Saturday, April 28, 2018

Solving Rubik's Cube Intuitively

Introduction

There's mainly two ways to solve a Rubik's Cube puzzle - 
  1. By memorizing a lot of moves (called algorithms) and applying the correct one whenever a pattern of pieces is detected.
  2. By trying to solve the cube intuitively i.e. by understanding each move rather than relying on algorithm memorization and pattern recognition.
The algorithmic methods are relatively fast and are used by 'speed cubers'.

This post describes an Intuitive method to solve the cube.

Rubik's Cube


Overview

I'm no expert on the Cube. The method I describe in this post is based on stuff I picked up from the Internet and books, along with some of my own insights.

I'm not much interested in solving the cube using pattern recognition and memorized algorithms as 'solving' in such a way isn't satisfying - but that's just me. On the other hand, if your aim is to solve the cube in the shortest possible time then memorized algorithms (70+ algorithms) and lots of practice is the way to go.


Terminology & Notation

A cube has 6 faces - White, Yellow, Green, Blue, Red, Orange. The centres of each face are fixed relative to each other. To solve a cube, all the pieces on a face should have the same colour as the centre piece of the face. 

There are 8 Corner (having 3 faces)  pieces and 12 Edge (having 2 faces) pieces.

When the cube is held in front of the user, the faces are denoted as Front (F), Back (B), Up (U), Down (D), Left (L), Right (R). When facing the face, clockwise rotations of the face are denoted by F, B, U, D, L, R while anti-clockwise rotations are denoted by F', B', U', D', L', R'. Double rotations are denoted by appending '2' e.g. R2, F'2. 

Piece positions are denoted by the faces the piece lies on e.g. the top-front-right corner is denoted by UFR.

Moving pieces from one position to another is called Permutation while rotating them is called Orientation.

Commutators & Conjugates

A bit of cube theory is essential for solving intuitively. If X is a series of moves then X' is its inverse i.e. the inverse of each move in X in reverse order. e.g. If X = R D' F then X' = F' D R'. Performing a move X immediately followed by its inverse X' will leave the cube state unchanged.

Commutators are a series of moves of the form X Y X' Y'. Pieces that are at the intersection of X and Y will have their state changed while the rest of the cube will remain unchanged. A practical use case for a commutator would be to permute or orient corners in the last layer while keeping the rest of the cube unchanged.

Conjugates are are a series of moves of the form X Y X'. A practical use for a conjugate would be a setup move (X) followed by a commutator (Y) followed by the inverse of the setup move (X').

It all sounds very abstract and maybe even a bit intimidating, but actually doing the moves is quite simple and intuitive as will be shown later.

The following sections describe each of the steps of the intuitive solution method.

White Cross

The idea is to solve the 4 Edges of the first layer such that their face colours match the corresponding cube face centres. This is usually done on the White layer, hence the 4 edges form a white cross.

Tips

  • Hold the cube with the white centre on top (U layer).
  • If an edge is permuted and oriented correctly then nothing more needs to be done for it.
  • If an edge is on the U layer but oriented wrongly then move it down to the middle layer then rotate U layer as required then move the edge back to the U layer in the correct orientation.
  • If an edge is in the D layer with White on the side then move it to the middle layer and follow the moves shown in previous point.
  • If an edge is in the bottom (D) layer with White on the bottom then rotate D to the correct position then rotate the edge to the U layer.
Its a good idea to try to solve the steps intuitively and only refer to the Tips if stuck - they are tips, not mandatory steps that need to be memorized :)

After the white cross is formed, flip the cube such that the Yellow centre is on top.

First Two Layers (F2L)

The idea is to solve the remainder of the first two layers except for one slot, the keyhole.

There are 4 Slots, each consisting of a corner on the bottom (D) layer and an edge on the middle layer immediately above its corner.

The top (U) layer is used as a scratchpad for forming each corner-edge pair then moving it into its slot on the bottom two layers. Solve 3 slots and leave the 4th slot unsolved. It will be used as a scratchpad (buffer) for solving the last layer's pieces.

