Algorithm Transformations

Algorithm transformation is the process of adjusting or altering an existing sequence of cube moves. It includes manipulations such as reverse, inversion, permutation, and symmetries.

While these concepts may sound challenging, they are fundamental to understanding algorithms. They're part of the broader concept of cube group theory1, which provides a mathematical foundation for studying cube symmetries and transformations.

The goals of algorithm transformations and symmetries include, among others:

  • Finding new algorithms
  • Reducing cube solving times
  • Optimizing finger tricks
  • Enhancing solving methods
  • Customizing existing algorithms to suit solvers' preferences
  • Identifying better memorization patterns

All these algorithm transformations can be applied within the CubePal app.

You can download the CubePal app for free on the App Store.

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Reverse

This transformation involves reversing all moves by flipping the entire sequence and inverting each move. In fact, it is identical to undoing the initial sequence of moves.

For example, if your original sequence is R' B' U, its reverse would be U' B R.

By applying the reverse operation to certain algorithms, we can obtain a new algorithm sequence that performs the same permutations or orientations as demonstrated in the following PLL T-permutation algorithm:

To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Transform...Reverse

Inversion

Inverting each move in a sequence means replacing each move with its opposite. It may sound similar to reversing, but its concept differs from the reverse transformation.

For example, if you have a sequence of moves like R' B' U, its inversion would be R B U'.

By applying the inversion operation to certain algorithms, we can obtain a new algorithm that performs the same transformation as demonstrated in the following 2 Checkers 4H pattern:

To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Transform...Invert

Permutations

Permutation involves manipulating the first and last moves in a sequence while keeping the rest of the algorithm intact. In this case, the sequence of moves can be shifted left or right, or the first and last moves can be inverted or swapped.

Sequence shifting

Shifting operation involves permuting all moves of a sequence to the left or right and rearranging the position of the first or last moves. Please note that half turns, like R2, are initially split into two 90-degree turns (R and R) before being rearranged to the beginning or end of the sequence.

Shift Left

This permutation means moving each move in the sequence one position to the left, and moving the first move to the end of the sequence.

For example, if your original sequence is R' B' U, its shifting to left permutation results in B' U R'.

By applying the Shift Left operation to certain algorithms, we can obtain a new algorithm that creates a new patten as demonstrated in the following 4 Crosses Pattern that after Shift Left operation transforms into a 6 Crosses Pattern:

To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Transform...Permute...Shift Left

Shift Right

This permutation means moving each move in the sequence one position to the right, and moving the last move to the beginning of the sequence.

For example, if your original sequence is R' B' U, its shifting to right permutation results in U R' B'.

By applying the Shift Right operation to certain algorithms, we can obtain a new algorithm that performs the same transformation as demonstrated in the following 4H Pattern:

Shift Right
Shift Right
Open in CubePal app

To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Transform...Permute...Shift Right

Flipping (inverting) the first and last moves

Flipping or inverting the similar first and last moves results in another permutation. This permutation is based on the fact that two similar moves, inverted or not, will result in a half-turn move: R + R and R'+ R' are both equivalent to R2 half turn. 

An example of this would be flipping first and last moves in the sequence R B U R which will result in R' B U R'.

By applying the flip operation to certain algorithms, we can obtain a new algorithm that performs the same transformation as demonstrated in the following Superflip2 pattern:

To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Transform...Permute...Flip First and Last

Swapping first and last moves

This permutation only deals with cases where the first and last moves are along the same axis of a cube. This permutation swaps the first and last moves.

For example R B U L will result in L B U R.

By applying the swap operation to certain algorithms, we can obtain a new algorithm that performs the same transformation as demonstrated in the following 4 Crosses pattern:

To apply this transformation, tap the menu button within the CubePal app and navigate as follows:
Transform...Permute...Swap First and Last


Notes:

  1. Cube group theory is a branch of mathematics that studies the symmetries and transformations inherent in a cube. It provides a formal framework for understanding how sequences of cube moves can be analyzed mathematically, aiding in the development of solving methods and algorithms.

  2. Superflip is a 3x3x3 cube configuration where all 12 edge pieces are flipped relative to its solved orientation, while all corner pieces remain correctly positioned and oriented. This configuration is the farthest from solved state, requiring 20 moves using the half-turn metric.


 

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