gen like this, as this is the final step of HTR-CP.

Edges

Given solved HTR-CP, a simple but effective method is to focus on the (red/orange) side edges while finishing corners:

1. DL edge: Solve a side edge at DL using S2
2. LOBE (Line Of Bad Edges): Put 2 side edges into DF and DB using M2
3. DR edge: Solve the final side edge at DR using 2-gen R/U while solving corners
4. Finish: Solve LOBE with U and M2 moves

Tricks:

• Cancel moves; Rw2 from R2 and M2 can combine the DR edge and Finish steps
• Can do F2/B2 just before the final R2 move without affecting HTR-CP
• Learn to recognise cases that skip LOBE
• Can swap the DL edge and LOBE steps around

HTR-CP

Corner Pair Cases

Take any solved HTR state; there are four corner pairs in DL, DR, UL, and UR. Look at any one of these pairs, say in UR, and compare the two corners for the same/opposite colours. Then it can be in one of four states:

• Only the Yellow/White stickers are different
• Only the Red/Orange stickers are different
• Only the Blue/Green stickers are different
• All stickers are different

When looking at all four corner pairs, there are three cases:

• All Match Case: All four corner pairs have the same state
• Two Match Case: Two corner pairs are in one state, and the other two pairs are in a different state
• No Match Case: All four corner pairs are in a different state

Solved HTR is in the All Match Case. Fake HTR can be in either the Two Match Case or the No Match Case.

HTR-CP

Note the corner pairs in Solved F/B and do U R2 (U’ R2 U R2)x2 to go to 1 Bar. Notice that this keeps the corner pairs identical but swaps the 2 corners within each pair, except in DL. Therefore, the corner pairs in 1 Bar correspond to the corner pairs in Solved F/B.

The method is:

1. Intuitively go to the 1 Bar case with 6 side colours (UL, UR, and DR) and a solved corner pair in DL
2. Solve HTR-CP depending on the corner pair case:
   1. All Match: CP is solved
   2. Two Match: Move the pair matching DL:
      1. to UB and do F2
      2. to UF and do B2
      3. to DR and do y2 (then U/U’)
   3. No Match: Do y2 (then U/U’) to turn it into a Two Match case
3. Solve 2-gen R/U

Also, if starting from Solved F/B we position corner pairs before moving to 1 Face to solve HTR-CP:

• All Match: CP is solved
• Two Match: Move the pair matching DL to DR. Then U/U’ to get to 1 Face to solve HTR-CP with: R2 U2 F2 U/U’ F2
• No Match: Do U/U’ such that DL matches UR. Then solve HTR-CP with: F2 U F2

So far, I have not seen much advantage in solving HTR-CP from Bars, so I usually just navigate to 1 Bar and only use Solved if I am already very close to it (and too far from 1 Bar).

Influence Edges

The edges can be influenced during HTR-CP. For example, around doing F2/B2 for HTR-CP in 1 Bar:

• Create the LOBE by using the correct F2/B2 move
• Having a side edge at UR (or otherwise UL) is better for solving the DR edge, so use Fw2/Bw2 to put a side edge from DL to UR

The edges can also be influenced before the CP solving move(s), like F2/B2. For example, for HTR-CP in 1 Bar, place the corner pair matching DL at UL and use R2 to move a side edge from DR to UR. This side edge will now be in the right place to create the LOBE during F2/B2. See the first example below.


Example Solves

Scrambles and optimal solutions found using Nissy:

Scramble: D2 R2 U' L2 D F2 U' R2 U F2 D' R2 U2
Optimal: U R2 U L2 U' R2 U R2 U (9)
HTR-CP: D B2 U2 R2 U’ B2 U’ R2 U Rw2 Uw (11)

• 1 Bar: D B2 (2)
• HTR-CP: U2 R2 U’ B2 (4)
• 2-gen finish: U’ R2 U Rw2 Uw (5)

Scramble: U L2 D2 R2 F2 U R2 B2 L2 F2 U' B2 U L2 U'
Optimal: U' R2 U L2 B2 U R2 U' F2 D (10)
HTR-CP: U’ B2 Rw2 U R2 U’ R2 S2 U Rw2 Uw (12 HTM, 11 STM)

• 1 Bar: U’ B2 U (3)
• HTR-CP Skip: U’ (-1)
• 2-gen finish: Rw2 U R2 U’ R2 S2 U Rw2 Uw (10 HTM, 9 STM)

Scramble: U2 B2 R2 L2 U R2 U' R2 U2 F2 U B2 U' R2 U
Optimal: U F2 U' B2 U F2 U2 R2 U' F2 U (11)
HTR-CP: D2 L2 y’ U’ R2 U’ R2 U Rw2 U’ Rw2 Uw (11)

• 1 Bar: D2 L2 y’ (2)
• HTR-CP: Skip
• 2-gen finish: U’ R2 U’ R2 U Rw2 U’ Rw2 Uw (9)

Scramble: L2 U' F2 U B2 U' R2 U' L2 D R2 D' L2 D'
Optimal: U' F2 U L2 U R2 L2 U R2 D (10)
HTR-CP: R2 U L2 y U’ B2 U’ R2 S2 U Rw2 Uw (12 HTM, 11 STM)

• 1 Bar: R2 U L2 y (3)
• HTR-CP: U’ B2 (2)
• 2-gen finish: U’ R2 S2 U Rw2 Uw (7 HTM, 6 STM)

Scramble: F2 R2 U2 R2 U F2 R2 U' L2 U
Optimal: U' R2 U F2 R2 U (6)
HTR-CP: U' R2 U B2 Rw2 U (6)

• 1 Bar: B2 U (2)
• HTR-CP: U’ B2 (-2)
• 2-gen finish: U' R2 U B2 Rw2 U (6)

Scramble: U2 B2 U' B2 U F2 L2 U L2 D' R2 F2 U' L2 U L2 U
Optimal: U2 F2 U F2 U2 R2 U' R2 U2 R2 F2 U (12)
HTR-CP: Rw2 U R2 U2 Fw2 U F2 U’ S2 U’ Rw2 U (13 HTM, 12 STM)

• Solved F/B: Skip (0)
• HTR-CP: Rw2 U R2 U2 Fw2 U F2 (7)
• 2-gen finish: U’ S2 U’ Rw2 U (6 HTM, 5 STM)

To Do

I may not get around to these for a while, so I am happy for others to look into anything themselves:

• List of algorithms for the 2-gen finish for each edge case, starting from each way to solve HTR-CP
• Further exploration of HTR-CP from other F/B cases
• Advanced edge influencing, such as while navigating to the 1 Bar F/B case

Feel free to ask any questions, and have a good day