Tips

  • If an edge and its corresponding corner are in the upper layer and are adjacent (stuck) to each other but not in the correct orientation then the first step is to separate them to allow each to be individually manipulated.
  • When moving faces, remember not to move previously solved slots to the top layer as it will break the slot.
  • If there are no corners left on the U layer with a white face then bring them up from the D layer using a move like R U R'.
  • If any corner or edge is on the U layer but its corresponding edge or corner is not in the U layer then bring them up to the U layer with a move like R U R', being careful not to disturb previously solved slots.
  • If a corner has white on the top then match the side face of its corresponding edge to colour of a cube face centre. Move that edge to the middle layer away from the face that contains its other colour. Then make a U move to place corner over the edge then bring pair fully into U layer. Bring the corresponding slot to the top layer and replace it with this pair then move the slot back to its proper position.
  • If the top face colours of edge and its paired corner are different then place corner such that white is on the side (left or right) and edge is on the face opposite to the non-white corner of the corner. Then move the corner next to the edge and bring the pair down into the correct slot.
  • If the top face colours of the corner and paired edge are the same then rotate the corner about its white face and move it to the D layer then  move the paired edge to its proper place on the U layer then bring the corner back to the top layer. Now bring the corresponding slot to the top layer and replace it with this pair then move the slot back to its proper position.
Once 3 slots have been solved, hold the cube such that the unsolved slot is at front-right position.

Solve Last 5 Edges

This step will solve the 4 edges of the Last Layer (U layer) and the middle layer edge of the unsolved slot.

Use the middle layer unsolved edge as a scratchpad (buffer) to position and orient the last (U) layer edges. Solve all 4 top layer edges and middle layer edge in the unsolved slot. 

Tips

  • If an edge has the correct orientation (yellow on top), then it can be permuted by using a series of moves like R U R' or F U' F'. 
  • If edge is not oriented properly then it should brought into the buffer using R face and brought out using F face (or vice versa). This will rotate the edge into the correct orientation.
  • When the final top layer edge is left to be solved then solve it using only the U layer and the face (R or F) that brought it to the top layer. Solving it will also simultaneously solve the remaining middle layer edge. e.g. a series of R U R' or F U' F' should solve these last 2 edges.

Solve Last 5 Corners

This step will solve the 4 corners of the Last Layer (U layer) and the D layer corner of the unsolved slot.

Use the D layer unsolved corner as a scratchpad (buffer) to position and orient the last (U) layer corners. Solve all 4 top layer corners and D layer corner in the unsolved slot. This is done using commutators. Sometimes a conjugate may be needed in addition to a commutator to orient the last 2 corners.

Tips

  • Hold the cube such that the unsolved bottom layer buffer corner is in DFR position.
  • Rotate U layer such edges surrounding UFR corner match the colours of the corner piece in the DFR position.
  • If DFR corner has yellow on the right then use R D' R to move it to UFR in the correct orientation. This is X of the commutator. Do some U move to bring some other unsolved piece to UFR. This is Y. Now do X' = R D R'. Y' is not required until the last 2 corners are solved. This will solve UFR corner while leaving the rest of the cube undisturbed. Repeat for each unsolved corner. 
  • If yellow is on the front in DFR corner then use F D F' instead of R D' R of the previous point.
  • If yellow is on the bottom of DFR corner then firstly move it to the right using R D R' D2 after which the previous step can be used as usual to move it to UFR. Remember to add the inverse of the extra moves to the commutator.
  • The final 2 corners can be oriented using various commutators. One of the simplest is X = F D2 F' R' D2 R or X = R' D2 R F D2 F' depending on the required rotation of the corner. I find it easy to remember which one to use depending on which direction the top face of the corner needs to move to.
  • If the final 2 corners are on U and D layers then hold the cube such that both corners are on U layer then perform the commutators as usual. Sometimes this won't be possible because the corners are at diametrically opposite corners. In this case use a conjugate (e.g. R2) to bring the other corner to the U layer then do the commutator as usual. Finally, do the inverse part of the conjugate, to solve the cube.
The cube should now be solved, as shown in the photo at the beginning of this post. It's very satisfying to see a solved cube :)

References


Utilities







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